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PLEASURE TEST REVISION SERIES – 01

CLASS XII [2012-2013] By OP GUPTA (+91-9650 350 480)

Max.Marks: 100 Time Allowed: 160 Minutes

SECTION – A (Question numbers 01 to 10 carry one mark each.)

Q01. What is the value of

1 12 2cos cos sin sin

3 3? Q02. Evaluate:

25 2

dx

x.

Q03. Find the number of all onto functions from the set 1, 2,3,...,n to itself.

Q04. Let A be a square matrix satisfying 2A I . What is the inverse of matrix A? Q05. If A is a square matrix of order 3 such that | A| 64adj , then find |A| .

Q06. If radius of a circle increases from 5cm to 5.1cm, find the increase in its area. Q07. Using determinants, find the equation of line containing the points (1, 2) and (3, 8).

Q08. Find a vector perpendicular to both the vectors ˆˆ ˆ4 3i j k and ˆˆ ˆ2 2i j k .

Q09. The equation of a line AB is

2 3 52 3

2 4

y zx , find the d.r.’s of a line parallel to AB .

Q10. Write a unit vector in the direction of the resultant of the vectors ˆˆ ˆ2 2 3i j k and ˆˆ ˆ2 j i k .

SECTION – B (Question numbers 11 to 22 carry four marks each.)

Q11. Find the value of k , for which

log(1 ) log(1 ), 0

( )

, 0

ax bxif x

f x xk if x

is continuous at 0x .

Q12. Evaluate

22

2 2

0

cos

cos 4 sin

xdx

x x, using properties of definite integrals.

OR Prove that:

3

2

1

2 2log

3 31

dx

x x.

Q13. Find the shortest distance between the lines whose vector equations are given below:

ˆˆ ˆ(1 ) ( 2) (3 2 )r i j k ,

ˆˆ ˆ( 1) (2 1) (2 1)r i j k .

Q14. Solve the equation:

1 1sin 2cos cot 2tan 0x .

Q15. Evaluate: 2 sin 3xe xdx . Q16. Evaluate

3

0

1xe x dx as the limit of sums.

Q17. The two sides of an isosceles triangle with the fixed base b are decreasing at the rate of 3cm/s. How fast is the area decreasing when two equal sides are equal to the base of the triangle?

OR Find the intervals in which 3

3

1( ) , 0f x x x

x is (i) increasing, and (ii) decreasing.

Q18. Let A N N, N being the set of natural numbers and * :A A A be defined as

( , )a b * ( , ) ( , )c d ad bc bd for all ( , ),( , ) Aa b c d , then show that:

(i) * is commutative, (ii) * is associative. Hence find the identity element (if it exists). Q19. An urn contains 4 Apples and 6 Mangoes. Four fruits are drawn at random (without replacement) from the urn. Find the probability distribution of the number of apples. How is the intake of fruits beneficial for the health? Explain in only one line.

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Q20. The two adjacent sides of a parallelogram are ˆˆ ˆ2 4 5i j k and ˆˆ ˆ2 3i j k . Find the unit vector parallel

to its diagonal. Also find the area of this parallelogram.

OR Show that if

,a b and c are coplanar vectors then,

,a b b c and

c a are also coplanar.

Q21. If 2cos cos 2x and 2sin sin 2y , find 2

2

d y

dx at

2.

OR If 2 1m

y x x , then show that: 2

2 2

2( 1) 0

d y dyx x m y

dxdx.

Q22. Using properties of determinants, prove that:

3

3 3

3

a a b a c

a b b c b a b c ab bc ca

a c b c c

.

SECTION – C (Question numbers 23 to 29 carry six marks each.)

Q23. Solve: 4 1 2

3 7 14, 2 3 4, 3 0x xz x xz x xzy y y

.

Q24. Using integration, find the area of the region: 2( , ) : 0 ,0 2,0 3x y y x y x x .

Q25. An insurance company insured 2000 cyclists, 4000 scooter drivers and 6000 motorbike drivers. The probability of an accident involving a cyclist, scooter driver and a motorbike driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? Which mode of transport would you suggest to a student and why? Q26. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of

radius r is 4

3

r.

OR A square piece of tin of side 18cm is to be made into a box without top by cutting a square piece from each corner and folding up the flaps. What should be the side of the square to be cut off, so that the volume of the box be maximum? Also, find the maximum volume of the box. Q27. Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point (1,3, 4)

from the plane 2 3 0x y z . Also, find the image of the point in this plane.

Q28. (i) Find the integrating factor of the differential equation: 2 0, 0ydx x y dy y .

(ii) Solve: 2 2 2 0x y dx xydy .

OR Show that the following differential equation is homogeneous:

sin cos

y yxdy ydx y ydx xdy x

x x. Hence solve it.

Q29. A dealer in rural area wishes to purchase a number of sewing machines. He has only `5760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him `360.00 and a manually operated sewing machine `240.00. He can sell an Electronic Sewing Machine at a profit of `22.00 and a manually operated sewing machine at a profit of `18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a linear programming problem and solve it graphically. Keeping the rural background in mind justify the ‘values’ to be promoted for the selection of the manually operated machine.

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PLEASURE TEST REVISION SERIES – 02 [ ]CLASS XII 2012 2013

( )By OP GUPTA 91 9650 350 480 Max.Marks: 100 Time Allowed: 160 Minutes

SECTION – A [Question numbers 01 to 10 carry one mark each.]

Q01. Write the value of x: 1 1sin cot (1 ) cos tanx x .

Q02. Evaluate:

3

0

3log 1 dx

x.

Q03. If A = {1, 2, 3} and B = {a, b}, write the total number of function from A to B.

Q04. If x, y, z are in geometric progression, evaluate:

0

xp y x y

yp z y z

xp y yp z

.

Q05. Evaluate:

. ( ) ( )A B C A B C .

Q06. What are the points at which the function ( ) | | 1f x x is not differentiable?

Q07. Determine the value of ‘c’ of Rolle’s Theorem for the function 4/3( )f x x on 1 1x .

Q08. If D is the mid-point of side BC of a ABC , then prove that AB AC 2AD

.

Q09. Find the value of k such that the line

24

1 1 2

yx z k

lies in the plane 2 4 7x y z .

Q10. If |adjA| = 36 then, find 1| 3A | if A is a square matrix of order 3.

SECTION – B [Question numbers 11 to 22 carry four marks each.]

Q11. A function ( )f x is defined as follows:

sin( ) , 0

2, 0

xf x if x

xif x

.

Is ( )f x continuous at 0x ? If not, what should be the value of ( )f x at 0x so that ( )f x becomes

continuous at 0x ?

Q12. Evaluate:

3 5

2 4

cos cos

sin sin

x xdx

x x. OR Evaluate:

2

2( )

xdx

a bx.

Q13. A plane which is perpendicular to two planes 2 2 0x y z and 2 4x y z , passes through the

point (1,–2, 1). Find the distance of the plane from the point (1, 2, 2).

Q14. Prove that:

1 1 1 1 18 8 300cot 2 tan cos tan 2 tan sin tan

17 17 161.

OR Prove that:

1 1θ cosθ2 tan tan cos

2 cosθ

a b a b

a b a b.

Q15. Evaluate:

cosθ sinθcos 2θ log θ

cosθ sinθd . Q16. Evaluate:

2

1/

loge

e

e

xdx

x.

Q17. Find the intervals in which (1 )( ) x xf x xe is (i) increasing, and (ii) decreasing.




Q18. Let T be the set of all triangles in a plane with R as relation in T given by 1 2 1 2R (T ,T ) : T T . Show

that the relation R is an equivalence relation. Q19. The probability of a man hitting a target is 1/4. How many times must he fire so that the probability of his hitting the target at least once is more than 2/3? In recent past, it has been observed that India has done quite well (as compared to other sports) at various International Shooting Contests including Olympics. What may be the reasons for this?

Q20. Let ˆ â i j ,

ˆˆb j k

and ˆ ˆc k i . If

d

is a unit vector such that

. 0 [ ]a d b c d then, find

d .

OR Anisha walks 4km towards west, then 3km in a direction 60 east of north and then she stops. Determine her displacement with respect to the initial point of departure.

Q21. Using first principle of derivative, differentiate: logcot2x .

OR If /( ) y xx a bx e then, prove that

223

2

d y dyx x y

dxdx.

Q22. For positive numbers x, y and z, find the numerical value of the determinant:

1 log log

log 1 log

log log 1

x x

y y

z z

y z

x z

x y

.

SECTION – C [Question numbers 23 to 29 carry six marks each.]

Q23. Solve the system of equations: 2 4, 3 2 6, 2x y z x y z x z .

OR Using elementary column operations, find the inverse of matrix

0 0 1

3 4 5

2 4 7

.

Q24. If the area enclosed between 2y mx and 2 ( 0)x my m

is 1 sq. unit then, find the value of m.

Q25. By examining the chest X-ray, the probability that T.B. is detected when a person is actually suffering is 0.99. The probability that the doctor diagnosis incorrectly that a person has T.B. on the basis of X-ray is 0.001. In a certain city, 1 in 1000 suffers from T.B. A person is selected at random and is diagnosed to have T.B. What is the probability that he actually has T.B.?

‘Tuberculosis (T.B.) is curable.’ Comment in only one line.

Q26. For what value of ‘a’ the volume of parallelopiped formed by ˆˆ î aj k , ˆj ak and ˆâi k is minimum?

Also determine the volume.

OR Show that the condition that the curves 2 2 1ax by and 2 2 1mx ny should intersect

orthogonally is given by: 1 1 1 1

a b m n.

Q27. Find the equation of the plane passing through (2, 1, 0), (4, 1, 1), (5, 0, 1). Find a point Q such that its distance from the plane obtained is equal to the distance of point P(2, 1, 6) from the plane and the line joining P and Q is perpendicular to the plane.

Q28. A) If ( )y t is a solution of (1 ) 1

dyt tydt

and (0) 1y then, what is the value of (1)y ?

B) Write the degree of the differential equation representing the family of curves 2 2y c x c ,

where c is a positive parameter. Q29. A farmer owns a field of area 1000m2. He wants to plant fruit tress in it. He has sum of `2400 to

purchase young trees. He has the choice of two types of trees. Type A requires 10m2 of ground per tree and costs `30 per tree and, type B requires 20m2 of ground per tree and costs `40 per tree. When full grown, a type A tree produces an average of 20kg of fruits which can be sold at a profit of `12 per kg and a type B tree produces an average of 35kg of fruits which can be sold at a profit of `10 per kg. How many of each type should be planted to achieve maximum profit when trees are fully grown? What is the maximum profit?

‘India is a land of farmers.’ Comment.

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PLEASURE TEST REVISION SERIES – 03 CLASS XII [2012-2013]

By OP GUPTA [+91-9650 350 480] Max.Marks: 100 Time Allowed: 160 Minutes

General Instructions: (a) AVOID PRACTICE OF UNFAIR MEANS (CHEATING) FOR YOUR OWN BENEFIT.

(b) The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each.

(c) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. (d) There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

(e) Use of calculators in not permitted. You may ask for logarithmic tables, if required.


Q01. Find the equation of tangent to the curve 2

4y x

x which is parallel to X- axis.

Q02. Prove that:

2 31 1

3 2

3tan 3 tan

3

a x x x

aa ax. Q03. Evaluate:

2 2 2 2

cos sin

sin cos

x xdx

a x b x.

Q04. If

2 1 5

2 3 0 13

x y x z

x y z w, write the values of x and y.

Q05. What is tangent of the angle which the vector ˆ ˆ î j 2 k makes with the Z-axis?

Q06. Elements of a matrix A of order 2 2 are given by

2

i j

(i 2j)a

2. Write the element

21a of matrix A.

Q07. Prove that: 0 0

( ) ( )p p

f x dx f p x dx . Q08. Solve for x:

1 1 5sin cosec5 4 2

x.

Q09. Find the area of a parallelogram having diagonals ˆˆ ˆ3i j 2k and ˆˆ î 3 j 4k .

Q10. Discuss the divisibility of such that

1 1 1

1 1 1 ; 0, 0

1 1 1

x x y

y

.


Q11. Let ˆâ j k and

ˆˆ ˆc i j k . Then find a vector b satisfying

a b c 0 and

a.b 3 .

OR If a vector a makes an angle of /4 with the positive directions of each of X and Y- axis, then

find the angle which is made by it with positive direction of Z- axis. Hence write the unit vector of a .

Q12. Solve: 2tan sec (1 ) 0x xe y dx y e dy .

Q13. A car starts from a point P at time 0t seconds and stops at point Q. The distance x, in the metres,

covered by it, in t seconds is given by

2 23

tx t . Find the time taken by it to reach Q and also find

distance between the points P and Q.

OR Find the intervals in which 1tan sin cosx x is rising and/or falling.

Q14. Suppose 15% of men and 36% of women have grey hair. The probability of dying hair by men is 21% and by women is 63%. A dyed hair person is selected at random, what is the probability that this person is a women? Excessive use of dyes to colour the hair can prove harmful. Elaborate.




Q15. Form the differential equation of all circles passing through origin and having their centers on X- axis. Q16. Find the value of ‘c’ for which the conclusion of Mean Value Theorem holds for the function

( ) loge

f x x on the interval [1, 3]. OR If log

( )x

f xx

then, show that 3

2 log 3( )

xf x

x

.

Q17. Find the angle between the pair of lines:

12 3

2 7 3

yx z and

2 82 5

1 4 4

yx z. Also check

whether the lines are parallel or perpendicular.

Q18. Find x such that:

22

21

xdt

t t. OR Evaluate:

cos 3 sin

dx

x x.


3

2 2

2 2

2 2

a b c a a

b b c a b a b c

c c c a b

.

Q20. Prove that

2sin tan2

xy , if it is given that

1 1cot cos tan cosy x x .

Q21. Let the function : N Nf is defined as:

1, when is odd

2( )

, when is even2

nn

f nn

n

for all Nn .

State whether the function f is bijective function. Justify your answer.

Q22. If

2 a b , if 0 2

( ) 3 2 , if 2 4

2a 5b , if 4 8

x x x

f x x x

x x

is a continuous function on [0,8] then, find value(s) of ‘a’ and ‘b’.


Q23. Find the area bounded between the curves 2y x and | |y x .

Q24. A dealer wishes to purchase a number of fans and sewing machines. He has only `5760 to invest and has space for at most 20 items. A fan costs him `340 and a sewing machine costs him `260. His expectation is that he can sell a fan at a profit of `22 and sewing machine at `18. Assuming that he can sell all the items he buys, how he should invest his money in order to maximize his profit? Q25. A window is in the form of a rectangle surmounted by a semicircular opening. Total perimeter of the window is 10m. Find the dimensions of window to admit maximum light through it. OR A 20m steel wire is to be cut into two pieces. The first piece is transformed into a circle and the other one into an equilateral triangle. Find out what should be the length of two pieces so that the combined area of both is minimum? Q26. Three shopkeepers A, B and C go to a store to buy electric equipments. A purchases 12 dozens 100watt bulbs, 5 dozen tube-lights and 6 dozens CFLs. B purchases 10 dozens 100watt bulbs, 6 dozen tube- lights and 7 dozens CFLs. Also, C purchases 11 dozens 100watt bulbs, 13 dozen tube-lights and 8 dozens CFLs. One 100watt bulb costs `40, one tube-light costs `65 and one CFL costs `120. Use matrix multiplication to calculate each individual’s bill. In your opinion, which equipment would you prefer to buy and why? Give two reasons. Q27. Find the equation of the plane which contains the line of intersection of the planes given as:

ˆˆ ˆ.(i 2 j 3k) 4r and

ˆˆ ˆ.(2i j k) 5 0r and which is perpendicular to plane ˆˆ ˆ.(5i 3 j 6k) 8 0r .

Q28. A bag contains five bananas, four oranges and three guavas. Three fruits are taken from it one after the other. Find the probability distribution of number of oranges if the fruits were taken

(i) with replacement and, (ii) without replacement. Intake of fruits is beneficial for health. Comment in short.

Q29. Evaluate 2

2

0

(3 5 7 2)x x x dx as limit of sums. OR Evaluate:

1/2

1/2

cos2

xx dx .

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By OP GUPTA [+91-9650 350 480] Please visit at: http://www.theOPGupta.wordpress.com , http://www.theopgupta.blogspot.com


SECTION – A

VERY SHORT ANSWER TYPE 01 MARKS

Q01. For a non-singular matrix A, find T( A )adj if 1

1 / 5 0 0

A 0 1 0

0 0 1

.

Q02. Let f be a function defined as 1

( )2 sin3

f xx

. Write the range of ( )f x .

Q03. If cosQ sinQ

Asin Q cosQ

, find Q, 0 Q2

, where TA A = I+ .

Q04. Write the principal value of 1sin (sin10) . Q05. Evaluate the integral:

2

cos2

sin cos

xdx

x x .

Q06. Check if the vectors ˆ ˆˆ ˆ ˆ ˆ4 6 2 , 4 3

a i j k b i j k and ˆˆ ˆ8 3 c i j k are coplanar vectors.

Q07. Find the distance between the planes 2 1x y z and 2 2 4 3 0x y z .

Q08. If vectors a

and b

are such that 3a

, 2

3b

and a b

is a unit vector then, find the angle between

the vectors a

and b

. Q09. Evaluate 3/2

0

x dx , where [.] is greatest integer function.

Q10. A matrix X has (a + b) rows and (a + 2) columns while the matrix Y has (b + 1) rows and (a + 3) columns. Both the matrices XY and YX exist. Find the values of a and b.

SECTION – B

SHORT ANSWER TYPE 04 MARKS

Q11. Let A N N and * be a binary operation on A defined by ( , )a b * ( , ) ( , ) c d a c b d . Show that * is

commutative and associative. Also, find the identity element for * on A, if any.

OR Let : [1, ) [1, ) f be defined as ( 1)( ) 2 x xf x and is invertible. Find 1( )f x .

Q12. If y x...x

x

then, show that: 2

2 log

dy yx

dx y x

. Q13. Evaluate:

2

4 23 16

xdx

x x

.

Q14. Show that the normal at any point θ to the curves cosθ θsinθ, sinθ θcosθx a a y a a is at a

constant distance from the origin. OR A water tank has the shape of an inverted right circular cone with its axis vertical and vertex

lowermost. Its semi-vertical angle is 1tan (0.5) . Water is poured into it at a constant rate of 5m3 per hour.

Find the rate at which the level of water is rising at the instant when the depth of water in the tank is 4m.

Q15. Discuss the continuity and differentiability of ( ) 3 Rf x x x at 3x . What do you conclude

from the observation? Q16. Find equation of a plane passing through (1,2,1) and perpendicular to the line joining the points (1,4,2) and (2,3,5). Also, find the perpendicular distance of the plane from origin.

Q17. Prove that: 1 1α π β sinα.cosβ2 tan tan tan tan

2 4 2 cosα sinβ

.

CLASS XII [Session 2012-2013]




Q18. Evaluate: 2

21 1

dx

x

. OR Evaluate: /2

3/2

/3

1 cos

(1 cos )

x

dxx

.

Q19. If a and

b are two vectors, then show that:

2 . .

. .

a a a b

a ba b b b

.

Q20. A) Which equation of curve would satisfy sin(10 6 )dy

x ydx

such that it passes through origin?

B) Write the order and degree of the differential equation:

32

5 22 32

2 33

3

4

1

d y

dxd y d yx

dx dxd y

dx

.

Q21. A die is thrown again and again until the number 6 is obtained three times. Find the probability that the third six comes in the seventh toss. “Outdoor games should be preferred over indoor games.” Why? OR Anshu and Manshu throw a die alternatively till one of them gets a ‘six’ and wins the game. Find their respective probabilities of winning, if Manshu starts the game. “Outdoor games should be preferred over indoor games.” Why?

Q22. If 0 1

A0 0

then, prove that 1I A I A

n n na b a n a b , Nn and I is the identity matrix of order 2.

SECTION – C

LONG ANSWER TYPE 06 MARKS

Q23. If

2 3 5

A 3 2 4

1 1 2

, find 1A . Hence use it to solve: 2 3 5 16, 3 2 4 4 , 2 3.x y z x y z x y z

OR If a , b , c are real numbers, and it is known that 0

b c c a a b

c a a b b c

a b b c c a

then, show that

either 0a b c or, a b c . Q24. A point on the hypotenuse of a right angled triangle is at the distances a and b from the sides of the

triangle. Show that the minimum length of hypotenuse is 2/3 2/3 3/2( )a b .

Q25. Find the area of the region: 2, : 1 5x y x y x .

Q26. A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively 3/10, 1/5, 1/10 and 2/5. The probabilities that he will be late are 1/4, 1/3, and 1/12, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probability that he comes by train? ‘Public transport should be encouraged.’ Why?

Q27. Find the line of intersection of the planes ˆˆ ˆ.( 2 3 ) 0r i j k

and ˆˆ ˆ.(3 2 ) 0r i j k

. Show that this line

is equally inclined to i and k and makes an angle 11sec (3)

2 with j .

Q28. A farmer mixes two brands P and Q of cattle feed. Brand P, costing `250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing `200 per bag, contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag? “Animals also require balanced diet for their growth.” Do you agree? Why or, why not?

Q29. Evaluate: 3/2 3/2

3 3

sin cos

sin cos sin( )

x xdx

x x x a. OR Evaluate:

11 2

0

tan (1 )x x dx .





CLASS XII [2012 2013] By OP GUPTA [+91-9650 350 480]


General Instructions: (a) The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each.

(b) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(c) There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions. (d) Use of calculators in not permitted. You may ask for logarithmic tables, if required.


Q01. Write the principal value of 1cosec ( 2) .

Q02. Express as a single matrix: 1 3 6 41

421 4 4 6

.

Q03. If 1,3 , 2, 5 , 3,7f and 3,–1 , 5,8 , 7,0g . Find (2)gof .

Q04. Write the value of integral 2sin xdx .

Q05. Given a matrix A of order 3 3 . Find |A . adj A|, if |A| = 9. Q06. Evaluate: 1

1

x dx

.

Q07. Find x, if 3 4 2 4

5 2 5 3

x

.

Q08. Find the value of λ , if ˆ ˆˆ ˆ ˆ ˆ(2 i j 5 k) (4 i λ j 10 k) 0

.

Q09. Write the value of p, such that the lines 1 2 2 p

yx z

and

3

2 4 1

yx z

are perpendicular.

Q10. Find the projection of vector ˆˆ ˆ3 i 4 j 5 k on the vector ˆˆ ˆ7 i j 8 k .


Q11. Discuss the continuity of the function ( )f x

cos π,

π 2

2π

1,2

xif x

x

if x

at π

2x .

Q12. Prove that: 1 1 112 4 63sin cos tan π

13 5 16 . OR Solve for x: 1 1 π

tan 2 tan 3 , 04

x x x .

Q13. Show that the equation of the perpendicular from the point (1, 6, 3) to the line 1 2

1 2 3

yx z is

61 3

0 3 2

yx z

and the foot of perpendicular is (1, 3, 5) and the length of the perpendicular is

13 units .




Q14. Solve the differential equation: 2 2(1 )x dy xydx xy dx .

OR Solve the differential equation: 3 2 2( 1) 2dy

x x x x xdx

, given that 1y when 0x .

Q15. Evaluate: 2 sin 4 2

1 cos 4x xe dx

x

. Q16. Evaluate 1

2 2 2 2

0( )( )

dx

x a x b .

Q17. A bag contains 3 red coloured crackers and 4 white coloured crackers. Two crackers are selected one after the other without replacement. If the second selected cracker is given to be white, what is the probability that the first selected cracker is also white? ‘Playing with crackers should be avoided.’ Why?

Q18. Consider the function : R (4, )f given by 2( ) 4f x x . Show that f is invertible, where R

represents the set of non-negative real numbers. Also find 1f .

Q19. The volume of a cube is increasing at a constant rate. Prove that the rate of increase in surface area varies inversely as the length of edge of the cube.

OR Find the intervals in which the function 4 3 23 4 36( ) 3 11

10 5 5f x x x x x is

(i) strictly increasing, and (ii) strictly decreasing.

Q20. Express the vector ˆˆ ˆa 5 i 2 j 5 k

as the sum of two vectors such that one is parallel to the vector

ˆˆb 3 i k=

and the other is perpendicular to b

.

OR Find a unit vector perpendicular to plane ABC, where A, B, C are the points (3,–1, 2), (1,–1,–3) and (4,–3, 1) respectively.

Q21. If 1sin( sin )y m x , then show that: 2

2 2

2(1 ) 0

d y dyx x m y

dxdx .

Q22. If cosx sin x

p(x)sin x cosx

=

then, prove that p(x).p(y) p(x y)= + .


Q23. Using matrix method, solve the system of equations: 2 6 2, 3 8, 2 3x y x z x y z .

OR Using properties of determinants, prove that: 2

b c q r y z a p x

c a r p z x b q y

a b p q x y c r z

.

Q24. Use integrals to find area of the region enclosed between the curves 2 2 9x y and 2 2( 3) 9.x y

Q25. Srishti is known to speak the truth 3 times out of 5 times. She throws a die and reports that it is ‘1’. What is the probability that it is actually a ‘1’? How does ‘being a liar’ impact our character development? Q26. Find the area of the largest isosceles triangle having perimeter 18 metres.

Q27. Find the distance of the point (–2, 3,–4) from the line 2 32 3 4

3 4 5

yx z

measured parallel to the

plane 4 12 3 1 0x y z .

OR Find the coordinates of the reflection of the point (1, 2, 3) in the plane 2 4 38x y z .

Q28. Prove that: ( ) ( )b b

a a

f x dx f a b x dx . Hence, evaluate: 6

1

7

7

x

x xdx

.

Q29. A factory owner wants to purchase two types of machines, Machine A and Machine B for his factory. Machine A requires an area of 1000m2 and 12 skilled men for running it and its daily output is 50 units, whereas Machine B requires 1200m2 area and 8 skilled men, and its daily output is 40 units. If an area of 7600m2 and 72 skilled men are available to operate the machines, set up L.P.P. to maximize the daily output and, hence solve it. “Use of machines has made the production of goods easier.” Comment.

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1 13/10/1 P.T.O.

Compiled By : OP Gupta [+91-9650 350 480 | +91-9718 240 480] For more stuffs on Maths, please visit : www.theOPGupta.com

Time Allowed: 180 Minutes Max. Marks: 100

SECTION A [Question numbers 1 to 10 carry 1 mark each.]

Q01. If y = 5

log (log x) then, write the value of dy/dx.

Q02. If the order of two matrices A and B are 3 4 and 4 3 respectively then, write the order of (AB)T. Q03. If A is a square matrix of order 3 and |A–1| = 9 then what is the value of |A|? Q04. Let M = {1, 2, 3, 4} and P = {1, 2}. Find the number of onto functions from M to P? Q05. Find the direction ratios of the line 5x – 3 = 15y + 7 = 2 – 6z.

Q06. Evaluate : x x

1dx

e e. Q07. Find the value of 1 1

sin 2tan3

.

Q08. How are the two non-zero vectors a

and b

related, if a b 0 = ?

Q09. What is the approximate change in the volume V of a cube of side x metres caused by increasing the

side by 2%? Q10. If |a b|=|a| |b| + + then, what is the angle between a

and b

?

SECTION B [Question numbers 11 to 22 carry 4 marks each.] Q11. Find the value of k so that the following function is continuous at x = 2 :

f

3 2

2

x x 16x 20; x 2

(x) (x 2)

k ; x 2

+ +

=

=

.

OR If 2 21 x 1 y a(x y) then, show that 2

2

dy 1 y

dx 1 x

.

Q12. Show that :

1 1a b θ acosθ b2tan tan cos

a b 2 a bcosθ. Q13. Evaluate :

1 1

2 3/2

0

x tan xdx

(1 x ).

Q14. Show that f : N N , defined by f (x) =

x 1; if x is odd

x 1; if x is even is a bijective function.

Q15. If a and b are unit vectors and θ is the angle between them, then show that : ˆˆθ | a b |

tanˆ2 ˆ| a b |

.

OR Establish the relationship between scalar product and vector of two non-zero vectors.

Q16. Evaluate :

2 1x tan x dx . OR Evaluate :

1

2

sin xdx

x.

Q17. If

cosθ sinθA

sinθ cosθ then prove that An

cosnθ sin nθ

sin nθ cos nθ for all natural numbers n.

Q18. Form the differential equation corresponding to the family of curves 2 2(x a) (y b) 36 .

Q19. Find the equation of line of shortest distance between the lines whose equations are given as below :

ˆ ˆ ˆ ˆ ˆ ˆr 3i 8j 3k λ(3i j k)

and ˆ ˆ ˆ ˆ ˆ ˆr 3i 7 j 6k μ(2 j 3i 4k)

.

Q20. Verify Rolle’s theorem for the function f (x) = x (x – 3)2, x[0, 3]. Q21. Using approximation, evaluate (3.968)3/2. If 1000(3.968)3/2 is the number of people who took part in a blood donation camp organized by an NGO, find the approximate number of these people.

Series : PTS/3 Code No. 13/10/1

2 1 3 1 2

Roll No.

Candidates must write the Code on the title page of the answer-book.




2 13/10/1

What is the importance of blood donation? OR A ladder 20m long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. Show that when the lower end of the ladder is 16m away from

the wall, the upper end is moving 4

3 times as fast as the lower end.

Q22. A mathematics teacher in a school has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

SECTION C [Question numbers 23 to 29 carry 6 marks each.] Q23. Two store-rooms A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to three ration shops D, E and F whose requirements are 60, 50 and 40 quintals respectively. Cost of transportation per quintal from the store-rooms to the shops are listed in the following table:

From / To A B D 6 4 E 3 2 F 2.50 3

How should the supplies be managed in order to minimize the transportation cost? Also find what is the minimum cost? While transportation of grains from store-room A to shop D, five packs of grains got stolen. You know the thief, who is a poor friend of yours. How would you inform the store room owner of the theft at the same time saving your friend? Q24. Using integrals, find the area of the triangle formed by positive x-axis, and the tangent and the normal

to the circle 2 2x y 4 at (1, 3 ) .

Q25. Let the point P(5, 9, 3) lies on the top of Qutub Minar, Delhi. Find image of the point P(5, 9, 3)

on the line : y 2x 1 z 3

2 3 4

. “Conservation of monuments is important.” Why?

OR Considering the earth as a plane having equation 5x + 9y – 10z + 153 = 0, a monument is standing vertically such that its peak is at the point (1, 2,–3). Find image of this point. Also find height of the monument. How can we save our monument? Mention any two points. Q26. An expensive square piece of golden color board of side 24cm is to be made into a box without top by cutting a square from each corner and folding the flaps to form a box. What should be the side of the square piece to be cut from each corner of the board to hold maximum volume and minimize the wastage? What is the importance of minimizing the wastage in utilizing the resources? How can a student utilise the resources?

Q27. Evaluate 2

2 x

0

(x 6x e )dx+ as a limit of sum.

Approximately 30 times the absolute value of this definite integral is the number of trees planted by Palash, a local NGO, in a locality. How would you stress this as a welcome step for the society?

Mention the ways in which trees help us. OR Evaluate : 3sin x dx .

Q28. In a group of students from a certain school, 200 students attend coaching classes, 400 students attend school regularly and 600 students study themselves with help of peers (self study). The probability that a student will succeed in life who attend coaching classes, attend school regularly and study themselves with the help of peers (self study) are 0.1, 0.2 and 0.5 respectively. One student is selected who succeeded in life, what is the probability that he studied himself with the help of peers? What type of study can be considered for the success in life and why?

Q29. If a ,b and c are positive and unequal then, show that

a b c

b c a 0

c a b

.

Transportation Cost Per Quintal (in `)

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3 13/10/1

HINTS & ANSWERS for PTS XII – 03 [2013 - 2014]

Q01. e

1

x(log 5)(log x) Q02. 3 3 Q03. |A| = 1/9

Q04. 4

2P 12 . See OPG Vol. 2 P62 Q. No. 51(e) for explanation.

Q05. 6, 2, –5 Q06. 1 xtan e k Q07. 3/5 Q08. a b

Q09. 0.6x3 cubic metres Q10. 0o Q11. k = 7 Q13. 1 π

142

Q14. See OPG Vol. 2 P61 Q. No. 20 Q15. See OPG Vol. 2 P09 Q. No. 30 OR See OPG Vol. 2 P06 Point No. 02

Q16.

3 1 2

2x tan x x 1log|1 x | C

3 6 6 OR

2 1

2

1 1 1 x sin xlog C

2 x1 1 x

Q18. 2 3 2[1 (y ) ] 36(y ) Q19. ˆ ˆˆ ˆ ˆ ˆr 3i 8j 3k (6i 15j 3k)= + + + +

Q20. c = 1 Q21. 7.968; No. of people is 7968 Q22. 5/9 Q23. From A : 10, 50, 40 units; From B : 50, 0, 0 units to D, E and F respectively. Minimum cost = `510.

[For solution, visit www.theOPGupta.com & download NCERT Solutions Miscell. Ex. Q. No. 6]

Q24. 2 3 sq.units Q25. Foot of perpendicular : (3, 5, 7) and Image : (1, 1, 11) OR

Foot of perpendicular : (–4, –7, 7) and Image : (–9, –16, 17) Q26. 4cm

Q27. 2 31e

3 OR Put 2x t x t dx 2tdt .

So, 3I sin t . (2t)dt . Integrate using By Parts.

3 3d

I 2t sin t dt (2t) sin t dt dtdt

...(i) 3 2

1Consider I sin t dt (1 cos t)sint dt

2 3

1

1Put cost = u sin tdt = du I (1 u ) du cos t cost

3=

3 31 2

By (i), I 2t cos t cos t cos tdt 2 cos tdt3 3

+ …(ii)

3 2

2Consider I cos t dt (1 sin t)cost dt

2 3

2

1Put sin t = v cos tdt dv I (1 v ) dv sin t sin t

3= =

3 31 2 1By (ii), I 2t cos t cos t sin t sin t 2sin t C

3 3 3+

3 32 2 2I x cos x 2 x cos x sin x sin x 2sin x C

3 3 9

i.e., 3 32 2 4I sin x x cos x 2 x cos x sin x C

9 3 3

Q28. 3/4. For explanation, see OPG Vol. 2 P105 Q. No. 16 Q29. See OPG Vol. 1 P18 Q. No. 09

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1 13/9/1 P.T.O.

Compiled By : OP Gupta [+91-9650 350 480 | +91-9718 240 480]

For more stuffs on Maths, please visit : www.theOPGupta.WordPress.com Time Allowed: 180 Minutes Max. Marks: 100

General Instructions :

(a) All questions are compulsory.

(b) The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each.

(c) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(d) There is no overall choice. However, internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

(e) Use of calculators in not permitted. You may ask for logarithmic tables, if required.

SECTION A

Question numbers 1 to 10 carry 1 mark each.

Q01. A matrix A of order 3 has determinant value 7. What is the value of |3A|?

Q02. If A

i j

2 3 5

[a ] 1 4 9

0 7 2

and

i j

2 1 1

B [b ] 3 4 4

1 5 2

then, find the value of 22 21

a b .

Q03. Evaluate the determinant: cosθ sinθ

sinθ cosθ.

Q04. What is the principal value of 2π 2π

cos cos sin sin3 3

1 1 ?

Q05. Show that the points (2, 3, 4), (–1,–2, 1) and (5, 8, 7) are collinear.

Q06. Evaluate the integral of 2 x

tan dx2

.

Q07. If ˆˆ ˆ2 i + 4 j k is perpendicular to ˆˆ ˆ3 i m j 2 k , then write the value of m .

Q08. If A = [ai j] be a square matrix of order 3 and Ci j denotes the cofactor of [ai j] in A. If |A| = 5, then

write the value of 31 31 32 32 33 33

a C a C a C .


Roll No.


Please check that this question paper contains 3 printed pages. Code number given on the right hand side of the question paper should be written on the

title page of answer-book by the candidate.

Please check that this question paper contains 29 questions.

Please write down the Serial Number of the question before attempting it. 15 minutes time has been allotted to read this question paper. The question paper will be

distributed at 07.00 a.m. From 07.00 a.m. to 07.15 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.




2 13/9/1 P.T.O.

Q09. Radius of a circle is increasing at the rate of 0.7cm/s. Find the rate of increase of its circumference?

Q10. Find the value of μ where it is given that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆμ i.( j k) j.(k i) k.(i j) = .

SECTION B

Question numbers 11 to 22 carry 4 marks each.

Q11. Let f (x) = |x + 2|. Show that f (x) is not differentiable at x = –2.

OR

Given that for the function 3 2(x) x bx a x, x [1,3]f , Rolle’s Theorem holds with 1

c 23

= .

Find the values of a and b.

Q12. Write in the simplest form :

1 cos x sin xtan ,x

cos x sin xπ .

Q13. If

xy x log

a bx+ then, prove that

223

2

d y dyx x y

dxdx.

OR

Find dy

dx for y x(cos x) (cos y) .

Q14. On the set R–{–1}, a binary operation is defined by a * b = a + b + ab for all a, b ∈ R–{–1}. Prove that * holds both commutative & associative properties on R–{–1}. Find the identity element and prove that every element of R–{–1}is invertible.

Q15. For any two vectors a and

b , show that:

2 2 2 2(1 |a| )(1 |b| ) (1 a.b) |a b a b|× .

OR

If a and

b are two vectors such that

|a b|+ =

|a|, then prove that vector

2a b+ is perpendicular to

vector b .

Q16. Evaluate :

π

0

1dx

5 4 cos x.

OR

Evaluate :

π/ 2

0

x sin xdx

1 cos x.


3 3

x y x y

y x y x 2(x y )

x y x y

.

Q18. Solve the differential equation : dy/dx + y cot x = 2x + x2 cot x, x = π /4, y = 1. Q19. Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to the each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 7.

Q20. Evaluate :

cosθ + sinθ

cos 2θ log dθcosθ sinθ

.

Q21. Obtain the equation of tangent to the curves 3 3x asin θ, y bcos θ= = at π

θ4

.

Q22. There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective items? Naresh is knowingly producing defective items in his factory with an aim of earning more money. How would you stop him doing that by making him conscious of his wrong act?




3 13/9/1 P.T.O.

SECTION C

Question numbers 23 to 29 carry 6 marks each.

Q23. One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solve it graphically. Explain the importance of balanced diet in food.

Q24. Using integration, find the area of the region : 2 2{(x,y) : 9x + y 36, 3x + y 6} .

What is the importance of integration in life. Q25. Prove that the image of (3,–2, 1) in the plane 3x – y + 4z = 2 lies on the plane x + y + z + 4 = 0.

OR

Find the distance of the point (–1,–5,–10) from the point of intersection of the plane ˆˆ ˆr.(i j k) 5

and the line ˆ ˆˆ ˆ ˆ ˆr 2i j 2k λ(3i 4 j 2k) .

Q26. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of

radius R is 2R

3. Also find the maximum volume.

Q27. Evaluate : 1

1 3

0

sin (x 1 x x x ) dx .

Q28. Assume that the chances of a patient having a heart-attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart-attack by 30% and prescription of certain drug reduces its chances by 25%. At a time, a patient can chose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart- attack. Find the probability that the patient followed a course a course of meditation and yoga. Interpret the result & state which of the above stated methods is more beneficial for the patient.

OR A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e., if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive? Why do you think that these kind of medical tests must be more accurate?

Q29. If

3 4 2 3 4 26

A 2 3 5 , B 3 1 11

1 0 1 3 4 17

then, find the product AB.

Using this solve the following system of equations : 3x 4y 2z 1, 2x 3y 5z 7, x z 2 .

This sample test paper named as PLEASURE TEST SERIES XII has been prepared by award winning teacher OP Gupta. He may be contacted on +91-9650 350 480 or +91-9718 240 480. Email id : [email protected] Visit at : www.theOPGupta.blogspot.com # For chapter-wise Solutions of NCERT books, Solved Previous Years CBSE Papers, Sample Papers, Value Based Questions, Advanced Level Questions & much more, you may visit :

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4 13/9/1 P.T.O.

HINTS & ANSWERS for PTS XII – 01 [2013-2014]

Q01. Use |kA| = knA, where n is order of A. So, |3A| = 189. Q02. 1 Q03. 1

Q04. 2π π

π3 3 Q05. Let A(2, 3, 4), B(–1,–2, 1) & C(5, 8, 7). The d.r.’s of AB : 3, 5, 3; the d.r.’s of

BC : 6, 10, 6. Since 3 5 3

6 10 6, i.e., the d.r.’s of AB and BC are proportional so, AB and BC are

parallel. But B is a common point so, A, B and C must be collinear. Q06.

x2 tan x k

2+

Q07. m = 2 Q08. 5 Q09. 4.4cm/s Q10. 3 Q11. OR a 11, b 6= =

Q12. π

x4

Q13.

y x

y x

dy y(cos x) tan x (cos y) log cos y

dx (cos x) log cos x x(cos y) tan y

OR y x logx log(a bx)+

yay ...(i)

a bx x

axxy y

a bx

2

(a bx).a ax(b)x y y

(a bx)y +

22

2

yaxy y

x(a bx) [By (i)] 3 2x y (xy y)

Q14. Identity Element : 0 is the identity element for * defined on R–{–1}.

Also, tnverse of an element a is : –

a

a 1 ∈ R–{–1} Q16.

π

3 OR

π

2

Q18.

22 16

y x cosec x16 2

Q19. 7x – 8y + 3z + 25 = 0

Q20.

sin 2θ π 1log tan θ log sec2θ k

2 4 2 Q21. 2 2x 2 2y a b=

Q22. Required probability =

929 19

20 20. I would tell Naresh that if he keeps on producing defective items

then the good will of his company will suffer in long run which will result in heavy loss in future. Q23. Maximum no. of cakes are: 30; 20 of kind one & 10 cakes of another kind. Q24. (π )3 2 sq.units

Q25. Find the image of given point in 1st plane and then show that it satisfies the 2nd plane. OR The point of intersection is (2,–1, 2) so, required distance is 13units.

Q26. πR

cubic units

34

3 3 Q27.

π

41

Q28. Let A : the patient follows a course of meditation and yoga, B : he takes a certain drug. Then, P(A) = 1/2, P(B) = 1/2. Let E : the patient suffers a heart-attack. Also P(E|A) = (70 / 100).(40 / 100) , P(E|B) = (75 / 100).(40 / 100) .

By Bayes’ Theorem, we get :

P(E|A)P(A)P(A|E)

P(E|A)P(A) P(E|B)P(B)

14

29.

Interpretation of result : It is evident that if a patient follows a course of meditation and yoga, then he is less likely to get heart-attack. [Since P(B|E) = 15/29.] So, clearly a course of meditation and yoga is more beneficial as compared to the intake of drugs.

OR Let A : the person actually has the disease, A : the person doesn’t have the disease. Then,

P(A) = 0.1%, P( A ) = 99.9%. Let E : blood test results in positive. P(E|A) = 99%, P(E| A ) = 0.5%.

By Bayes’ Theorem, we get :

P(E|A)P(A) 22P(A|E)

P(E|A)P(A) P(E|A)P(A) 133.

These medical tests must be accurate in order to avoid wrong prescription of medicines and also inaccurate tests may cause unnecessary tension. Q29. –9I; x 3,y 2,z 1 .




1 13/9/2 P.T.O.


For more stuffs on Maths, please visit : www.theOPGupta.WordPress.com Time Allowed: 180 Minutes Max. Marks: 100

SECTION A

Q01. Let 1 2 3 nA d d d ... d= be a diagonal matrix. What is the value of det.(A)?

Q02. If

2 3A

k 2 and A. adjA 12 I then, find the value of k .

Q03. Under what condition (A – B) (A + B) is equal to A2 + B2 such that orders of A and B are same? Q04. Let : R Rf be defined by f (x) = (3 – x3)1/3. Determine fof (x).

Q05. If the plane 4x + 4y – λ z = 0 contains the line

y 1x 1 z

2 3 4, find the value of λ.

Q06. Evaluate the integral of 10

16

5 xdx

x

+. Q07. Evaluate : sin cos–1(1) + cos sin–1(1).

Q08. Find the value of m if the lines x 3 y 1 z 4

3 5m 4

and

x 1 y 4 z 4

1 1 2

are perpendicular to

each other. Q09. Find the unit vector in the direction of sum of vectors ˆ î j and ˆ ˆk j .

Q10. Find the integrating factor for the linear differential equation : 2

2

dy 2 dy(y 1) 2xy

dx y 1 dx

.

SECTION B

Q11. Let f

x a, x π/4

( ) x cot x b, π/4 x π/2

a cos2x b sin x, π/2 x π

x

<

2 0

2 is a continuous function on 0 x π . Then, determine

the values of ‘a’ and ‘b’. What are your views about “learning”? Is “learning” a continuous process?

Q12. Solve : π

sin sinx x 2

1 15 12. Q13. Evaluate :

π

πx sin x sin cosx

2dx

x π

0

2

2.

Q14. Let * be a binary operation on N defined by a*b = HCF of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N? OR Let : R Rf be defined as f (x) = 10x + 7. Find a function : R Rg such that we

have RIgof fog .

Q15. Express ˆ ˆ î j k 2 3 as the sum of a vector parallel and perpendicular to ˆ ˆ î j k 2 4 2 .

Q16. Evaluate : 4

2

xdx

(x 1)(x 1) . OR Evaluate :

2 2

2

2

x sin xsec x dx

1 x+.

Q17. Using properties of determinants, evaluate:

x sinθ cosθ

sinθ x 1

cosθ 1 x

.


2 1 3 1 2

Roll No.





2 13/9/2 P.T.O.

Q18. Form the differential equation of the family of circles in 2nd quadrant and touching coordinate axes. OR Form the differential equation of the family of curves given by (a + bx) ey/x = x. Q19. Find the equation of the plane parallel to x-axis and which contains the line of intersection of

the palnes ˆˆ ˆr.(i j k) 1

and ˆˆ ˆr.(2i 3 j k) 4 0

.

Q20. If xy log(x y) 1 then, prove that 2

2

dy y x y x y

dx x xy x y

.

OR If 2

2

1 1y x 1 log 1

x x

then, prove that

2dy x 1

dx x

.

Q21. If cos y = x cos (a + y) then, prove that 2sina (dy/dx) cos (a y) .

Q22. Two thirds of the students in a class are boys and the rest are girls. It is known that the probability of a girl getting first class is 0.25 and that of a boy is getting a first class is 0.28. Find the probability that a student chosen at random will get first class marks in the subject.

SECTION C Q23. Two trainee carpenters A and B earn `150 and `200 per day respectively. A can make 6 frames and 4 stools per day while B can make 10 frames and 4 stools per day. How many days shall each work if it is desired to produce at least 60 frames and 32 stools at a minimum labour cost? Solve graphically.

Q24. Triangle AOB is made in first quadrant of 22

2 2

yx1

a b where OA = a and OB = b. Find the area

enclosed between the chord AB ad arc AB of ellipse. “Differentiate the wastage of seconds; integrate the number of hours in a day.” Comment. Q25. Find the distance of the point (2, 3, 4) from the plane 3x 2y 2z 5 0 measured parallel to the

line x 3 y 2 z

3 6 2

. Q26. Evaluate :

2

2

1 xdx

1 x.

Q27. A farmer wants to construct a circular well and a square garden in his field. He wants to keep sum of their perimeters fixed. Then prove that the sum of their areas is least when the side of square garden is double the radius of the circular well. Do you think good planning can save energy, time and money?

OR Find the condition for the curves 22

2 2

yx1

a b and 2xy c to intersect orthogonally.

Q28. Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that

kx, if x = 0, or 1

P(X x) 2kx, if x = 2

k(5 x), if x = 3 or 4

, k is a positive constant.

Find the mean and variance of the probability distribution. Q29. For keeping fit, X people believe in morning walk, Y people believe in yoga and Z people join gym. Total no. of people are 70. Further 20%, 30% and 40% people are suffering from any diseases who believe in morning walk, yoga and gym respectively. Total number of such people is 21. If morning walk costs `0, yoga costs `500/month and gym costs `400/month and total expenditure is `23000. (i) Formulate a matrix problem. (ii) Calculate the number of each type of people.

(iii) Why exercise is important for health?

OR Find the inverse of

1 2 2

1 3 0

0 2 1

using elementary transformations.

This sample test paper named as PLEASURE TEST SERIES XII has been prepared by award winning teacher OP Gupta. He may be contacted on +91-9650 350 480 or +91-9718 240 480. Email id : [email protected] Visit at : www.theOPGupta.blogspot.com # For chapter-wise Solutions of NCERT books, Solved Previous Years CBSE Papers, Sample Papers, Value Based Questions, Advanced Level Questions & much more, you may visit : www.theOPGupta.WordPress.com Various works on Maths by OP Gupta can also be obtained from : www.cbseGuess.com

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3 13/9/2 P.T.O.


Q01. 1 2 3 n

|A| d d d . ... .d Q02. k = –3 Q03. AB BA Q04. (x) xfof

Q05. λ = 5 Q06.

3/2

10

1 51 k

75 x Q08. m = –11/5

Q07. 0 Q09. ˆ ˆ(i k)1

2 Q10. I.F. = 2(y 1)

Q11. π π

a , b( ) ( )

= = 2 1 2 2 4 1 2 2

. Yes, ‘learning’ is a continuous process. A person learns at

every moment of life from the daily activities happening around him. Q12. x = 13

Q13. π2

8 Q14. See NCERT Solutions Chapter 01 Ex. 1.4 Q. No. 8, visit

www.theOPGupta.WordPress.com OR x

(x)g

7

10 Q15. ˆ ˆ ˆ ˆ ˆ(k i j) (i j)

1 52

2 2

Q16. 2x 1 x 1 1

2log | x 1 | log k2 4 x 1 2(x 1)

OR 1tan x tan x k Q17. 3x

Q18. (x y) [(y ) ] (x yy ) 2 2 21 OR 2

3

2

d y dyx x y

dx dx

2

Q19. ˆˆr.( j 3k) 6 0

Q22. 0.27

Q23. To minimize : Z ` ( x y)150 200 ; Subject to constraints : x y , x y ; x, y 6 10 60 4 4 32 0 .

Minimum value of Z = `1350 at (5, 3). Q24. π

ab Sq.Units

2

4 Q25. 7 Units

Q26. Let

2 2 2

2 2 2 2 2

1 x 1 x 1 xI dx dx dx

1 x (1 x ) 1 x (1 x ) 1 x

2

2 2

1 x 2dx

(1 x ) 1 x

2

2 2 2 2

1 x 2I dx dx

(1 x ) 1 x (1 x ) 1 x

22

1dx I

1 x.

2 22 2 2

1 1 2tNow put x in I dx . So I dt

t t (t 1) 1 t= = .

Now put 2 2t 1 y 2t dt 2ydy= .

We have, 2 2

2yI dy

(y 2)y

2 2

12 dy

y ( 2)

2

2

y 21 1 t 1 22 log log

2 2 y 2 2 t 1 2

2

2 2

1 x 1 x 2i.e., I log

2 x 1 x 2. So,

22

2

1 x 1 x 2I log x 1 x log k

2 x 1 x 2+

Q27. Yes, every work done in a planned way proves to be more fruitful. If a student makes a planning for his studies he can do wonders. OR a2 = b2 Q28. 19/8, 47/64 Q29. (i) x + y + z = 70, 2x + 3y + 4z = 210, 5y + 4z = 230 (ii) x = 20, y = 30, z = 20 (iii) Exercise keeps

fit and healthy to a person. OR

3 2 6

1 1 2

2 2 5

.




1 13/10/2 P.T.O.



SECTION – A

Q01. Find |A |, if |adjA| = 225 s.t. A = [ aij

]3 3

. Q02. Find

1cosθ sinθ

sinθ cosθ.

Q03. Write the value of x y z if

1 0 0 x 1

0 1 0 y 1

0 0 1 z 0

.

Q04. Evaluate 1 11 12cos 3sin

2 2. Q05. Evaluate:

1 1 1tan 2cos 2sin

2.

Q06. The contentment obtained after eating x-units of a new dish at a trial function is given by the function

3 2C(x) x 6x 5x 3 . If the marginal contentment is defined as rate of change of C(x) with respect

to the number of units consumed at an instant, then find the marginal contentment when three units of

dish are consumed. Q07. Write the degree of

22 2

2 2

d y d y dy2 1 0

dxdx dx.

Q08. If a

and b

are two vectors of magnitude 3 and 2

3 respectively such that a b

is a unit vector, write

the angle between a

and b

. Q09. Find the projection of ˆˆ ˆ7i j 4k on ˆˆ ˆ2i 6 j 3k .

Q10. Write the distance between the parallel planes 2x – y + 3z = 4 and 2x – y + 3z = 18.

SECTION – B

Q11. Show that :

21 1

2

x 1sin[cot {cos(tan x)}]

x 2.

OR Solve for x : x x x π

sin cos tanx x x 3

21 1 1

2 2 2

2 1 23 4 2

1 1 1.

Q12. Prove that the function f : NN, defined by 2(x) x x 1f is one-one but not onto.

Q13. Two schools A and B decided to award prizes to their students for three values honesty (x), punctuality (y) and obedience (z). School A decided to award a total of `11000 for the three values to 5, 4 and 3 students respectively while school B decided to award `10700 for the three values to 4, 3 and 5 students respectively. If all the three prizes together amount to `2700, then : i. Represent this situation by matrix equation and form linear equations using matrix multiplication. ii. Is it possible to solve the system of equations so obtained using matrices? iii. Which value you prefer to be rewarded most and why?

Q14. If x a(θ sinθ) and y a(1 cosθ) , find 2

2

d y

dx.

Q15. If

1

2

sin xy

1 x, show that

22

2

d y dy(1 x ) 3x y 0

dxdx.


2 1 3 1 2

Roll No.





2 13/10/2

Q16. For what values of a and b, the function

2x ax b, 0 x 2

(x) 3x 2, 2 x 4

2ax 5b, 4 x 8

f is continuous on [0, 8]?

OR Differentiate

2 21

2 2

1 x 1 xtan

1 x 1 x with respect to 1 2cos x .

Q17. Evaluate :

3

2

x x 1dx

x 1. OR Evaluate :

x 1 sin xe dx

1 cosx.

Q18. Evaluate : 2 2

2xdx

(x 1)(x 2). Q19. Evaluate :

π /

log( tan x)dx4

0

1 .

Q20. Let ˆˆ ˆa 4i 5 j k

, ˆˆ ˆb i 4 j 5k

and ˆˆ ˆc 3i j k

. Find a vector d

which is perpendicular to

both a

and b

and satisfying d.c 21

.

Q21. Find the distance between the point P(6, 5, 9) and the plane determined by the points A(3,–1, 2), B(5, 2, 4), and C(–1,–1, 6). OR Find the equation of the perpendicular drawn from the point P(2, 4,–1) to the line

y 3x 5 z 6

1 4 9. Also, write down the coordinates of foot of the perpendicular from P to the line.

Q22. There is a group of 50 people who are patriotic out of which 20 believe in non-violence. Two persons are selected at random out of them, write the probability distribution for the selected persons who are non-violent. Also find the mean of the distribution. Explain the importance of Non-violence in patriotism.

SECTION – C

Q23. If

1 2 3

A 2 3 2

3 3 4

, find A–1. Hence solve : x 2y 3z 4 , 2x 3y 2z 2 , 3x 3y 4z 11 .

Q24. Find the equations of tangent & normal to the curve

x 7y

(x 2)(x 3) at the point where it cuts x-axis.

OR Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone.

Q25. Find the area of the region enclosed between the two circles 2 2x y 1 and 2 2(x 1) y 1 .

Q26. Find the particular solution of : (x sin y)dy (tany)dx 0 ; given that y(0) 0 .

Q27. Find the vector and Cartesian equations of the plane containing the two lines whose vector equations are

given as ˆ ˆˆ ˆ ˆ ˆr 2i j 3k λ(i 2 j 5k)

+ + + and ˆ ˆˆ ˆ ˆ ˆr 3i 3 j 2k μ(3i 2 j 5k)

+ + .

Q28. A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profit from crops A and B per hectares are estimated as `10500 and `9000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment? Q29. In a game of gambling, a card from a pack of 52 cards is lost. From the remaining cards, two cards are drawn at random and are found to be hearts. Find the probability of the missing card to be a heart. Why do you think that gambling is a curse on society? OR In a class, 50% of the boys and 10% of the girls have an I.Q. of more than 150. In the same class 60% of the students are boys. If a student is chosen at random and found to have an I.Q. of more than 150, find the probability that the student is a boy. What measures would you suggest to those having low I.Q.s so that they also are competent enough to succeed in life?

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3 13/10/2


Q01. |A | 15 Q02.

cosθ sinθ

sinθ cosθ Q03. x y z 1 ( 1) 0 0 Q04.

7π

6

Q05. π

4 Q06. 68 units Q07. Degree = 2 Q08.

π

6

Q09. 8

7 Q10. 14 units Q11. OR

πx tan

1

6 3

Q12. See OPG Vol. 2 Solved Mathematics Sample Question Paper

Q13. i.

5 4 3 x 11000

4 3 5 y 10700

1 1 1 z 2700

5 4 3 11000,

4 3 5 10700,

2700.

x y z

x y z

x y z

ii. Since |A| 3 0 i.e., so 1A exists and the equations have a unique solution.

iii. Any answer of the three values with proper reasoning will be considered correct.

Q14. 41 θcosec

4a 2 Q16. a = 3 and b = 2 OR

1

2

Q17. 2x 3 1

log|x 1| log|x 1| k2 2 2

OR x xe cot k

2

Q18. 2 2log|x 1| log|x 2| k Q19. π

log8

2 Q20. ˆˆ ˆd 7i 7 j 7k

Q21. 6

units34

OR Foot of perpendicular drawn from the point P on the line L is Q(–4, 1,–3).

And, the equation of PQ is :

y 4x 2 z 1

6 3 2 Q22. Mean =

196

245

Q23.

1

6 17 131

A 14 5 867

15 9 1

; x = 3, y = –2, z = 1

Q24. Tangent : x 20y 7 0 and Normal : 20x y 140 0 Q25. 2π

sq.units3

3

2

Q26. 2x = sin y

Q27. Vector equation : ˆˆ ˆr.(10i 5 j 4k) 37

. And Cartesian equation : 10x 5y 4z 37

Q28. Max.Z = `495000 at (30, 20) Q29. 11

50 OR

15

17.

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1 13/10/3 P.T.O.



SECTION – A Q01. If f : [0, 1] [0, 1] and and g : [0, 1] [0, 1] be the functions defined by f (x) = x2 & g (x) = 1 – x,

then find fog .

Q02. Find a skew symmetric matrix using A and AT where A =

2 5

3 4.

Q03. Without expanding, evaluate :

109 102 95

6 13 20

1 6 13

. Q04. Find the domain of the function f (x) = log x2 .

Q05. Evaluate : 33 x (1 x )dx . Q06. Evaluate :

π

cos x dx400

0

1 2 .

Q07. Total cost C(x) associated with the provision of free mid-day meals to x students of a school in primary

classes is given by 3 2C(x) 0.005x 0.02x 30x 50 . If the marginal contentment is given by rate of

change dC

dx of total cost, then write the marginal cost of food for 300 students. What value is shown here?

Q08. If OP 3a 2b

and OQ a b

, find the position vector OR

of a point R which divides the join of the

points P and Q in the ratio of 2:1 internally. Q09. For what value of x, ˆ ˆ ˆx(i j k) is a unit vector?

Q10. Show that the points (2,3,4),( 1, 2,1) and (5,8,7) are collinear.

SECTION – B

Q11. Show that : x y y x

tan tan sinx y x y

1 1 1

2 2

1 1

1 1 1 1.

OR Solve for x : π

tan x x sin x x 12

+ + + + 1 2 1 2 .

Q12. Discuss the commutativity and associativity of binary operation * defined on Q by the rule given as : a b a b ab* for all a, b R .

Q13. Find the equation of a plane which passes through (–1, 3, 2) and is perpendicular to each of the planes given as x + 2y + 3z = 5 and 3x + 3y + z = 0.

Q14. Discuss the differentiability of (x) x | x |f at 0x .

OR Let f (x) = sin x, g (x) = x2 and h (x) =e

log x . If V (x) = hofog (x), then prove that :

2

2 2 2 2

2

d V2cot x 4x cosec x

dx.

Q15. If θ

x a cosθ log tan2

and y asinθ then, find

2

2

d y

dx at

π

6θ .


2 1 3 1 2

Roll No.





2 13/10/3

Q16. Discuss the continuity of f (x) at x = 0, where

2

2 2

cosax cos bx, when x 0

x(x)b a

, when x 02

f ?

Q17. Evaluate : sinx

dxsin4x

. OR Evaluate : 3

3

cot x cot x

1 cot xdx

.

Q18. Evaluate : sin(x α)

dxsin(x α)

. Q19. Evaluate :

π /2

0

1sin 2x tan (sin x)dx .

Q20. Using properties of determinants, prove that :

2 2 2 2

2 2 2 2 2 2 2

2 2 2 2

b c a a

b c a b 4a b c

c c a b

+

+

+

.

Q21. If a, b,c

are three mutually perpendicular vectors of equal magnitude, prove that a b c

is equally

inclined with vectors a ,b

and c

.

OR If a, b,c

are non-coplanar vectors and b c c a

p , q[a b c] [a b c]

and a b

r[a b c]

then,

show that p.(a b) q.(b c) r.(c a) 3

.

Q22. Out of a group of 30 honest people, 20 always speak the truth. Two persons are selected at random from the group. Find the probability distribution of the number of selected persons who speak the truth. Also find the mean of the distribution. What values are described in this question?

SECTION – C Q23. Using matrix method, solve the following system of equations : x + 2y + z = 7, x – y + z = 4, x + 3y +2z = 10. Suppose x represents the number of persons who take food at home, y represents the number of persons who take junk food in market and z represent the number of persons who take food at hotel. Which way of taking food you prefer and why? OR In a survey of 20 richest person of three cities A, B and C it is found that in city A, 5 believe in honesty, 10 in hardwork, 5 in unfair means while in city B, 5 believe in honesty, 8 in hardwork, 7 in unfair means and in city C, 6 believe in honesty, 8 in hardwork, 6 in unfair means. If the per day income of these richest persons of cities A, B and C are `32.5K, `30.5K and `31K respectively, then find the per day income of each type of person by matrix method. (Here `1K means `1000.) (i) Which type of persons have more per day income? (ii) According to you, which type of person is better for country? Q24. A water tank with rectangular base and sides, open at the top is to be constructed so that its depth is 2m and volume is 8m3. If building of tank costs `70 per m2 for the base and `45 per m2 for the sides, what is the cost of least expensive tank? What is the importance of ‘save water’ movement? Q25. Using integrals, find the area of the triangular region the equations of whose sides are given as follow : y = 2x + 1, y = 3x + 1 and x = 4.

Q26. Find the particular solution of : 2 2(3xy y )dx (x xy)dy 0 ; given that for x = 1, y = 1.

OR Show that y dy y

x cos ycos xx dx x

is a homogeneous differential equation. Hence solve it.

Q27. Show that the lines ˆ ˆ ˆ ˆ ˆr i j k λ(3i j)

and ˆ ˆ ˆ ˆr 4i k μ(2i 3k)

intersect. Hence find their point of

intersection and, the equation of plane containing them. Q28. An aeroplane can carry a maximum of 200 passengers. A profit of `1000 is made on each executive class ticket and a profit of `600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit? Q29. In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probability of getting a special chest diseases are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the diseases. What is the probability that the selected person is a smoker and non-vegetarian? What value is reflected in this question?

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3 13/10/3


Q01. 2(1 x) Q02.

0 8

8 0

Q03. Apply 1 1 3 2 2 3 1 3

R R R R R R R and R are proportional. So 0 .

Q04. R – {0} Q05. 2x x x C 3 Q06. 0 Q07. `1368 Q08. 5a

3

Q09. 1

3 Q11. OR x , 0 1

Q12. Neither commutative nor associative Q13. 7x – 8y + 3z + 25 = 0

Q14. Function f (x) is differentiable at x = 0 OR Obtain V (x) = 2log sin(x )e

and then proceed.

Q15. 8

9a Q16. Function f (x) is continuous at x = 0

Q17.

1 1 2 sin x 1 1 sin xlog log C

8 1 sin x4 2 1 2 sin x

OR

2 11 1 1 2cot x 1log|cot x 1| log|cot x cot x 1| tan C

3 6 3 3

Q18.

1 2 2cos xcosα sin sinα log|sin x sin x sin α| C

sinα Q19.

π

21

Q21. See Hints/Answers in OPG Vol. 2 Q22.

Q23.

1

5 1 31

A 1 1 03

4 1 3

; x = 3, y = 1, z = 2 OR Per day income of person who believe in

honesty = `1500, Per day income of person who believe in hardwork = `2000 and, Per day income of person who believe in unfair means = `1000. (i) The persons who believe in hardwork has more per day income. (ii) A person who believes in hardwork and honesty, is better for country. Q24. C = ` [280 + 180 (l + b)], where 2lb = 8. Minimum Cost = `1000.

Q25. 8sq.units Q26. x2y(y + 2x) = 3 OR sin

y

x = log |x| + C

Q27. Point : (4, 0, –1); 3x + 9y – 2z = 14 Q28. Assume that the number of tickets to be sold for the executive class and economy class be respectively x

and y.

To maximize: Z = ` 1000 600x y

Subject to constraints: 200, 20, 4 , 0, 0x + y x y x x y

Max.Z `136000 at (40,160) Q29. 28

45.


Good Luck & God Bless You!!!

Values described here :

Honesty, Truthfullness.

Probability Distribution :

X 0 1 2 P(X) 90/870 400/870 380/870




1 13/10/4 P.T.O.



SECTION – A

01. Show that the binary operation defined by a*b = ab + 1 on Q is commutative.

02. Find a matrix X such that B – 2A + X = O, where A = 5 3

3 1

and 0 -2

B3 1

=

.

03. The income I of Dr.Rastogi is given by I(x) = ` (x3–3x2+5x). Can an insurance agent ensure him for the

growth of his income? 04. Evaluate : tan cos

11 5

2 3.

05. Write the values of x – y + z from the following equation:

x y z

x z

y z

9

5

7

.

06. Evaluate : )xlog1(cosx

dx2

. 07. Evaluate :

/2

5 4

/2

sin xcos xdx.

08. The Cartesian equation of a line AB is y 22x 1 z 3

2 33

. Find the direction cosines of a line

parallel to AB. 09. If | a | 5 ; | b | 13 and |a b | 25 , find a . b

.

10. If a i j k; b 2i j 3k and

c i 2j k , then find a unit vector parallel to the

vector 2a b 3c+

. SECTION – B

11. If

5

3R

5

7R:f

be defined as

7x5

4x3)x(f

and

5

7R

5

3R:g be defined as

3x5

4x7)x(g

. Show that Agof I and Bfog I where B =

5

3R and A = R–

5

7.

12. Using properties of determinants, prove that :

a +b+2c a b

c b+c+2a b

c a c+a+2b

= 2(a + b + c)3.

OR Using properties of determinants, prove that :

2

2

2

a ab ac

ba b bc

ca cb c

= 4 2 2 2a b c .

13. Solve the following differential equation : xtanydx

dy)x1( 12 .

14. For what value of k is the function f (x)

2x 1; x 2

k; x 2

3x 1; x 2

+ <

=

>

continuous at x = 2?

OR Show that |x + 4| is not differentiable at x = –4. Is it continuous at the same point?


2 1 3 1 2

Roll No.





2 13/10/4

15. Suppose X has a binomial distribution 1

B 6,2

. Show that X 3 is the most likely outcome.

16. Find the intervals in which the function f (x) = sin x – cos x; 2x0 is (i) increasing and/or (ii) decreasing.

17. Prove that : 1 1 1 31tan tan2A tan (cot A) tan (cot A) 0

2

, π π

A4 2 .

OR Solve for x : 1 1 πsin (1 x) 2sin x

2 . 18. Prove that :

22

4 4

0

x sin x cosxdx

cos x sin x 16

.

19. Find the particular solution of differential equation : ,0x

yeccos

x

y

dx

dy

if y(1) = 0.

20. If a

and b

are two vectors such that b.a = c.a , caba and 0a , then prove that cb .

21. If 2

2

d y πx 2 cosθ cos2θ and y 2 sinθ sin2θ, find at θ

dx 2 .

OR If y = [log(x + 22 )]x1 , show that 2

2

2

d y dy(1 x ) x 2 0.

dx dx

22. Find the shortest distance between the lines 1

z

1

1y

2

1x

and

x 2.

3

y z

1 1

5 2

SECTION – C

23. Given that A = 4 4 4

7 1 3

5 3 1

and B = 1 1 1

1 2 2

2 1 3

then, find the product AB. Hence using

this product solve the system of equations : x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1. 24. A milkman is having a vessel in the shape of a right circular cylinder, which is open at the top and has a given surface area. Show that the vessel will acquire the maximum amount of milk if its height is equal to the radius of its base. “Intake of milk proves good for health.” How? 25. Make a rough sketch of the region given below and find the area using the method of integration :

2x,y : 0 y x 3,0 y 2x 3,0 x 3

OR Using the method of integration, find the area of the region bounded by the following lines :

2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0. 26. Evaluate : [ tan x cot x]dx .

27. In a bolt factory, machines A, B and C, manufacture respectively 25%, 35%, 40% of the total bolts. Of their output 5%, 4% and 2% respectively are defective bolts. A bolt is drawn at the random and is found to be defective. Find the probability that it is manufactured by machine B. ‘Machines have proved beneficial for mankind.’ Comment. 28. An aeroplane can carry a maximum of 200 passengers. A profit of `400 is made on each first class ticket and a profit of `300 is made on each economy class ticket. The airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class to by the first class. Determine how many of each type ticket must be sold in order to maximize the profit for the airline. What is the maximum profit? Frame an L.P.P. and solve it graphically. 29. Find the image Q of the point P(1, 2, 3) in the plane x + 2y + 4z = 38. Also find the perpendicular distance from the point to the plane. Hence write the vector equation of PQ. OR A line makes angles α, β, γ and δ with the diagonals of a cube, prove that :

sin2α + sin2β + sin2γ + sin2δ = 8/3.

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3 13/10/4


Q02.

10 8

9 1 Q03. Yes. Show that I (x) > 0 for all xR.

Q04.

3 5

3 5 Q05. 1 Q06. tan (1 + log x) + k

Q07. Show that sin5x cos4x is an odd function. So, using the property:

( ) 0a

a

f x dx if ( )f x is an

odd function, we get :

/2

5 4

/2

sin xcos xdx 0=

Q08. Parallel lines have same set of d.c.’s. So, d.c.’s of line parallel to AB are: 3 4 6

, ,55 55 55

Q09. 60 Q10. 1

(3i 3 j 2k)22

Q11. NCERT Part I Ex 17, Page 13

Q13.

11 tan xy tan x 1k e Q14. k = 5 OR Continuous at x = –4

Q16. (i) [0, 3 / 4] [7 / 4,2 ] (ii) [3 / 4,7 / 4] Q17. x = 0

Q19. log x = 1 – cos (y/x) Q21. –3/2 Q22. 10

units59

Q23. AB = 8I, x = 3, y = –2, z = –1 Q25. 50/3sq.units OR 7/2sq.units

Q26. NCERT Part II Example 41, Page 350 :

1 tan 12 tan

2 tan

xk

x

Q27. 196/449 (Refer NCERT Part II Page 553 Example 19. The method chosen there is incorrect.)

Q28. `64000 Q29. Q (3, 6, 11), 21units . Eq. of PQ:

r i 2 j 3k (3i 6 j 11k)= + .


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1 13/11/1 P.T.O.



SECTION – A

Q01. Write the value of x : 1 1sin cot (1 ) cos tanx x . Q02. Evaluate :

2cos

1 x

xdx

a.

Q03. If A = {1, 2, 3} and B = {a, b}, write the total number of function from A to B.

Q04. If x, y, z are in geometric progression, evaluate :

0

xp y x y

yp z y z

xp y yp z

.

Q05. Evaluate :

. ( ) ( )A B C A B C .

Q06. What are the points at which the function ( ) | | 1f x x is not differentiable?

Q07. Determine the value of ‘c’ of Rolle’s Theorem for the function 4/3( )f x x on 1 1x .

Q08. If D is the mid-point of side BC of a ABC , then prove that AB AC 2AD

.

Q09. Find the value of k such that the line

24

1 1 2

yx z k

lies in the plane 2 4 7x y z .

Q10. If |adjA| = 36 then, find 1| 3A | if A is a square matrix of order 3.

SECTION – B Q11. A function ( )f x is defined as follows :

sin( ) , 0

2, 0

xf x if x

xif x

.

Is ( )f x continuous at 0x ? If not, what should be the value of ( )f x at 0x so that ( )f x

becomes continuous at 0x ?

Q12. Evaluate :

3 5

2 4

cos cos

sin sin

x xdx

x x. OR Evaluate :

2

2( )

xdx

a bx.

Q13. A plane which is perpendicular to two planes 2 2 0x y z and 2 4x y z , passes through

the point (1,–2, 1). Find the distance of the plane from the point (1, 2, 2).

Q14. If 1 1 1cosec [tan {cos(cot sec(sin ))}]x a and 1 1 1sec [cot {sin(tan cosec(cos ))}]y a , then

find a relation between x and y in terms of a.

OR Prove that :

1 1 1 1 18 8 300cot 2 tan cos tan 2 tan sin tan

17 17 161.

Q15. Evaluate :

cos 5 cos 4

1 2cos 3

x xdx

x. Q16. Evaluate :

2

1/

loge

e

e

xdx

x.

Q17. Find the intervals in which (1 )( ) x xf x xe is (i) increasing, and (ii) decreasing.

Q18. Let A be the set of all students of class XII in a school and R be the relation having the same sex (i.e., male or female) on set A, then prove that R is an equivalence relation. Do you think, co-


2 1 3 1 2

Roll No.





2 13/11/1

education may be helpful in child development and why? Q19. The probability of a man hitting a target is 1/4. How many times must he fire so that the probability of his hitting the target at least once is more than 2/3? In recent past, it has been observed that India has done quite well (as compared to other sports) at various International Shooting Contests. What may be the reasons for this?

Q20. Let ˆ â i j ,

ˆˆb j k

and ˆ ˆc k i . If

d

is a unit vector such that

a is perpendicular to

d and

[ ] 0 b c d then, find the vector

d .

OR Anisha walks 4km towards west, then 3km in a direction o60 east of north and then she stops. Determine her displacement with respect to the initial point of departure. Q21. Using first principle of derivative, differentiate : log cot 2x.

OR If 2 21 1x y a x y then, show that 2

2

1

1

dy y

dx x

.

Q22. For positive numbers x, y and z, find the numerical value of :

1 log log

log 1 log

log log 1

x x

y y

z z

y z

x z

x y

.

SECTION – C Q23. In a Legislative assembly election, a political party hired a public relation firm to promote its candidate in three ways : telephone, house calls and letters. The numbers of contacts of each type in three cities A, B & C are (500, 1000, 5000), (3000, 1000, 10000) and (2000, 1500, 4000), respectively. The party paid `3700, `7200, and `4300 in cities A, B & C respectively. Find the costs per contact using matrix method. Keeping in mind the economic condition of the country, which way of promotion is better in your view?

OR Using elementary column operations, find the inverse of matrix

0 0 1

3 4 5

2 4 7

.

Q24. If the area enclosed between 2y mx and 2x my , ( 0)m

is 1 sq. unit then, find the value of m.

Q25. By examining the chest X-ray, the probability that T.B. is detected when a person is actually suffering is 0.99. The probability that the doctor diagnosis incorrectly that a person has T.B. on the basis of X-ray is 0.001. In a certain city, 1 in 1000 suffers from T.B. A person is selected at random and is diagnosed to have T.B. What is the probability that he actually has T.B.?

‘Tuberculosis (T.B.) is curable.’ Comment in only one line.

Q26. For what value of ‘a’ the volume of parallelopiped formed by ˆˆ î aj k , ˆj ak and ˆâi k is

minimum? Also determine the volume.

OR Show that the condition that the curves 2 2 1ax by and 2 2 1mx ny should intersect

orthogonally is given by: 1 1 1 1

a b m n.

Q27. Find the equation of the plane passing through (2, 1, 0), (4, 1, 1), (5, 0, 1). Find a point Q such that its distance from the plane obtained is equal to the distance of point P(2, 1, 6) from the plane and the line joining P and Q is perpendicular to the plane. Q28. A) If ( )y t

is a solution of (1 ) (1 )t dy ty dt and (0) 1y then, what is the value of (1)y ?

B) Write the degree of the differential equation representing the family of curves 2 2 ( )y c x c ,

where c is a positive parameter. Q29. A farmer owns a field of area 1000m2. He wants to plant fruit trees in it. He has sum of `2400 to

purchase young trees. He has the choice of two types of trees. Type A requires 10m2 of ground per tree and costs `30 per tree and, type B requires 20m2 of ground per tree and costs `40 per tree. When full grown, a type A tree produces an average of 20kg of fruits which can be sold at a profit of `12 per kg and a type B tree produces an average of 35kg of fruits which can be sold at a profit of `10 per kg. How many of each type should be planted to achieve maximum profit when trees are fully grown? What is the maximum profit? ‘India is a land of farmers.’ Comment.




3 13/11/1


Q01.

1 1

2 2

1 1 1sin sin cos cos

22 2 1x

x x x

Q02. Let

2cosI

1 x

xdx

a…(i). Use

b b

a a

f x dx f a b x dx to get,

2cos ( )

I1 x

xdx

a…(ii)

Adding (i) & (ii), we have :

2 2

0 0

1 1 1 cos 2I cos 2 cos

2 2 2

xxdx xdx dx

2

Q03. Total number of function from A to B is 23 = 8

Q04. As x, y, z are in GP, so 2 ...(i)y xz . Then apply 1 1 2

C C Cp 3 3 1

C C C . Then expand

along 3

C and use (i) to get 0

Q05. .

A B A B B B C C A C B C C

.( ) .(0) .( ) .( ) .( ) .(0)

A B A A A B C A C A A C B A

[ ] [ ] [ ] [ ] 0 [ ] 0 [ ] 0

A B A A B C A C A A C B A B C A B C

Q06. The function ( ) | | 1f x x is not differentiable at x = 0. Also for 0x , we have ( ) | 1|f x x if

x > 0 and ( ) | 1|f x x if x < 0 which reflects their nature of not being differentiable at x = 1, –1

respectively. So, the function f (x) is not differentiable at x = –1, 0, 1. Q07. 0c

Q08. Let OA , OB , OC .a b c

We have OD2

b c

.

Now LHS : AB AC ( ) ( ) ( 2 ) 22

b cb a c a b c a a

2 OD OA 2AD

= RHS.

Q09. Obtain the coordinates of random point M (say) on the given line then, M must satisfy the equation of plane 2 4 7x y z . So we get k = 7.

Q10. Use |adjA| = |A|3–1 to find |A| = 6 then |A–1| = 1

6 . So finally |3A–1| = 33|A–1| = 27

1 9

6 2

.

Q11. We have RHL = 1. Also f (0) = 2. Since RHL f (0) so, f (x) is discontinuous at x = 0. In order to make it continuous, the value of f (x) at x = 0 should be 1.

Q12. I =

3 5

2 4

cos cos

sin sin

x xdx

x x

2 4

2 4

(cos cos )cos

sin sin

x x xdx

x x. Put sin cosx t xdx dt

2 2 2

2 4

[1 (1 ) ]I

t tdt

t t

2

2 4

2 41

tdt

t t

2

2 2

2 4

(1 )

tt dt

t t…(i)

Consider

2

2 2

2 42 4

(1 ) 1(1 )

yt A B

y y y yt t where 2y t so, equation (i) becomes,

1

2 2

2 6 2I 6 tan C

1t dt t t

tt t

1I sin 2cosec 6 tan sin Cx x x .

OR Put

1t a

a bx t x dx dtb b

. So,

2

2

1 1I

t adt

b bt

2

3 2

1 21

a adt

tb t

2

3

1I ( ) 2 log( ) C

aa bx a a bx

a bxb

2

2 3 3

2I log| | k

( )

x a aa bx

b b b a bx, where

3k C

a

b.

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4 13/11/1

Q13. Let the d.r.’s of required plane be A, B, C. Since required plane is perpendicular to the given planes

so, 2A 2B C 0 and A B 2C 0

A B C

3 3 0. So the required equation of plane is :

3( 1) 3( 2) 0( 1) 0x y z i.e., 1 0x y . And its distance from (1, 2, 2) is 2 2 units.

Q14. 23x y a OR OPG Vol.1 Q No.08 (l)

Q15. See C-30 on Indefinite Integrals Q No.25. Download it from www.theOPGupta.com/ in the section Class XII Advanced Level Questions.

Q16. 2 21

1/ 1/ 1

log log logI ...(i)

e e

e e e

e e

x x xdx dx dx

x x x. Consider

2log (log )

2e x x

dxx

. So by (i), 5

I2

.

Q17. (1 ) 2 1( ) [1 2 ]

2x xf x e x x x , 1.

Q18. The relation R is reflexive, symmetric and transitive. Co-education is very helpful because it leads to the balanced development of the children and in future they become good citizens. Q19. Let p = probability of hitting the target = 1/4. So q = 1 – p = 3/4. Let the man fires ‘n’ times.

According to question, 2 2

P(r 1) 1 P(r 1) 1 P(0)3 3

= 1

P(0)3

i.e., n 0 n 0 n

0C (1/4) (3/4) 1/3 (3/4) 1/3 . That is, the least value of n is 4. Better coaching,

training and more exposure to the shooters along with good quality equipments are responsible for good show of shooters at international level.

Q20. Let ˆˆ ˆd xi yj zk 2 2 2 1x y z …(i). As . 0

a d a d x y…(ii).

Also [ ] 0 b c d 0x y z …(iii).

Solving (i), (ii) & (iii), we get :

2 1 1ˆ ˆ ˆ6 6 6

d k i j .

OR Similar question on Page 03 of OPG Vol.2 Q No.02

Q21. Let ( ) logcot 2f x x . So,

2

0

logcot2( ) logcot2 2cosec 2( ) lim

cot 2h

x h x xf x

h x

OR OPG Vol.1 Page 50 Q No. 60

Q22. M 1 :

1 log log

log 1 log

log log 1

x x

y y

z z

y z

x z

x y

log log log

log log log

log log log

log log log

log log log

log log log

x y z

x x x

x y z

y y y

x y z

z z z

[Using log

loglog

ba

b

pp

a

Take log x , log y and log z common from 1

C , 2

C and 3

C respectively. We have :

1 1 1

log log log

1 1 1log log log

log log log

1 1 1

log log log

x x x

x y zy y y

z z z

Again take 1

logx,

1

log y and

1

logz common from

1

R , 2

R and 3

R respectively. We have :

1 1 1log log log

1 1 1log log log

1 1 1

x y z

x y z= 0.

M 2 : 1(1 log log ) log (log log log ) log (log log log )z y x y z y x y z zy z y x x z z x y x

(1 1) log log log log log log log log log logx y z x y y z x x zy x x y z x y z z x

–ve ve –ve – –1/2 1 +

Hints & Answers For PTS XII – 07 [2013-14] By OP Gupta [+91-9650 350 480] | www.theOPGupta.com/ Page 4




5 13/11/1

log log1 1log log log log

log log log log

y xx z z x

x y x x

z zy x y z

y x y z

1 log log log log 1z x z yx z y z 0 [

1log

log

a

b

ba

, log

loglog

ba

b

pp

a].

Q23. Cost per Contact : Telephone = `0.40, House calls = `1.00, Letters = `0.50. Telephone is better medium for promotion as it is cheap.

OR Let A

0 0 1

3 4 5

2 4 7

.

Since A = A I (Using column operations), we have :

0 0 1 1 0 0

3 4 5 A 0 1 0

2 4 7 0 0 1

Follow the following steps of properties :

I : 1 1 3

C C C II : 2 2 3

C C C III : 2 2 1

C C C IV : 3 3 1

C C C

V : 1 1 3 2

C C C C VI : 3 3 2

C C 3C VII : 1 1 3

C C C VIII : 1 1 3

1C C C

4

IX :

3 3

1C C

4 X :

2 2 3C C 2C .

Now since AA–1 = I so,

1

2 1 1

A 11/4 1/2 3/4

1 0 0

.

Q24. On solving given eqs., we have : x = 1/m, 0. Required Area 1/m 1/m

2

0 0

1xdx mx dx

m

1

3m .

Q25. Let E : A person is diagnosed to have T.B., A : The person actually has T.B.

So, P(A) = 1/1000, P( A ) = 999/1000, P(E|A) = 990/1000, P(E| A ) = 1/1000.

By Bayes’ Theorem, P(A|E)

P(E|A)P(A) 110

P(E|A)P(A) P(E|A)P(A) 221. Although T.B. is a dangerous

disease still it can be cured with proper medicines (DOTS) under the supervision of medical expert.

Q26. Use Scalar Triple Product of vectors to obtain the volume. Volume, V 3 11

3a a a . Also

the minimum volume is 2

V 13 3

cubic units.

OR OPG Vol.1 Page 61 Q No. 10 Q27. Equation of plane : 2 3x y z . Note that Q is the Image of point P in the plane. So Q(6, 5,–2).

Q28. A)

1

(1 ) (1 )

dy ty

dt t t. I.F. (1 ) tt e so, solution is (1 ) Ct te t y e . Use (0) 1y to

get :

1

(1 )y

t and then,

1(1)

2y .

B) 2 3/ 22 2y cx c …(i). On differentiating we get : c yy . Put value of c in (i), we have :

2 3 /22( ) 2( )y yy x yy

22

32( )

2

y xyyyy . It is clear that degree is 3.

Q29. Z = `(240x + 350y). Also 2 100x y ; 3 4 240x y ; x , 0y . Max.Z = `20100 at (40, 30).


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[SECTION – A] Q01. Let * be a binary operation defined by a*b = 2a + b – 3, find the value of 3*4.

Q02. Write the value of :

1 3πsin sin

5. Q03. What is the value of , if

a + ib c + id

c + id a ib

?

Q04. Check if the function 3

2x 3x x

3 2 is decreasing in R.

Q05. For what value of ‘m’ and ‘p’, is the matrix

0 5 3

5 m 4

p 4 0

skew-symmetric?

Q06. Show that a powerful bomb shot along the line of fire

y 2x 1 z 3

2 3 4 will never hit a helicopter

flying in the plane 2x + 4y – 4z + 11 = 0. Q07. Write the number of binary operations that can be defined on the set {1, 2}.

Q08. Let a and

b are non-collinear vectors. For what value of x, the vectors

c (x 2)a b and

d (2x 1)a b are collinear?

Q09. Write a unit vector perpendicular to the vectors a and

b both, if it is given that

ˆˆ ˆa = 3i 2 j 6k and

ˆˆ ˆb i 2 j 2k . Q10. Write the value of 2

3cos xdx

2sin x .

[SECTION – B]

Q11. Discuss the differentiability of f (x)

1 x, if x < 1

= (1 x)(2 x), if 1 x 2

2 x, if x > 2

at x = 2.

(OR) A car driver is driving a car on the dangerous path given by

f (x)

m1 x, if x 1

1 xm 1, if x 1

, m ∈ N.

Find the dangerous point (point of discontinuity) on the path. Whether the driver should pass that point or not? Justify your answer.

Q12. Let ˆ ˆˆ ˆ ˆa = 2i + k, b = i + j + k and

ˆˆ ˆc = 4i 3j + 7k be three vectors. Determine a vector r which satisfies

the condition

r b c b and r .a = 0 .

Q13. Express 2

1

2

1 x 1cos

2 1 x

in simplest form. (OR) Solve : 2 1 2 1sec tan 2 cosec cot 3 x .

Q14. Evaluate :

π

2

0

x sinxdx

1 cos x. Q15. Solve the differential equation : 1 2(x tan y)dy (y 1)dx 0 .

Q16. If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its volume. (OR) Water is dripping out from a conical funnel at a uniform rate of 4cm3/s through a tiny hole at the vertex in the bottom. When the slant height of the water is 3cm, find the rate of decrease of the slant height of the water-cone. Given that the vertical angle of funnel is 120o. Q17. A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?

13/11/2 Page 01 P.T.O.

PLEASURE TEST SERIES XII – 08 Series : PTS/8 Code No. 13/11/2





Q18. Find the distance of the point (– 2, 4, – 5) from the line y 4x 3 z 8

3 5 6

.

Q19. Evaluate : dx

sin(x α)sin(x β)

. (OR) Evaluate : 3 1/3

4

(x + x )dx

x

.

Q20. Let f, g : R→R be defined as f (x) = |x| and g (x) = [x], where [x] denotes greatest integer less than or equal

to x. Evaluate :

5 5(gof) (fog)

3 3

5(fo(gof))

3

. Q21. If 2y 4ax , then evaluate : 2 2

2 2

d y d x.

dx dy

.


2 2 2

2 2 2 3 3 3 2

2 2 2

2bc a c b

c 2ca b a =(a + b + c 3abc)

b a 2ab c

.

[SECTION – C] Q23. Using integrals, find area of region bounded by the following curve after making a rough sketch: y 1 | x 1 | , | x | 3 , y 0 .

Q24. Three friends A, B and C visited a Super Market for purchasing fresh fruits. A purchased 1kg apples, 3kg grapes and 4kg oranges and paid `800. B purchased 2kg apples, 1kg grapes and 2kg oranges and paid `500. While C paid 700 for 5kg apples, 1kg grapes and 1kg oranges. Find the cost of each fruit per kg by using matrix method. Why are the fruits good for health?

(OR) Using elementary operations, find the inverse of

1 1 0

2 3 4

0 1 2

, if it exists.

Q25. A manufacturer has three machine operators A (skilled), B (semi-skilled) and C (non-skilled). The first operator A produces 1% defective items whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of time, B in the job for 30% of the time and C is on the job for 20 % of the time. A defective item is produced, what is the probability that it was produced by B? What is the value of skill in industries?

Q26. A bird at A(7, 14, 5) in space wants to reach a point P on the plane 2x + 4y – z = 2 when AP is least. Find the position of P and also the distance AP travelled by the bird.

Q27. Evaluate π/2

0

log cosec x dx , by using properties of the definite integral.

Q28. If a class XII student aged 17 years, rides his motor cycle at 40km/hr, the petrol cost is `2 per km. If he rides at a speed of 70km/hr, the petrol cost increases to `7per km. He has `100 to spend on petrol and wishes to ` cover the maximum distance within one hour.

(i) Express this as an L.P.P., and solve graphically. (ii) What is benefit of driving at an economical speed?

(iii) Should a child below 18years be allowed to drive a motorcycle? Give reasons. Q29. If PA and QB be two vertical poles of height 16m and 22m at points A and B respectively such that

AB= 20m then, find the distance of a point R on AB from the point A such that 2 2RP + RQ is minimum.

(OR) A point P is given on the circumference of a circle of radius r. The chord QR is parallel to the tangent line at P. Find the maximum area of the triangle PQR.

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13/11/2 Page 02




HINTS & ANSWERS for PTS XII – 08 [2013-14]

Q01. 7 Q02. 2

5 Q03. 2 2 2 2a b c d

Q04. Decreasing function Q05. 0, 3 Q06. Show that the line is parallel to the plane i.e., the line is at right angle the normal vector of the

plane. Q07. 2 22 16 Q08. 1/3

Q09. ˆ ˆ4k 8i

4 5 Q10.

3cosec

2x k

Q11 Not differentiable as LHD = 1 but RHD = –1 OR Point of discontinuity : x = 1. No, because life is precious so vehicles should be driven carefully.

Q12. ˆ ˆ ˆ2k 8 j i Q13. 11tan x

2 OR x = 15

Q14.

2

2 Q15.

1tan y1x tan y e 1= k Q16. 9.72cm3 OR

32

27cm

Q17. 42%. Since no one trusts a liar, so the statement of B will carry more weight as he speaks truth in more number of cases than A.

Q18. 37

10units Q19.

1 sin(x α)log

sin(α β) sin(x β)k OR

4/3

2

3 11

8 xk

Q20. –1 Q21. –2a/y3 Q23. 16 sq.units Q24. Let cost of each fruit be x, y, z respectively. Then solve the equations so formed by using matrix

method. The inverse of matrix will be

1 1 21

8 19 611

3 14 5

.

So, x = 100, y = 100, z = 100. Hence the cost of each fruit is `100 per kg. Importance of fruits : Fruits contain nutrients and vitamins which help our body in its proper

growth and maintenance. OR

1/3 1/3 2/3

2/3 1/3 2/3

1/3 1/6 5/6

Q25. 15/34

Q26. P(1, 2, 8), AP = 189 units Q27. π

log22

Q28. Max. Z = x + y. Subject to constraints: x/40 + y/70 1, 2x + 7y100; x, y0. Here x & y represents the distance travelled by the boy at speed of 40km/hr & 70km/h respectively. (i) x = 1560/41km, y = 140/41km. (ii) It saves petrol. It saves money. (iii) No, because according to the law driving license is issued when a person is above the 18 years of age.

Q29 10m OR 23 3 r

sq.units4

.

Good Luck & God Bless You!!! Hints & Answers For PTS XII – 08 [2013-14] By OP Gupta [+91-9650 350 480] | www.theOPGupta.com/






[SECTION – A]

Q01. If∗isabinaryoperationgivenby∗:R×RR,a∗b=a+b2,thenwhatisthevalueof–2∗5?

Q02. If 1sin : [ 1,1]

π 3π,

2 2isafunction,thenwritethevalueof

1 1sin

2.

Q03. It is given that

9 6 2 3 3 0

3 0 1 0 1 2. What do you get on applying elementary row

transformationR1R1–2R2onboththesides?

Q04. Writethedegreeofthedifferentialequation:

2/33 2

2

dy d y1

dx dx.

Q05. IfAandBaresquarematricesoforder3suchthat|A|=–1and|B|=4,thenwritethevalueof

|3(AB)|. Q06. Whatisthevalueof 2

ˆ ˆi j ?

Q07. Whatisthedistancebetweentheplanes3x+4y–7=0and6x+8y+6=0?

Q08. Evaluate: sinx + cosx

dx1+ sin2x

. Q09. Writethevalueofintegral:

π/2

83 123

π/2

(sin x + x )dx .

Q10.Writetheintegratingfactorforthelineardifferentialequation: 2dyx y x

dx.

[SECTION – B]

Q11. Discussthecontinuityoff(x)=|x+1|+|x+2|atx=–2.Q12. Determinethevectorequationofalinepassingthrough(1,2,–4)andperpendiculartothetwo

lines ˆ ˆˆ ˆ ˆ ˆr = 8i 16j + 10k + λ(3i 16j + 7k) and

ˆ ˆˆ ˆ ˆ ˆr = 15i 29 j + 5k + μ(3i 8 j 5k) .

Q13. Provethat: 1 1 1 1cot 7 cot 8 tan 3 tan 18 .

(OR) Solvetheequation: 1 1 1 2tan (2 x) tan (2 x) tan

3.

Q14. Findaparticularsolutionofthefollowingdifferentialequation:

x x

y y2y e dx (y 2xe )dy 0 giventhatx=0wheny=1.

Q15. Evaluate: π/6

4 3

0

sin x cos x dx .

Q16. Findtheequationoftangenttothecurvex=sin3t,y=cos2tatt=π

4.

(OR) Find the intervals in which the function f (x) = sin4 x + cos4 x, 0 < x <π

2 is strictly

increasingorstrictlydecreasing. Q17. Fourdefectivebulbsareaccidentlymixedwithsixgoodones.Ifitisnotpossibletojustlookat abulbandtellwhetherornotitisdefective,findtheprobabilitydistributionofthenumberof


2 1 3 1 2

Roll No. Candidates must write the Code on the title page of the answer-book.

13/11/3 (Page 1) P.T.O.




defectivebulbs,iffourbulbsaredrawnatrandomfromthislot.Q18.Evaluate: 2x(logx) dx .

Q19. OnthesetR–{–1},abinaryoperationisdefinedbya*b=a+b+abforalla,b∈R–{–1}.Check if*isbinary.Provethat*iscommutativeonR–{–1}.Findtheidentityelementandprovethat everyelementofthesetR–{–1}isinvertible. (OR) Letnbeafixedpositive integerandRbetherelationinZdefinedasaRbifandonlyif “a–bisdivisiblebyn”,∀a,b∈Z.ShowthatRisanequivalencerelation.

Q20. Finddy

dx,if

θ 1x = e θ

θand

θ 1y = e θ

θ.

(OR) Using mean value theorem, prove that there is a point on the curve y = 2x2 – 5x + 3 betweenthepointsA(1,0)andB(2,1),wheretangentisparalleltothechordAB.Also,findthat point.Q21. Find the differential equation of all the circles which pass through the origin and whose

centreslieonx-axis. Q22. Solvefor x :

x 2 x 6 x 1

x 6 x 1 x 2 = 0

x 1 x 2 x 6

.

[SECTION – C] Q23. Find the area of the triangle formed by positive x-axis, and the normal and tangent to the

circle 2 2x y 4+ = at(1, 3 ),usingintegration.

Q24. IfA=

1 3 2

2 0 1

1 2 3

,thenshowthatAsatisfiestheequationA3–4A2–3A+11I=O.HencefindA–1.

Q25. There are three coins. One is a biased coin that comes up with tail 60% of the times, the secondisalsoabiasedcointhatcomesupheads75%ofthetimesandthethirdisanunbiased coin. One of the three coins is chosen at random and tossed, it showed heads. What is the probabilitythatitwastheunbiasedcoin? If one of these biased coins is used for the toss ina cricketmatchthen,whichvalueoflifewillbedemolished?

Q26. Findtheequationoftheplanepassingthroughtheintersectionoftheplanesx+3y+6=0and 3x–y–4z=0andwhoseperpendiculardistancefromoriginisunity.

(OR) FindthedistanceofP(3,4,5)fromtheplanex+y+z=2measuredparalleltotheline

2x=y=z. Q27. Evaluate:

π/4

π/4

log(sin x + cos x)dx .

Q28. A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contains at least 8 units of Vitamin A and 10 units of Vitamin C. Food ‘I’ contains 2units/kgofVitaminAand1unit/kgofVitaminC.Food‘II’contains1unit/kgofVitaminAand 2units/kg of Vitamin C. It costs`50 per kg to purchase Food ‘I’ and `70 per kg to purchase Food ‘II’. Formulate this problem as a linear programming problem to minimize the cost of suchamixtureandsolveitgraphically.Justifytheimportanceofbalanceddiet.Q29. Anisoscelestriangleofverticalangle2θisinscribedinacircleofradiusa.Showthatthearea oftriangleismaximumwhenθ=π/6. (OR) A telephone company in a town has 500 subscribers on its list and collects fixed charges of `300/- per subscriber per year. The company proposes to increase the annual subscriptionanditisbelievedthatforeveryincreaseof`1/-,onesubscriberwill discontinue theservice.Findwhatincreasewillbringthemaximumprofit?

# Prepared By OP Gupta (+91-9650 350 480 | +91-9718 240 480) [Electronics & Communications Engineering, INDIRA AWARD WINNER] Visit at : www.theOPGupta.WordPress.com, www.theOPGupta.com Email id : [email protected] 13/11/3 (Page 2)




HINTS & ANSWERS For PTS XII – 09 [2013-14]

Q01. 23 Q02. 7π

6 Q03.

6 6 0 3 3 0

3 0 1 0 1 2 Q04. Degree:2

Q05. –108 Q06. 2 Q07. 2units Q08. x+C Q09. 0 Q10. 1/x Q11. Continuousatx=–2

Q12. ˆ ˆˆ ˆ ˆ ˆr = i 2j 4k + λ (2i 3 j + 6k)

Q13. (OR) x=3 Q14.

x

y2 e log|y| 2

Q15. 23/4480 Q16. 2 2x 3y 2 (OR) Increasingonπ

4<x<

π

2,decreasingon0<x<

π

4

Q17. X 0 1 2 3 4 P(X) 15/210 80/210 90/210 24/210 1/210

Q18.

2 2x (logx) x xlog C

2 e 4

Q19. *isbinary.Identityelement:0.Inverseofanelement:

a

1 a

Q20.

2 32θ

2 3

dy θ 1 θ θ= e

dx θ 1 θ θ (OR) (3/2,0)

Q21. 2 2dy2xy x y 0

dx Q22. x = –7/3

Q23. Eq.of Tangent : x 3y 4+ = , Eq.of Normal : y 3x .Area=2 3 sq.units

Q24. Obtain

3

28 37 26

A 10 5 1

35 42 34

.

1

2 5 31

A 7 1 511

4 1 6

Q25. 10/33

Q26. 2x+y–2z+3=0,x–2y–2z–3=0 (OR) 6units[OPGVol.2,Q59Page34]

Q27.

π 1log

4 2

Q28. TominimizeZ=`(50x+70y).Subjecttoconstraints: x 2y 10,2x y 8; , 0x y .Minimum

valueofZ=`380at(2,4)Q29. (OR) Maximumprofitwillbeobtainedforanincreaseof`100inannualsusbscription. Hello students, I have to convey a few messages to you. I get many Calls/SMSes/Emails asking about if doing NCERT books is enough to score ‘good marks’. Let me clear this : Firstly definition of ‘good marks’ varies from one student to other. A student will be happy to get 60% marks while other may be unhappy despite of scoring 95%. Many questions in the question paper is from NCERT, be it examples or exercise questions. But NOT ALL the questions are from NCERT books. In fact, it will be better to say that the Question Paper is completely based on NCERT ‘Concepts’. So decide yourself. Dear students/parents/teachers, I keep on urging you all for any feedback(s), error(s) or suggestion(s). Please take this on a Serious Note. It encourages me for doing this service for Maths students. I’ll be waiting for a response from all of you. Send me your responses on [email protected], +91-9650 350 480 | +91-9718 240 480.

Good Luck & God Bless You!

Your MathsGuru OP Gupta.

# Soon I’ll be uploading PTS XII – 10 (with Marking Scheme) which is basically Sample Paper Issued by CBSE for session 2013-14.




1 13/12/1 P.T.O.



SECTION – A Q01. For binary operation on Z defined by a * b = a + b + 1, find the identity element.

Q02. If θ is the angle between two vectors a

and b

, then a.b 0

only when :

(a) 0 θ π/2 (b) 0 θ π/2 (c) 0 θ π (d) 0 θ π Choose any one of the given options in Q02.

Q03. If a

and b

are two vectors of magnitude 3 and 2/3 respectively such that a b

is a unit vector.

Write the angle between a

and b

.

Q04. Find the d.c.’s of :

y 3x 2 z 4

9 6 2. Q05. Evaluate : 1 o 1 ocos cos320 sin sin 320 .

Q06. Find the total number of matrices of order 2 2 , each of whose elements are either 0 or 1.

Q07. Without actually expanding, evaluate :

2 7 65

3 8 75

5 9 86

.

Q08. If ijA [a ] is a 3 3 matrix and ijA denotes the co-factors of the corresponding elements ija ’s

then, what is the value of 21 11 22 12 23 13a A a A a A ? Q09. Evaluate : π

39 529

π

[sin x x ]dx+

.

Q10. If f (x) = nlog xdx

x+ C, write the value of C such that f (1) = 1/2.

SECTION – B Q11. Examine which of the followings is binary operation :

(i) a * b = a b

2; a, bN. (ii) a ~ b =

a b

2; a, bQ.

For the binary operation, check the commutativity and associativity. Q12. Using properties of determinants, show that :

2 2 3

2 2 3

2 2 3

a a bc 1 a a

b b ca 1 b b (a b)(b c)(c a)(ab bc ca)

c c ab 1 c c

.

Q13. Evaluate 2

2 x

1

(x e )dx as a limit of sum. OR Evaluate : 1

cosecx cosxdx

+

.

Q14. If p 4ˆ ˆ ˆi 5j k

, q ˆ ˆ ˆi 4 j 5k

and r ˆ ˆ ˆ3i j k

then, find a vector u

which is

perpendicular to both p

and q

such that u.r 21

.

OR Using scalar triple product of vectors, check if the points A(4, 5, 1), B(0,–1, 1), C(3, 9, 4) and D(–4, 4, 4) are coplanar or not. Q15. ABCD is a parallelogram. The position vectors of the points A, B and C are respectively

ˆ ˆ ˆ4i 5j 10k , ˆ ˆ ˆ2i 3j 4k and ˆ ˆ ˆ2 j i k . Find the vector equation of the diagonal BD.

Q16. Prove that : cot–199 – cot–170 + 4 cot–15 = π/4 .


2 1 3 1 2

Roll No.





2 13/12/1 P.T.O.

OR Solve : 1 1cos 3x cos x π/2 .

Q17. Obtain the differential equation of the family of curves 2x xxy ae be x .

Q18. Find the particular solution of differential equation : 2[x sin (y/x) y]dx x dy 0, y(1) π/4 .

Q19. An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:

P(A fails) = 0.2, P(B fails alone) = 0.15, P(A and B fail) = 0.15 Evaluate the following probabilities :

(i) P A fails|B has failed (ii) P A fails alone .

Q20. If

2x ax b, if 0 x 2

(x) 3x 2, if 2 x 4

2ax 5b, if 4 x 8

f

is a continuous function on the interval [0, 8] then, determine

the values of ‘a’ and ‘b’. Q21. If y =

1log x

x then, prove that

dy x 1

dx 2x(x 1).

Q22. Show that the curves xy = c2 and y2 = 4ax cut each other at right angles iff c4 = 32a4. OR Two particles start from the same point at same time and move along the same straight line.

If the distances x and y covered by them in t seconds are given by : 3 2x t 3t 6t 5 and

3 22 1y t t 2t 8

3 2 . When will the particles have same speed?

SECTION – C Q23. Find the equations of the two lines through origin such that each line is intersecting the line

x 3 y 3 z

2 1 1

at an angle of

π

3.

OR Find the equations of the perpendicular drawn from the point (2, 4,–1) to the line

1 1

x 5 (y 3) (z 6)4 9

and hence obtain the coordinates of the foot of this perpendicular.

Q24. A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2units/kg of vitamin A and 1unit/kg of vitamin C while food II contains 1unit/kg of vitamin A and 2units/kg of vitamin C. It costs `5.00 per kg to purchase food I and `7.00 per kg to purchase food II. Determine the minimum cost of such a mixture by formulating the given situation as a LPP. Q25. Evaluate area of the region enclosed between y | x 1 | and y 3 | x | .

Q26. A factory has two machines A and B. Past records show that the machine A produced 60% of the items of output and machine B produced 40% of the items. Further 2% of the items produced by machine A were defective and 1% produced by machine B were defective. If an item is drawn at random, what is the probability that it is defective? OR A letter is known to have come either from TATANAGAR or from CALCUTTA. On the envelope, just two consecutive letters TA are visible. What is the probability that the letters

came from TATANAGAR? Q27. Evaluate : 1 x

dx1 x

.

Q28. Mr.Nakul Saini has invested a part of his income in 10% (bond A) and another part of his income in 15% (bond B). His interest during a certain period is `4000. Had he invested 20% more in bond A and 10% more in bond B, his interest would have been increased by `500 for the same period. Then : (i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication. (ii) Is it possible to solve the system of equations so obtained by matrices? If yes, solve it too. Q29. A poster show organized in a school. A student of class XII makes a poster in the form of cone and

writes “Peace & Integrity” on it. Show that the semi-vertical angle of cone of maximum volume and

of given slant height is 1 1cos

3

. Write the value of “Peace & Integrity” in world.

For Any Clarification(s) Or Queries, Please Contact On : +91-9650 350 480, +91-9718 240 480 .




3 13/12/1 P.T.O.

HINTS & ANSWERS [PTS XII – 11 for 2013-14 ]

Q01. Identity element : –1 Q02. Option (b) Q03. π

6

Q04. 9 6 2

, ,11 11 11

Q05. o80 Q06. 24 = 16

Q07. Use C3 C3–C1. Then observe that C2 and C3 are proportional. So 0 .

Q08. 0 Q09. 0 Q10. C = 1

2

Q11. (ii) is a binary operation as a~b Q for all a,b Q

Q13. 27e e

3 OR 11 3 sin x cosx

log tan (sin x cosx) C2 3 3 sin x cosx

Q14. ˆ ˆ ˆ7i 7 j 7k OR No, the points are not coplanar.

Q15. ˆ ˆ ˆ ˆ ˆ ˆr 2i 3j 4k λ(13j i 19k)

Q16. OR 1

x2

Q17. xy2 + 2y1 – xy + x2 – 2 = 0 Q18. cot (y/x) = loge(ex) Q19. (i) 0.50 (ii) 0.05 Q20. a = 3, b = –2 Q22. OR 1 or 4 sec

Q23. x y z x y z

;1 2 1 1 1 2

OR x 2 y 4 z 1

;( 4,1, 3)6 3 2

Q24. To minimize : Z = ` (5x 7y)+ .

Subject to constraints : x,y 0;2x y 8;x 2y 10 + + . Minimum value of Z = `38 at (2,4)

Q25. 4 Sq.units Q26. 16

1000 OR

7

11 Q27. 1[ x 2] 1 x sin x C

Q28. Let x and y be the initial investments by Mr. Saini in bond A and bond B respectively.

(i)

2 3 x 80000

8 11 y 300000 (ii) x = `10000, y = `20000.

Dear Student/Teacher, I would urge you for a little favour. Please notify me about any error(s) you notice in this (or other Maths) work. It would be beneficial for all the future learners of Maths like us. Any constructive criticism will be well acknowledged. Please find below my contact info when you decide to offer me your valuable suggestions. I’m looking forward for a response.

Also I would wish if you inform your friends/students about my efforts for Maths so that they may also benefit.

Let’s learn Maths with smile :-)

For any clarification(s), please contact : MathsGuru OP Gupta [Electronics & Communications Engineering, Indira Award Winner] Contact Nos. : +91-9650 350 480 | +91-9718 240 480 Mail me at : [email protected] Official Web-page : www.theOPGupta.com Official Blog-page : www.theOPGupta.blogspot.com

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1 13/12/3 P.T.O.



SECTION – A

Q01. Show that the points having position vectors a 2b 3c

, 3b 2a 2c

and 13b 8a

are collinear

whatever a

, b

, c

may be. Q02. If A, B and C are three non-zero square matrices of same order then, find the condition on A such that AB AC B C . Q03. Is it possible to have the product of two matrices to be the null matrix while neither of them is the null matrix? If it is so, give an example.

Q04. If f (1) = 4, (1) 2f , find the value of the derivative of log (e )xf w.r.t. x at the point x = 0.

Q05. For what value of ‘a’, the function f (x) = a(x + sin x) + a is an increasing function?

Q06. Evaluate π

π

(x [x])dx

, where [x] is greatest integer function.

Q07. Write the order & degree (if defined) of the differential equation,

2dy dy

y x a 1dx dx

.

Q08. If p.r 0

= and p r 0

= then, prove that either p 0

= or r 0

= .

Q09. A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Find the probability that the number is divisible by 5. Q10. The probability that an event happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. Find the probability that the event happens at least once.

SECTION – B

Q11. Find the value of 1 1 11 5 2 12 tan sec 2 tan

5 7 8 .

Q12. If 0 1

A0 0

, prove that (a I + b A) n = an I + n an–1 b A where I is unit matrix of order 2 & n Z .

OR If x, y and z are real numbers such that x + y + z = then find the value of ,

sin(x y z) sin(x z) cosz

sin y 0 tan x

cos(x y) tan(y z) 0

.

Q13. If x = a sin pt and y = b cos pt, find the value of 2

2

d y

dx at t = 0.

Q14. The radius of a balloon is increasing at the rate of 10cm/sec. At what rate is the surface area of the balloon increasing when its radius is 15cm. Balloon if thrown from a distance may hurt anyone. How would you teach your younger brothers/sisters to not throw it on others in Holi?

Q15. Show that the function defined by g x x x is discontinuous at all integral points.

Q16. Evaluate : π/2

0

x sin xdx

1 cosx

. Q17. Evaluate : π/2

π/2

(sin | x | cos | x |)dx

.


2 1 3 1 2

Roll No.





2 13/12/3 P.T.O.

Q18. Show that the points A, B and C with the position vectors ˆ ˆ ˆ2i j k , ˆ ˆ ˆi 3j 5k and ˆ ˆ ˆ3i 4 j 4k

respectively, are the vertices of a right triangle. Also find the remaining angles of triangle.

OR Let a, b, c

be the position vectors of the vertices A, B, C of ABC respectively. Find an

expression for the area of ABC and hence deduce the condition for the points A, B, C to be collinear.

Q19. Evaluate : 4 4

sin x cosxdx

sin x cos x

+

+. OR Evaluate :

2

4 2

x 4dx

x x 16

.

Q20. Solve : 2 2 dy2xy y 2x 0; y(1) 2

dx . OR Solve : 2x dy y(x y)dx 0 for y 1, x 1 = .

Q21. In a bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their outputs 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product.

(i) What is the probability that the bolt drawn is defective?

(ii) If the bolt is found to be defective, find the probability that it is a product of machine B.

Q22. Two dice are thrown simultaneously. Let X denote the number of sixes, find the probability distribution of X. Also find the mean and variance of X, using the probability distribution table.

SECTION – C Q23. Let X be a non-empty set and P(X) be its power set. Let ‘*’ be an operation defined on the elements of P(X) by A * B =AB A, BP(X). Then (i) Prove that * is a binary operation in P(X). (ii) Is * commutative? (iii) Is * associative? (iv) Find the identity element in P(X) with respect to *. (v) Find all the invertible elements of P(X). (vi) If o is another binary operation defined on P(X) as A o B = AB then, verify that o distributes itself over *. Q24. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, find the dimensions of the rectangle that will produce the largest area of the window. OR A given quantity of metal to be cast into half cylinder with a rectangular base and semicircular ends. Show that in order that total surface area is minimum, the ratio of length of cylinder to the diameter of its semi-circular ends is π : (π 2) .

Q25. Using integrals, find the area enclosed in the region 2 2 2{(x, y) : y 4x,4x 4y 9} .

Q26. Every gram of wheat provides 0.1gm of proteins and 0.25gm of carbohydrates. The corresponding values for rice are 0.05gm and 0.5gm respectively. Wheat costs `4 per kg and rice `6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50gms and 200gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost. Frame an L.P.P. and solve it graphically. Q27. Ten students were selected from a school on the basis of values for giving awards and were divided into three groups. The first group comprises hard working students, second group has honest students and the third group contains obedient students. Double the number of students of first group added to the number in the second group gives 13, while the combined strength of first and second group is four times that of the third group. Using matrix method, find the number of students in each group. Apart from the values, hard-work, honesty and obedience, suggest two more values, which in your opinion the school should consider for awards.

Q28. Find the equation of the plane containing the lines, ˆ ˆ ˆ ˆ ˆr i j λ(i 2 j k)

= + + + and ˆ ˆ ˆ ˆ ˆr i j μ( j i 2k)

= + + .

Find the distance of this plane from origin and also from the point (1, 1, 1). OR Find the equation of the plane passing through the intersection of the planes

ˆ ˆ ˆr.(2i 3j k) 1 0

+ + = and ˆ ˆ ˆr.(i j 2k) 3 0

+ + = and which is perpendicular to ˆ ˆ ˆr.(3i j 2k) 4

.

Also find the inclination of this plane with the XY- plane. Q29. Prove that the image of point P(3,–2, 1) in the plane 3x – y + 4z = 2 lies on the plane x + y + z + 4 = 0.

For Any Clarification(s) Or Queries, Please Contact On : +91-9650 350 480, +91-9718 240 480




3 13/12/3 P.T.O.

HINTS & ANSWERS [PTS XII – 13 for 2013-14 ]

Q02. A = I Q03. Yes, for example : 1 0 0 0

,0 0 0 1

Q04. 1/2

Q05. a > 0 Q06. π Q07. Order : 1, Degree : 2 Q09. 6/24

Q10. 98/125 Q11. π /4 Q13. 2

b

a Q14. 1200 π cm2/sec

Q16. π

2 Q17. 4 Q18. 1 16 35

cos ,cos41 41

OR Area of triangle : 1

| a b b c c a |2

+ +

Sq.units, For collinearity : a b b c c a 0+ + =

Q19. 11 2 1 sin x cos x 1 sin x cos xlog tan C

2 2 2 1 2 1 sin x cos x 2 2 1 2 1

OR 2

11 x 4tan C

3 3x

Q20. 2x = y(1 – log |x|) OR y2(3x + y) = x3(4 + 3 log |x|)

Q21. (i) 3.45% (ii) 28/69 Q22.

X 0 1 2 P(X) 25/36 10/36 1/36

Q23. (ii) yes (iii) yes (iv) (v) Q24. Length : 12

6 3m, breadth :

6(3 3)

6 3

m

Q25. 12 9 1cos

6 4 3

Sq.units Q26. Min.Z = `2.80 for 400g of wheat and 200g of rice

Q27. 1st group : 5, 2nd group : 3, 3rd group : 2

Q28. Equation of plane : ˆ ˆ ˆr.(i j k) 0=

; Distance from origin : 0 units, Distance from (1, 1, 1) : 3

Units3

OR 7x + 13y + 4z = 9, 1 4cos

3 26

Q29. Obtain the image which comes out to be (0, –1, –3). Then see if this satisfies the second eq. of plane.




For any clarification(s), please contact : MathsGuru OP Gupta [Electronics & Communications Engineering, Indira Award Winner] Contact Nos. : +91-9650 350 480 | +91-9718 240 480 Mail me at : [email protected] Official Web-page : www.theOPGupta.com | www.theOPGupta.WordPress.com Official Blog-page : www.theOPGupta.blogspot.com

Mean = 12/36, Variance = 10/36




1 13/12/4 P.T.O.



SECTION – A

Q01. In the matrix equation 11 16 2 3 1 2

7 10 1 2 3 4

, apply 2 2 1C C C on both sides.

Q02. Let f be a function defined as 1

( )2 sin3

f xx

. Write the range of ( )f x .

Q03. For a non-singular matrix A, find T( A )adj if 1

1/5 0 0

A 0 1 0

0 0 1

.

Q04. Write the principal value of 1sin (sin10) . Q05. Evaluate the integral : 3cos

sin

xdx

x .

Q06. Check if the vectors ˆ ˆˆ ˆ ˆ ˆ4 6 2 , 4 3

a i j k b i j k and ˆˆ ˆ8 3 c i j k are coplanar vectors.

Q07. Let : [1, ) [1, ) f be defined as ( 1)( ) 2 x xf x and is invertible. Find 1( )f x .

Q08. Check if the points A(4, 2, 3), B(1, 3, 1) and C(–5, 5,–2) are collinear points? Use vectors.

Q09. Evaluate the integral : x

0loge

1

e dx

.

Q10. A matrix X has (a + b) rows and (a + 2) columns while the matrix Y has (b + 1) rows and (a + 3) columns. Both the matrices XY and YX exist. Find the values of a and b.

SECTION – B

Q11. A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as : a*b =a b, if a b 6

a b 6, if a b 6

+ <

+.

Show that ‘0’ is the identity element for this operation and each non-zero element ‘a’ of the set is invertible with ‘6 – a’ being the inverse of ‘a’.

Q12. If f (x) =

3 2

2

x x 16x 20, x 2

(x 2)

k, x 2

is continuous at x = 2, find the value of k.

Q13. If x y x y3 3 3 then prove that y xdy3 0

dx . Q14. Evaluate :

2

4 23 16

xdx

x x

.

Q15. Show that the normal at any point θ to the curves cosθ θsinθ, sinθ θcosθx a a y a a is at a

constant distance from the origin.

Q16. Let 2 1

A3 4

and

5 2B

7 4

. Find a matrix D s. t. CD – AB = O where

2 5C

3 8

.

Q17. Prove that: 1 1α π β sinα.cosβ2 tan tan tan tan

2 4 2 cosα sinβ

.

OR Solve for x : tan–1(x – 1) + tan–1x + tan–1(x + 1) = tan–1(3x).


2 1 3 1 2

Roll No.





2 13/12/4 P.T.O.

Q18. Evaluate : 2

21 1

dx

x

. OR Evaluate : π/2

1

0

sin2x tan (sin x)dx

.

Q19. If a and

b are two vectors, then show that :

2 . .

. .

a a a b

a ba b b b

.

Q20. A) Which equation of curve would satisfy sin(10 6 )dy

x ydx

such that it passes through origin?

B) Write the order and degree of the differential equation :

32

5 22 32

2 33

3

4

1

d y

dxd y d yx

dx dxd y

dx

.

Q21. A die is thrown again and again until the number 6 is obtained three times. Find the probability that the 3rd six comes in seventh toss. Why should the outdoor games be preferred over indoor games? OR In a game, a man wins `10 for a number more than 4 and loses `3 for any other number, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he wins. Find the expected value of the amount he wins/loses. Do you think gambling is a good habit? Why?

Q22. Show that the equation of the perpendicular from the point (1, 6, 3) to the line 1 2

1 2 3

yx z is

61 3

0 3 2

yx z

and the foot of perpendicular is (1, 3, 5) and the length of the perpendicular is

13 units . OR Find the perpendicular distance of (1, 1, 0) from x 1 y 1 z 10

2 3 8

.

SECTION – C

Q23. If a , b , c are real numbers, and it is known that 0

b c c a a b

c a a b b c

a b b c c a

then, show

that either 0a b c or, a b c . Q24. A point on the hypotenuse of a right angled triangle is at the distances a and b from the sides of the

triangle. Show that the minimum length of hypotenuse is 2/3 2/3 3/2( )a b .

OR An open box with square base is to be made out of a given iron sheet of area 27 sq.metres, show that the maximum value of the volume of the box is 13.5 cubic metres.

Q25. Find the area of the region: 2(x, y) : x 2 y 20 x .

Q26. A bag contains 5 balls. Two balls are drawn at random from the bag and are found to be white. What is the probability that 4 balls in the bag are white and 1 is non-white?

Q27. Find the line of intersection of the planes ˆˆ ˆ.( 2 3 ) 0r i j k

and ˆˆ ˆ.(3 2 ) 0r i j k

. Show that this

line is equally inclined to i and k and makes an angle 11sec (3)

2 with j .

Q28. A manufacturing company makes two type of teaching aids A and B of Mathematics of class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 hours, respectively. The profit on type A and B is `80 and `120 per piece, respectively. How many pieces of each type should be manufactured per week by the company to maximize its profit? What is the maximum profit per week? Is teaching aid necessary for teaching-learning process? Justify your answer.

Q29. Evaluate : 3/2 3/2

3 3

sin cos

sin cos sin( )

x xdx

x x x a. OR Evaluate :

11 2

0

tan (1 )x x dx .

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3 13/12/4 P.T.O.

HINTS & ANSWERS [PTS XII – 14 for 2013-14]

Q01. 11 5 2 3 1 1

7 3 1 2 3 1

Q02. [1/3, 1]

Q03. 25 Q04. 3π 10 Q05. 21log | sin x | sin x C

2

Q06. Coplanar Q07. 21 1 4log x

2

Q08. Non-Collinear

Q09. e 1 Q10. a 2, b 3 Q12. 7

Q14. 2 2

1

2

1 x 4 1 x 11x 4tan log C

2 5 5x 4 11 x 11x 4

Q16. 191 110

77 44

Q17. OR 1

0,2

Q18. 3log2 OR π

12

Q20. A) 11 5tan 4x 5xy tan

3 4 3tan 4x 3=

B) Order : 3, Degree : 2

Q21. 7

3125

6 OR

Q22. 2 6Units Q25. (5π 2) Sq.units Q26. 3

10

Q27. ˆ ˆ ˆr 0 λ(i 2 j k)

Q28. Max. Profit : `1680 at (12, 6)

Q29. 2 cosa sina cot x2 cosa tan x sina

Ccosa sin a

OR log2





X 10 7 4 –9 P(X) 9/27 6/27 4/27 8/27

Expected amount = `76

27




1 13/12/5 P.T.O.


Visit at : www.theOPGupta.com | www.theOPGupta.WordPress.com Time Allowed : 180 Minutes Max. Marks : 100

SECTION – A

Q01. If

1

A 2

3

then, find TA A . Q02. Evaluate :

x 2x 3x

a b c

y 2y 3y

.

Q03. Find the area of a parallelogram having diagonals ˆ ˆ ˆ3i j 2k and ˆ ˆ ˆi 3j 4k .

Q04. Let f : {1, 2, 3}{a, b, c} given by (1) a, (2) b and, (3) cf f f . Show that 1 1( )f f .

Q05. Given that xdy/dx ye and x = 0, y = e. Find the value of y when x = 1.

Q06. Vectors a and b

are inclined at angle of oθ 120 . Also it is known that |a| =1 and |b| =2

then, what

is the value of 2[(a + 3b) (3a b)]

? Q07. Evaluate : π

cot cot4

arc

.

Q08. Find the distance of (1, 1, 2) from the plane ˆ ˆ ˆr.(2i j 3k) 4

.

Q09. Find a.b

, if |a| 5

, |b| 13

and |a b| 25

. Q10. Write the I.F. of dy

(x y) 1dx

.

SECTION – B Q11. a) If f (x) = sin x, g(x) = cos x & h(x) = 2x; f, g, h are real valued functions, check if ho(fog) and foh are equal or not. b) Let * be a binary operation defined on the set of integers as a * b = a + b – 1 for a, bZ. (i) Show that 1 is an identity element. (ii) Find the value of x such that 2 * (x * 3) = 10.

Q12. If

2x ax b, if 0 x 2

(x) 3x 2, if 2 x 4

2ax 5b, if 4 x 8

f

is a continuous function on [0,8] then, find value(s) of a and b.

Q13. If xy x , show that:

22

2

d y dy y10

y dx xdx

. Q14. Evaluate :

a xdx

a x

.

Q15. In a group of 200 people, 50% believe that anger and violence will ruin the country, 30% do not believe that anger and violence will ruin the country and 20% are not sure about anything. If 3 people are selected at random, find the probability that 2 people believe and 1 does not believe that anger and violence will ruin the country. How do you consider that anger and violence will ruin the country? OR A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes. Also determine the mean.


2 2

2 2 3 3 2

2 2

a 2ab b

b a 2ab (a b )

2ab b a

.

Q17. Simplify the expression : 1 1sin(cot costan x) .

OR Prove that 2sin y tan (x/2) , if it is given that 1 1y cot cos x tan cosx .

Q18. Find the least value of ‘a’ such that the function f (x) = x2 + 2ax + 3 is strictly increasing on (3, 4).


2 1 3 1 2

Roll No.





2 13/12/5 P.T.O.

Q19. Show that if the vectors a

, b

and c

are coplanar vectors then, a b+

, b c

and c a

are also coplanar vectors. Q20. Form the differential equation corresponding to the family of all the ellipses having major axis and minor axis of 2a and 2b along the coordinate axes respectively.

Q21. Evaluate 2

x 2

0

(e 5x )dx as limit of sums. OR Evaluate : 1/2

1/2

πxxcos dx

2 .

Q22. A bird at A(7, 14, 5) in space wants to reach a point P on the plane 2x + 4y – z = 2 when AP is least. Find the position of P and also the distance AP travelled by the bird. OR Two bikers are running at the speed more than allowed speed on the road along the lines

ˆˆ ˆ ˆ ˆr i j k (3i j)

and ˆ ˆˆ ˆr 4i k (2i 3k)

. Using Shortest Distance formula, check if they

meet to an accident or not? While driving, the driver should maintain the speed limit as allowed. Why?

SECTION – C Q23. Three shopkeepers A, B and C go to a store to buy electric equipments. A purchases 12 dozens 100watt bulbs, 5 dozen tube-lights and 6 dozens CFLs. B purchases 10 dozens 100watt bulbs, 6 dozen tube-lights and 7 dozens CFLs. Also, C purchases 11 dozens 100watt bulbs, 13 dozen tube- lights and 8 dozens CFLs. One 100watt bulb costs `40, one tube-light costs `65 and one CFL costs `120. Use matrix multiplication to calculate each individual’s bill. In your opinion, which equipment would you prefer to buy and why? Give two reasons. Q24. The lengths of the sides of an isosceles triangle are 9 + x2, 9 + x2 and 18 – 2x2 units. Calculate the area of the triangle in terms of x and find the value of x which makes the area maximum.

OR Find the maximum area of an isosceles triangle inscribed in the ellipse 22

2 2

yx1

a b with its

vertex at one end of major axis. Q25. Find the area bounded between the curves y2 = x and y = |x| using integrals. Q26. In a self-assessment survey, 60% persons claimed that they never indulged in corruption, 40% claimed that they always speak truth and 20% say that they neither indulged in corruption nor tell lies. A person is selected at random out of this group. (i) What is the probability that the person is either corrupt or tells lie? (ii) If the person never indulged in corruption, find the probability that she/he tell the truth. (iii) If the person always speaks truth, find the probability that she/he claims to have never indulged in the corruption. (iv) What values have been discussed in this question? (v) Why is it must for all to practice values in our life? Q27. Find the equation of a plane which is perpendicular to 2x 3y 3z 2= and 5x 4y z 6 and

passes through the point (–1,–1, 2).

Q28. Evaluate : 3x 3e cos x dx

. OR Evaluate :

1tan x

2 2

edx

(1 x )

.

Q29. An NGO is helping the poor people of earthquake hit village by providing medicines. In order to do this, they set up a plant to prepare two medicines A and B. There is sufficient raw material available to make 20000 bottles of medicine A and 40000 bottles of medicine B but there are 45000 bottles into which either of the medicines can be put. Further it takes 3 hours to prepare enough material to fill 1000 bottles of medicine A and takes 1 hour to prepare enough material to fill 1000 bottles of medicine B. There are 66 hours available for the operation. If the bottle of medicine A is used for 8 patients and bottle of medicine B is used for 7 patients. How the NGO should plan its production to cover maximum patients? How can you help others in case of natural disasters?

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3 13/12/5 P.T.O.


Q01. [14] Q02. 0 Q03. 5 3 Sq.units Q05. ee

Q06. 300 Q07. 3π

4 Q08.

314 Units

14 Q09. 60

Q10. ye Q11. (a) Not equal (b) ii. x 7

Q12. a 3, b 2 Q14. 1 2 2xasin a x C

a

Q15. 0.225 OR

Q17. 2

2

1 x

2 x

Q18. a 3 Q20.

22

2

d y dy dyxy x y 0

dx dx dx+

Q21. 2 37e

3 OR

2 [π 4 4 2]

π

Q22. P(1, 2, 8), AP = 3 21 Units OR S.D. = 0, this means they meet to an accident. Q23. Bill for A : `18300, Bill for B : `19560, Bill for C : `26940

Q24. Area : 6x (9 – x2) Sq. units, x 3 OR 3

3 ab Sq.units4

Q25. 1

Sq.units6

Q26. (i)8

10 (ii)

2

6 (iii)

1

2 Q27. 9x 17y 23z 20

Q28. 3xe

4sin3x 4 cos3x 9sin x 27cos x96

+ C OR

1tan x

2

e 3 2x(1 x)C

1 x 5

Q29. 10500 bottles of medicine A and 34500 bottles of medicine B and they can cover 325500 patients.




For any clarification(s), please contact : MathsGuru OP Gupta [Electronics & Communications Engineering, Indira Award Winner] Contact Nos. : +91-9650 350 480 | +91-9718 240 480 Mail me at : [email protected] Official Web-page : www.theOPGupta.com | www.theOPGupta.WordPress.com Official Blog-page : www.theOPGupta.blogspot.com Follow me on Twitter @theopgupta

X 0 1 2 3 4 P(X) 625/1296 4C1 (125/1296) 4C2 (25/1296) 4C3 (5/1296) 1/1296

Mean = 864

1296




1 13/1/1 P.T.O.



SECTION – A

Q01. Evaluate : 1 11 1sin cos

2 2

. Q02. Find the value of 2log x

dxx .

Q03. If cosα sinα

Asinα cosα

, then for what value of is A an identity matrix?

Q04. What is the cosine of the angle which the vector ˆ ˆ ˆ2 i j k makes with y-axis?

Q05. Find [a b c]

, if [a b b c c a] 72+ =

. Q06. Find the range of the function | x 1 |

(x)(x 1)

f

?

Q07. Find the minor of the element of second row and third column ( 23a ) in :

2 3 5

6 0 4

1 5 7

.

Q08. Write the vector equation of the line : x 5 y 4 6 z

3 7 2

.

Q09. What is the degree of the differential equation : 2 2

2

dy d y5x 6y log x

dx dx

?

Q10. If 1 2 3 1 7 11

3 4 2 5 k 23

, then write the value of k.

SECTION – B

Q11. If

1 px 1 px; 1 x 0

x(x)2x 1

;0 x 1x 2

f

is continuous function at x = 0 then, find value of p.

OR Find dy

dx, if 1 2y sin [x 1 x x 1 x ] .

Q12. There are two bags, Bag I and Bag II. Bag I contains 4 white and 3 red balls while the other bag contains 3 white and 7 red balls. One ball is drawn at random from one of the bags and it is found to be white. Find the probability that it was drawn from Bag I.

Q13. Show that the relation S in the set R of real numbers, defined as 3S {(a,b) : a,b R and a b } is

neither reflexive, nor symmetric nor transitive.

Q14. Find the equations of normal(s) to the curve 3y x 2x 6 which are parallel to x 14y 4 0 .

Q15. Using elementary row operations, find :

14 7

1 2

. Q16. Evaluate : 1

dx1 sin x .

Q17. Find the position vector of a point R which divides the line joining two points P and Q whose

position vectors are (2a b)

and (a 3b)

respectively, externally in the ratio 1 : 2. Also, show

that P is the midpoint of the line segment RQ.


2 1 3 1 2

Roll No.





2 13/1/1 P.T.O.

Q18. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3,–1, 2) and

parallel to the line x 4 y 3 z 1

1 4 7

.

Q19. If 1 1y cot tanθ tan tan θ then, show that

2

2

y1 tanπ y 2tan θ tan

y4 2 1 tan2

.

OR Prove that : 1 1 1tan (1) tan (2) tan (3) π .

Q20. Evaluate 3

x

1

(x e )dx as limit of sums. OR Evaluate : 1

2

1tan x dx

x

.

Q21. Find the particular solution of the following differential equation satisfying the given conditions : 2 2x dy (xy y )dx 0; y 1 when x 1 .

Q22. Find the particular solution of the following differential equation satisfying the given conditions : dy

y tan xdx

; given that y 1 when x 0 .

OR Find the general solution of the differential equation : dy 2

x log x y log xdx x

.

SECTION – C

Q23. Using integration, find area of the region bounded by curve 2x 4y and the line x 4y 2 .

OR Using integration, find the area of the region : 2 2x y x y

(x, y) : 19 4 3 2

.

Q24. A small firm manufactures fancy plastic bags and paper bags. The total number of plastic bags and paper bags manufactured per day is at most 24. It takes 1 hour to make a plastic bag and 30 minutes to make a paper bag. Maximum number of hours available per day is 16. If the profit on a plastic bag is `30 and that on a paper bag is `19, find the number of bags of both kinds that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically. Keeping the ‘save environment’ factor in mind, state which kind of bags should be promoted? Justify your answer. Q25. In a game of gambling, a card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to both clubs. Find the probability of the lost card being of clubs. Is gambling a good habit? Why or, why not? OR From a lot of 10 CFLs, which includes 3 defectives, a sample of 2 CFLs is drawn at random. Find the probability distribution of the number of defective CFLs. Also find mean and variance of the distribution. What is the benefit of using CFLs in present times?

Q26. Using properties of determinants, show that :

2

2 3

2

(b c) ab ca

ab (a c) bc 2abc(a b c)

ca bc (a b)

+

+

+

.

Q27. Find the value(s) of x for which 2(x) [x(x 2)]f is an increasing function. Also find the

points on the curve, where the tangent is parallel to x-axis. Q28. If the sum of the surface areas of a cube and a sphere is constant, what is the ratio of the edge of the cube to the diameter of the sphere, when the sum of their volumes is least? Q29. The points A(4,5,10), B(2,3,4) and C(1,2,–1) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.

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3 13/1/1 P.T.O.


Q01. π

2 Q02. (log x)2 + C Q03. 0o Q04. 1/2

Q05. 36 Q06. {–1, 1} Q07. 13

Q08. ˆ ˆ ˆ ˆ ˆ ˆr 5i 4 j 6k λ(3i 7 j 2k)

Q09. 1 Q10. 17

Q11. 1

p2

OR 2

1 1

2 x 1 x1 x

Q12.

40

61

Q14. x 14y 254,x 14y 86 0+ = + + = Q15. 2 7

1 4

Q16. x π

2 log tan C4 8

Q17. 3a 5b

Q18. 3x – 19y – 11z = 0

Q20. 1

2tan x 1log x log |1 x | C

x 2

Q21. 3x2y = y + 2x

Q22. y sec x OR 2

ylog x (1 log x) K+x

Q23. 9

Sq.units8

OR π

3 1 Sq.units2

Q24. No. of plastic bags : 8 & No. of paper bags : 16

Q25. 11

50 OR

Q27. Increasing in (0, 1) (2, ). Points : (0, 0), (1, 1), (2, 0) Q28. 1:1

Q29. ˆ ˆ ˆ ˆ ˆ ˆEquation of AB : r 4i 5j 10k λ(2i 2 j 6k)

, ˆ ˆ ˆ ˆ ˆ ˆEquation of BC : r 2i 3j 4k μ(i j 5k)

And coordinates of D are (3, 4, 5).





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X 0 1 2 P(X) 7/15 7/15 1/15

Mean = 9

15, Variance =

28

75




1 13/1/2 P.T.O.



SECTION – A

Q01. Write the value of 1 1 1 31 π πtan tan 2θ tan (cot θ) tan (cot θ), θ

2 4 2

.

Q02. Evaluate the integral : 2

n

sec x tan x sec xdx

(sec x + tan x)

.

Q03. Solve for x, if 5x y y 4 1

2y x 3 3 3

.

Q04. Write the direction cosines of the line joining the points (2, 3, 4) to its image in XY-plane.

Q05. If a

is a unit vector and (b a).(b a) 80=

then, find | b

|.

Q06. Let * be a binary operation on N given by a*b = HCF(a, b); a, bN. Write the value of 16*4.

Q07. Evaluate the determinant by using properties :

a b b c c a

b c c a a b

c a a b b c

.

Q08. Write the value of

1

2

20

1dx

1 x .

Q09. For what value of p, the line : x 2 y 1 z 3

9 p 6

is perpendicular to 3x – y + 2z = 7.

Q10. If A is a matrix of order ‘2 by 3’ and B is of ‘3 by 5’, what is the order of (AB)T.

SECTION – B Q11. Everyone wants to be a perfect ideal human being. Let us assume that dishonesty is one of the factors that affects our perfectness and perfectness has an inverse square relationship with dishonesty. For any value x of level of dishonesty, we have a unique value y of perfection. (i) Write down the equation that relates y with x. (ii) Does this relationship from x (0, ) to y (0, ) , form a function?

(iii) For what level of dishonesty one can achieve (1/4)th level of perfection? (iv) What will be the change in level of perfection when the level of dishonesty changes from 4 to 2? (v) What are the values depicted in this question? What are their importance in life?

Q12. The probability of a student A passing an examination is 3/5 and of student B passing is 4/5. Find the probability of passing the examination by (i) both the students A and B (ii) at least one of the students A and B. What ideal conditions a student should keep in mind while appearing in an examination? OR A can solve 80% of the problems and B can solve 60% of the problems given in the MathsMania, a question bank for CBSE Exams. They decide to help each other in solving the remaining sums. If a problem is selected at random, what is the probability that at least one of them will solve the problem? What value(s) has/ve been indicated in this question?


2 1 3 1 2

Roll No.





2 13/1/2 P.T.O.

Q13. If

ax bxe e, x 0

x

(x) 4, x 0

a log (1 x), x 0

x

f

is continuous function at x = 0 then, find value of ‘a’ and ‘b’.

Q14. Show that the curves 2x = y2 and 2xy = k cut at right angle if k2 = 8.

OR Find the intervals of increasing & decreasing for the function 22x log x .

Q15. Using properties, prove that :

a b c c b

c a b c a 2(a b)(b c)(c a)

b a a b c

+

.

Q16. The scalar product of ˆ ˆ ˆi j k with a unit vector along the sum of vectors ˆ ˆ ˆ2i 4 j 5k and

ˆ ˆ ˆλi 2 j 3k is equal to one. Find the value of λ .

Q17. Show that the lines x 5 y 7 z 3

4 4 5

and

x 8 y 4 z 5

7 1 3

intersect each other. Also find

their point of intersection.

Q18. Prove that 1 1 sinθ 1 sinθ θcot

21 sinθ 1 sinθ

;π

θ 0,4

.

OR Simplify : 2 2

1 1

2

1 x 1 xsin tan cos

2x 1 x

.

Q19. Evaluate : π/2 2

n

0

sec xdx

(secx + tanx) .

Q20. Evaluate : 1x sin x dx

.

Q21. If 1

2

sin xy

1 x

, show that 2

2

2

d y dy(1 x ) 3x y 0

dx dx .

OR Find dy

dx, if y x(cosx) (sin y) .

Q22. Solve the differential equation : x(dy/dx) y x tan(y/x) .

SECTION – C Q23. A poor deceased farmer had agriculture land bounded by the curve y = cos x, between x = 0 and the line x = 2π. He had two sons. Now they want to distribute this land in two parts as decided by their deceased father, such that both of them have equal share of land. Find the area of each part. Do you think that the decision taken by the farmer before his death was good? Justify your answer.

OR Using integration, find the area of the region : 2 2{(x, y) : x y 1 x y} .

Q24. A village has 500 hectares of land to grow two types of plants X and Y. The contribution of total amount of oxygen produced by plant X and Y are 60% and 40% per hectare respectively. To control weeds, a liquid herbicide has to be used for the plants X and Y at the rate of 20 litres and 10 litres per hectare, respectively. Further no more than 8000 litres of herbicides should be used in order to protect aquatic animals in a pond which collects drainage from this land. How much land should allocated to each crop so as to maximize the total production of oxygen? (i) How do you think excess use of herbicides affect our environment? (ii) What are the general implications of this question towards planting trees around us?

Q25. Evaluate 3

x 2

1

(3 x 4)dx , using first principle of integrals.

Q26. Coloured balls are distributed in three bags as shown in the following table :




3 13/1/2 P.T.O.

Bag

Black White Red Bag I 1 2 3 Bag II 2 4 1 Bag III 4 5 3

A bag is selected at random and then two balls are randomly drawn from the selected bag. They happened to be black and red. What is the probability that they came from the bag I?

Q27. Using elementary operations, find the inverse of :

3 0 1

2 3 0

0 4 1

.

Q28. If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between then is π/3 . Q29. Let the point P(5, 9, 3) lies on the top of Qutub Minar, Delhi. Find the image of the point on

the line : y 2x 1 z 3

2 3 4

.

Do you think that the conservation of monuments is important? Why?

OR Find the coordinates of the image of the point (2,–1,3) in the plane 3x 2y z 9 .

Colour of the ball




4 13/1/2 P.T.O.


Q01. 0 Q02. 1 n(secx tan x)

C1 n

Q03. 1 Q04. 0,0, 1

Q05. 9 Q06. 4 Q07. 0 Q08. π

4

Q09. –3 Q10. 5 by 2

Q11. (i) 2

1y ,x 0

x (ii) Yes (iii)

2

1 1 1For y , we have x 2

4 4 x (iv)

1 1 3y

4 16 16

Q12. (i) 12/25 (ii) 23/25 OR 92% Q13. a = 4, b = 0

Q14. OR Increasing in 1 1

,0 ,2 2

, decreasing in 1 1

, 0,2 2

Q16. 1 Q17. (1, 3, 2) Q18. OR 1

Q19. 2

n

1 n Q20.

12 2sin x x

(2x 1) 1 x C4 4

Q21. OR logsin y y tan x

log cosx x cot y

Q22.

yxsin C

x

Q23. First part of the land from x = 0 to x = π having area of 2 Sq. units and Second part from x = π to

x 2π= having area of 2 Sq. units OR π 1

Sq.units4 2

Q24. Maximum production of Oxygen will be achieved when plant X and Y are planted in 300 hectares and 200 hectares of land respectively.

Q25. 24 2

log 3 3 Q26.

231

551 Q27.

3 4 3

2 3 2

8 12 9

Q29. (1, 1, 11) OR 26 15 17

, ,7 7 7





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