AHLCON PUBLIC SCHOOL ASSIGNMENT -1 CLASS XII … · 2017-09-11 · AHLCON PUBLIC SCHOOL ASSIGNMENT...

AHLCON PUBLIC SCHOOL

ASSIGNMENT -1

CLASS XII MATHEMATICS (SESSION: 2017-18)

TOPIC: RELATIONS AND FUNCTIONS

Q.1 Which one of the following graph represents an identity function? Why?

(a) (b)

Q.2 Which one of the following graph represents a constant function? Why?

a) b)

Q.3 Find the domain of f(x) = ][

1

xx

Q.4 Define a binary operation * on the set A={0,1,2,3,4,5} as a*b={ + 𝑖𝑓 + < 6+ − 6 𝑖𝑓 + ≥ 6 show that 0 is the

identity for this operation and each element a of the set is invertible with 6-a being the inverse of a.

Q 5 Let Q be the set of rational numbers and R be the relation on R ={ , : + > }. Prove that R is

reflexive and symmetric but not transitive.

Q.6 Show that the function RRf : defined by f(x) = x2 is not bijective.

Q.7 Given A = {-1, 0, 2, 5, 6}, B = { -2, -1, 0, 18, 28 } and f (x) = x2 – x – 2. Show that function f is not onto.

Q.8 Show that the function NNf : given by f(1) = f(2) = 1 and f(x) = x – 1 for every x > 2, is onto but

not one-one.

Q 9 Show that the function f:R R defined by f(x) = 2x3-7 for all x R is bijective.

Q 10 Show that the function f:R R defined by f(x) = x3 + 3x for all x R is bijective.

Q.11 Let f:

5

3 R be a function defined as

35

2)(

x

xxf , find f

-1

Q.12 If 54

35)(

x

xxf , show that )(xff is an identity function.

Q.13 Let Rf ,2: and Rg ,2: be two real functions defined by f(x) = 2x , 2)( xxg .

Find f+g and f-g.

Q.14 Let RRf

3

4: be a function defined by

43

4)(

x

xxf , Find f

-1

Q.15 Show that the function RRf : defined by f(x) = 3

12 x, x R is one – one and onto function. Also

find the inverse of the function f.

Q.16 Consider RRf : , given by f(x) = 7x +30. Show that f is invertible. Find the inverse of f.

.Q.17 If f(x) = x2 and g (x)=x+1 , show that fog gof.

Q.18 Let RRf : be defined by f(x) = 10x + 7. Find the function g: R R such that gof = fog = IR

Q.19 If

x

xxf

1

1log)( Show that f

21

2

x

x= 2f (x).

Q.20 If f (x) =

x1

1, find f

2

1f

Q.21 If f(x) =

x

1, g(x) = x and h (x) = 12 x , find

i) fogoh (x)

ii) hofog (x)

Q 22 If f:R R is given by f(x) = 3x +2 f or all x R and g : R R is given by g(x)= for all

x R find a) fog b)gof c)fof d)gog

Q 23 If f: R R is given by f(x)=(5 – x5)1/5

then find fof.

Q24 If f(x) = and g(x)= for all x R ,find

a) (gof)(-5/3) b) (fog)(-5/3)

Q 25 If f(x) = x+7 and g(x) = x-7 , find

a) (fog)(7) b) (gog)(7) c) (gof)(7) d) (fof)(7)

Q 26 If f: R-{2} R is a function defined by f(x) = then find f-1 :

range of f R-{2}.

Q 27 If f:R-{-3/5} R is a function defined by f(x) = , then find f-1 :

range of f R-{-3/5}.

Q.28 Examine which of the following is a binary operation:

i) Nbaba

ba

,;2

ii) Qbaba

ba

,;2

For binary operation check the cumulative and associative property.

Q.29 Let X be a non – empty set. P(x) be its power set. Let ‘*’ be an operation defined on elements of P(x) by,

BABA A, B P(x). Then,

i) Prove that * is a binary operation in P(x)

ii) Is * cumulative?

iii) Is * associative?

iv) Find the identity element in P(x).

Q.30 Let A = Q x Q. Let * be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b) then find i) identity

element

ii) invertible elements of (A, *).

Q 31 Let A be the set of all real numbers except -1 . Let * defined on A as a*b = a+ b + ab for all a, b R .

Prove that

a) * is binary.

b) The given operation is commutative and associative.

c) The number 0 is the identity.

d) Every element a of A has –a/(1+a) as inverse.

e) Solve the equation 2*x*5 = 4

Q32. Let A = {1,2,3,4….,9} and R be the relation in AxA defined by (a,B) R (c,d) if a+d = b+c for (a,b), (c,d) in AXA. Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)].

ANSWERS SHEET

1. a

2. a

3. R+

except whole number

11. f-1 = −

13. (i) 2x + 2x (ii) 2x - 2x

14. f-1 = −

15. f-1 =

+

18. g(y) = −7

20. −

21. (i √ 2+ 𝑖𝑖 +

22. a) 2+ +2+ b)

+9 2+ + c) 9x + 8 d) 2+2+ 2+ 2

23. x

24. a.) 1 b) 2

25. a) 7 b) 7 c) 7 d) 21

26. f-1 =

+−

27. f-1 = −

28. (i) not binary (ii) binary ; commutative but not associative

29. (ii) Yes (iii) Yes (iv) X

30. (i) (1,0 ) ii) ( , −

31. x = -13/18


ASSIGNMENT – 2

CLASS – XII MATHEMATICS (SESSION: 2017-18)

TOPIC :INVERSE TRIGONOMETRIC FUNCTIONS

Q.1 Write the principal value of

1)

2

3sin 1 5)

3

1tan 1 9) 2sec 1

2)

3

1tan 1

6) 2cos 1 ec 10) 2cos 1

ec

3)

2

3sin 1 7)

2

3cos 1 11)

3

1cot 1

4)

2

3cos 1 8)

3

1cot 1

12) 2sec 1

Q.2 Simplify each of following using principal value.

1)

3

2sec

3

1tan 11 7) 1sin1cot1tan 111

2)

2

3cos

2

1sin 11 8)

2

1sin3cot 11

3) 1cot1tan 11 9)

5

4sinsin 1

4)

2

1sin

2

1cos 11 10)

5

7coscos 1

5)

3

1cot3tan 11 11)

6

5tantan 1

6) 2sec2cos 11 ec 12)

4

3coscos 1

ecec

Q.3 Simplify

1) cos(tan− ) 5) tan( cos− √

2) sin(cos− ) 6) tan(2tan− - 𝜋 )

3) cos(cos− −√ + 𝜋 7) sin(𝜋 - sin− −√

4) sin( 𝜋 − sin− − 8) sin tan− −√ + cos− −√

)) Q.4 Prove

4cos1

cos1cot

sin1

costan 11

x

x

x

x

Q.5 Prove

a

xa

a

x

xa

x22

11

22

1 cossintan

Q.6 Prove

161

300tan

17

8sintan2tan

17

8costan2cot 11111

Q.7 Prove

01

tan1

tan1

tan 111

ac

ac

bc

cb

ab

ba

Q.8 Prove

153cotcos2tansec 1212 ec

Q.9 Prove

2

1tancoscotsin

2

211

x

xx

Q.10 Prove

48

1tan

3

1tan

7

1tan

5

1tan 1111

Q.11 Solve the equation for x

1) sin− 𝑥 + sin− − 𝑥 = cos− 𝑥

2)sin− 𝑥 ) + sin− 𝑥 ) = 𝜋

Q.12 Prove

(1) 5

3cos

2

1

9

2tan

4

1tan 111

(2) 43

1tan2

7

1tan 11

(3) 22

1cot

2

1tan

21

21

x

x

x

x

(4) 11

1cos

2

1tansin

2

21

21

x

x

x

x

Q.13 Write in simplest form

(1) 21 11sin xxxx (9) tan− √ −𝑠𝑖 𝑥+𝑠𝑖 𝑥

(2)

xa

xa1tan (10) tan− si 𝑥+c s 𝑥

(3) 21 1tan xx (11) tan− √ −√𝑥+√𝑥

(4)

2

11sin 1 xx

(12) tan− √ −𝑥2+𝑥2

(5

x

x

1

1tan2sin 1

(13tan− si 𝑥−c s 𝑥𝑠𝑖 𝑥+ 𝑠𝑥

6)

2

1

61

5tan

x

x

(14) tan− + 𝑥− 𝑥

(7) 21 1cot xx (15) tan− 𝑥+ 𝑥2

(8) tan− √ − 𝑠𝑥+ 𝑠𝑥 (16)sin2 cot− √ +𝑥−𝑥

ANSWERS SHEET

1. i) ) -π

ii) 𝜋 iii) 𝜋 iv)

𝜋 v)

−𝜋 vi)

−𝜋

vii) 𝜋 viii)

𝜋 ix)

𝜋 x)

𝜋 xi)

𝜋 xii)

𝜋

2. i) 0 ii) -π

iii)−𝜋

iv) 𝜋 v)

𝜋 vi)

𝜋

vii) 𝜋 viii) 𝜋 ix)

𝜋 x)

𝜋 xi)

−𝜋 xii)

𝜋

3. i) ii) iii) -1 iv) 1 v) −√

vi) −

vii) viii) 1

11. i) x = 0, ii) x = 13

13. i) sin-1

x - sin-1√𝑥 ii) cos− 𝑥

iii) 𝜋 + tan− 𝑥

iv) 𝜋 + cos− 𝑥 v) √ − 𝑥 vi) tan− 𝑥 + tan− 𝑥

vii) tan− 𝑥 + tan− − 𝑥 viii) 𝑥 ix)

𝜋 -

𝑥 x)

𝑥

xi) cos− √𝑥 xii) ) cos− 𝑥 xiii) x - 𝜋

xiv) tan− + tan− 𝑥 xv) tan− 𝑥 − tan− 𝑥 xvi) −𝑥


ASSIGNMENT NO. : 3


TOPIC : MATRICES

Q.1 Construct a 2 3 matrix A = [a ij], whose elements are given by aij = ji

ji

.

Q.2 Find x, y, z and w such as

yx

yx

2

wx

wz

2

2 =

12

5

15

3

Q.3 If

5

ba

ab

2 =

5

6

8

2 find the values of a and b.

Q.4 Find a matrix A such that 2A – 3B + 5C = 0, where B =

3

2

1

2

4

0 and

C =

7

2

1

0

6

2.

Q.5 Solve the matrix equation

2

2

y

x - 3

y

x

2 =

9

2

Q.6 (i)If A =

1

4

1

2, prove that (A – 2I ) (A – 3I ) = 0.

(ii)If A=[ ] then show that A2

-3A-7=0.

(iii)If A=[ ] then show that A2

-4A+I=0.

(iv)If A=[ ] then show that A2

-12A-I=0

Q.7 If A and B are two matrices such that AB=B and BA=A then find A2+B

2

Q.8 Let A =

7

2

5

3, B =

2

1

4

0, verify that

1) (2A)T = 2A

T 2) (A+B)

T = A

T + B

T

3) (A – B)T = A

T – B

T 4) (AB)

T = B

TA

T

Q.9 Let A be a square matrix. Then show

i) A + AT is a symmetric matrix.

ii) A - A T is a skew – symmetric matrix.

iii) A A T and A

T A are symmetric matrix.

Q.10 Let A and B be symmetric matrices of the same order. Then, show that

i) A+B is symmetric

ii) AB – BA is a skew – symmetric matrix.

iii) AB + BA is a symmetric matrix.

Q.11 Express A =

1

3

1

4 as the sum of a symmetric and a skew – symmetric matrix.

Q.12 Find the inverse of each of the following by using elementary row transformations:.

(1)

2

5

1

2 (2)

3

1

5

6

(3)

3

1

2

1

2

1

1

4

3

(4)

2

3

1

3

1

1

1

1

2

Q.13 Construct a 2 2 matrix whose elements are aij = i when i is odd

=j when i is even.

Q.14 If A =

32

4

x

1

2

x

x is symmetric matrix, find x.

Q.15 Given that A =[ 𝑎], show by induction An

= [ 𝑛𝑎] Q.16 Find the value of k if A

2=8A+KI , where A =[− ]

Q.17 If A =

122

212

221

prove that 0542 IAA

Q.18 Two shopkeepers A and B of a particular school have stocks of Physics, Chemistry and Mathematics

books as given by the matrix

Physics Chemistry Maths

Shop A 3 dozen 4 dozen 2 dozen

Shop B 2 dozen 1 dozen 5 dozen

If the selling price of these books all respectively Rs.300, Rs.250, and Rs.200 per book, find the total

amount (using matrices )received by each shopkeeper if all the books are sold.

Q.19 Find matrices X and Y if

1

0

2

6

4

62 YX and

7

5

1

2

2

32YX

Q.20 If

001

011

111

YX and

0811

411

153

YX find X and Y.

ANSWERS SHEET

CHAPTER : MATRICES

1. [ − / − // − / ]

2. x = 1, y = 2, w = 1, z = 1

3. a = 4 , b = 2 or a = 2 , b = 4

4. [− − − ]

5. x= 2, 1 y = 3 ± 3√

11. A= [ − /− / − ] + [ − /− / − ]

12. i) A-1

= [ −− ] ii) A-1 = [ − ] iii) A-1

= − [− −− −− − ] iv) A

-1 = [− −− −− − ]

13. A = [ ]

14. x = 5

16. k = -7

18. [ ]

19. X = [ −− − / ] , Y = [ − / ]

20. X = [ ] , Y = [− ]


ASSIGNMENT – 4


TOPIC: DETERMINANTS

Q.1 Write the minors and co. factors of each element of the first column of the following

matrices.

(1)

10

5

1

20 (2)

3

1

0

7

5

2

1

0

6

Q.2 For what value of x the matrix A =

1

1

1x

1

1

1

x

1

1

1

x

is singular?

Q.3 Prove that

yx

yx

yx

810

45

x

x

x

8

4

x

x

x

3

2 = 3x

Q.4 If a, b, c are all positive and are pth

, qth

and rth

terms of a G.P, then show that

c

b

a

log

log

log

r

q

p

1

1

1

= 0

Q.5 If x, y, z are the 10th

, 13th

, and 15th

terms of a G.P, find the value of

z

y

x

log

log

log

15

13

10

1

1

1

Q.6 Show that

3

2

1

x

x

x

4

3

2

x

x

x

cx

bx

ax

= 0 where a, b, c are in A.P.

Q.7 If A is skew symmetric matrix of order 3 3 what is the value of the determinant of A.

Q.8 If A = 2B, where A and B are square matrices of order 3 3 and B = 5 what is A .

Q.9 If A is a square matrix of order 3 3 . what is KA where K is a scalar.

Q.10 If A =

1

5

3

2, B =

2

1

1

3 verify that AB = A B

Q.11 Find the value of so that the points (1, -5), (-4, 5) and ( , 7) are collinear.

Q.12 If the point (a,b), // ,ba and // , bbaa are collinear. Show that baab// .

Q.13 If A is (2 2) matrix and if

A (adjA) =

0

12

12

0 find A

Q.14 If A =

2

4

1

2 find A5

Q.15 If A =

1

3

2

4 find 2 A

Q.16 If A =

3

0

0

0 what is A

20

Q.17 Verify that (AB) -1

= 11 AB where

A =

7

3

5

2, B =

3

4

2

6

Q.18 If A =

7

3

5

2 verify (A

T)

-1 =

(A

-1) T

Q.19 If A =

7

3

5

2, B =

3

4

2

6 verify that adj (AB) = (adjB) (adj A)

Q.20 Find the adjoint of A

2

0

1

4

5

2

3

0

3

and verify A(adj A) = A I3 = (adj A) A.

Q.21 Find A(adj A) without finding (adj A) if A =

1

3

1

0

1

2

3

2

3

.

Q.22 Let A be a non singular matrix of order 3 3 such that A = 5. What is adjA ?

Q.23 Show that A =

1

5

2

3 satisfies 732 xx =0 Thus, find 1

A .

Q.24 Show that A =

3

2

1

4

1

0

1

2

2

satisfies the equation 03 3

23 IAAA . Hence,

find 1A .

Q.25 Find the matrix X satisfying.

7

3

5

2 X

2

1

1

1 =

0

2

4

1

Q.26 Find the matrix X satisfying.

X

1

5

2

3 =

7

14

7

7

Q.27 Solve the following systems of equations,

72 zyx

113 zx

132 yx

Q.28 If A =

1

2

1

1

1

1

1

3

1

, find 1A and hence solve

42 zyx

0 zyx

23 zyx

Q.29 Determine the product

5

7

4

3

1

4

1

3

4

2

1

1

1

2

1

3

2

1

and use it to solve

4 zyx

922 zyx

132 zyx

Q30. Two schools P and Q want to award their selected students on the values of Discipline,

Politeness and Punctuality. The school P wants to award rupees x each, rupees y each and

rupees z each for the three respective values to its 3, 2 and 1 students with a total award

money of rupees 1,000. School Q wants to spend rupees 1500 to award its 4, 1 and 3

students on the respective values( by giving the same award for the three values as before).

If the total amount of awards for one prize on each value is rupees 600, using matrices, find

the award money for each value.

ANSWERS SHEET TOPIC: DETERMINANTS

1. i) M11 = 1 , M21 = 20 , A11 = 1 , A21 = -20

ii) M11 = 5 , M21 = -40 , M31 = -30 ii) A11 = 5 , A21 = 40 , A31 = -30

2. x = 1, 2

7. |A| = 0

8. 40

9. K3|𝐴|

11. 𝜆 = -5

13. |𝐴| = 12

14. 200

15. 20

16. [0 00 0]

20. Adj A = [ −−− ]

21. A(Adj A) = -14 [ ]

22. 25

23. 17 [− − ]

25. X = [− − ]

26. . X = [ − ]

27. x= 2, y= 1, z= 3

28. x = 9/5, y = 2/5 , z = 7/5

29. x= 3, y = -2 , z = -1


ASSIGNMENT NO :5


TOPIC : CONTINUITY AND DIFFERENTIABILITY

Q1 A function f(x) is defined as f(x) = { − −6− ≠ =

Show that f(x) is continuous at x=3.

Q2. If the function f(x) = { + > = − <

Is continuous at x = 1 find values of ‘a’ and ‘b’.

Q3 A function f(x) is defined by f(x) = { 𝑠𝑖 ≠ = Find whether the function f(x) is continuous at x=0

Q4. Discuss the continuity of the function f(x) at x = 0 if f(x) =

0,12

0,12

xx

xx

Q5. Let f(x) = { − 𝑠 < = √− +√ 6+√ >

Determine the value of so that the function is continuous at x = 0.

Q6. Verify Rolle’s theorem for the following function

1) f(x) = x − [0,3]

2) f(x) = cos2x [0,𝜋]

3) f(x) = sinx+cosx [0,2

]

Q7 . It is given that for the function f(x) = x3 + bx

2 + ax + 5 on [1, 3], Rolle’s theorem holds with

‘c’ = 2 + 3

1. Find the value of ‘a’ and ‘b’.

Q8 If f(x) is continuous at x=𝜋 find a and b where f(x) = {

−𝑠𝑖𝑠 < 𝜋 = 𝜋−𝑠𝑖𝜋− > 𝜋

Q9 Verify Langrange’s mean value theorem for the following functions.

i) f(x) = x (x – 1) (x – 2) in [0, 2

1]

ii) f(x) = 2sinx + sin2x in [0, ]

iii) f(x) = √ − in [2,4]

Q10 Find dx

dy when

i) y = 3x tan x

ii) y = 3

1

3

52 532

xx

iii) y = 2xaaa ; a being constant

iv) x = a cos3t, y = b sin

3 t.

v) (cosx)y = (cosy)

x

vi) y = (sinx)tanx

+ (cosx)secx

Q11. Find dx

dy at t =

4

when x = )cos(sin tte

t and y = )cos(sin ttet .

Q12. Prove that the derivative of

2

1

11tan

x

x w.r.t sin

-1x is independent of x.

Q13 If log

y

xyx

122 tan2 . Show that dx

dy =

xy

xy

.

Q14. If xyxy = a then show that dx

dy =

2

2

a

x

Q15. If √ − +√ − =a(x-y) prove that =√ −−

Q16. If x=2cosƟ − 𝑜𝑠 Ɵ and y =2 sinƟ -sin2𝜃 find 2

2

dx

yd at Ɵ=

2

.

ANSWERS SHEET

2. a = 3, b = 2

3. not continuous

4. continuous everywhere except x=0

5. . a=8

6. i) c = 1 ii) c = 𝜋

iii) c=𝜋/

7. a=11,b= -6

8. a=1/2,b=4

9. i) c=1-√6 ii) c=𝜋/ iii) c=√

10. i) 3 sec2x + 3

xlog3 tan x ii) 3

1 (2x

2 + 3)

2/3 (x + 5)

-4/3(18x

2+100x - 3)

iii) √ +√ +√ + √ +√ + +√ + iv) - tan t

v) l g c + al g c + a

vi) (Sin )tanx

(1 + log sin sec2

+ (COSX)secx

)(-tan sec + log cos sec tan )

11. - 1

16. -3/2

AHLCON PUBLIC SCHOOL, MAYUR VIHAR PH – 1 DELHI 110091

ASSIGNMENT – 6

CLASS – XII MATHEMATICS (SESSION: 2017-18) TOPIC : APPLICATIONS OF DERIVATIVES

Q.1 The volume of a cube is increasing at a constant rate. Prove that the increase in surface area

varies inversely as the length of the edge of the cube.

Q.2 A man is walking at the rate of 6.5 Km/hr towards the foot of a tower 120m high. At what rate is

he approaching the top of the tower when he is 50m away from the tower?

Q.3 Water is dripping out from a conical funnel of semi vertical angle 4

at the uniform rate of

2cm2/sec. in its surface area through a tiny hole at the vertex in the bottom. When the slant

height of the water is 4cm, find the rate of decrease of the slant height of the water.

Q.4 The time T of a complete oscillation of a simple pendulum of length l is given by the equation T =

g

l2 where g is a constant. What is the percentage error in T when l is increased by 1.1?

Q.5 If 104 xy and if x changes from 2 to 1.99. What is the approximate change in y?

Q.6 Prove that the tangents to the curve 652 xxy at the points (2, 0) and (3, 0) are at right

angles.

Q.7 Find the points on the curve 194 22 yx , where the tangents are perpendicular to the line

02 xy .

Q.8 Find the point on the curve 462 2 xxy at which the tangent is parallel to the x – axis.

Q.9 Find the points on the curve 04 xy at which the tangents are inclined at an angle of 45o

with the x – axis.

Q.10 Find the equation of the tangent line to the curve sin,cos1 yx at 4

Q.11 Find the equation of the tangent and the normal to the curve 32

7

xx

xy at the point,

where it cuts x – axis.

Q.12 Find the intervals in which the following functions are increasing or decreasing.

a) 3

)(3

4 xxxf

b) x

xxf

14)(

2

c) 1

2)(

x

xxf

d) x

xxf

2

2)(

e) x

xxf

log)(

f) 14)2log(2( 2 xxxxf

Q.13 Find the points of local maxima and local minima, if any, of each of the following functions. Find

also the local maximum and local minimum values, as the case may be.

1. ,2sin)( xxxf 22

x

2. ,2cos2

1sin)( xxxf

20

x

3. ,cossin)( 44xxxf

20

x

Q.14 Show that all the rectangles with a given perimeter, the square has the largest area.

Q.15 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 them is 3

Q.16 Show that the triangle of maximum area that can be inscribed in given circle is an equilateral

triangle.

Q.17 Show that the height of a cylinder, which is open at the top, having a given surface area and

greatest volume, is equal to the radius of the box.

Q.18 Prove that the radius of the right circular cylinder of greatest curved surface which can be

inscribed in a given cone is half of that of the cone.

Q.19 An open tank with a square base and vertical sides is to be constructed from a metal sheet so as

to hold a given quantity of water. Show that the cost of the material will be least when depth of

the tank is half of its width.

ANSWERS SHEET 2. 2.5 km/hr

3. √𝜋 cm/sec

4. 1.2

1

5. 5.68

7.

103

1,

102

3 and

103

1,

102

3

8. , − 7

9. (2, -2) & (-2, 2)

10. ( √ − ) − = ( √ – ) − 𝜋

11. x – 20y – 7 = 0 , 20x +y - 140 = 0

12. a) (- , 0) (0, 4

1)

b) ,2121,

2

1,00,

2

1

c) Increasing on R – (-1)

d) 22&,22,

e) ),( e and 1),0( e f) (2, 3] and [3, α )

13. i) Point of local minimum value is 6

x and value is

62

3 point of local

maximum is 6

x and Maximum value

62

3

ii) 4

3&6

x

2

x &

2

1

iii) ( 𝑐𝑎 𝑖 𝑖 4x 𝑎 𝑒 = 21 )


ASSIGNMENT – 7 (a)


TOPIC : INDEFINITE INTEGRALS.

Evaluate

1. xdx4sin 2. xdx

5sin

3. xdxx52 cossin

4. dx

x

x

2cos

4sin

5. dxxsin1 6.

dxxex 14log3 1

7.

dxx

ax

sin

)sin(

8.

dxax

x

)sin(

sin

9. dx

bxax )sin()sin(

1

10. )cos()sin( bxax

dx

11. dxbx

ax

)sin(

)sin(

12. dxx

tan1

1

13. dxx

cot1

1

14. dx

x sin1

1

15. dxx

cos1

1

16. dx

x

x

32

3

)1(

17. dxxxx 1)24( 2

18. xdxx sinlogcot

19. dxx

xx

555 555

20. dx

x

xx

6

312

1

tan

21. dxxba

x

2)cos(

2sin

22. dxax 1

12

23. dxx

x

4

12

2

24. dx

x 241

1

25. dx

x

1)2(

1

2

26.

dxx

x

1

12

4

27. dxxxx

2log7)(log6

12

28.

dxe

e

x

x

21

29. dxxx

)1(

13

30.

dx

xx

4

1

3/23/2

31.

dx

xx

x

2sincos

2sin24

32.

33.

dx

sin4cos6

cos2sin22

34.

dxxx

xx

23

352

2

35. dx

x

x

1

36. dx

xx 22 cos8sin31

1

37. 2)cos3sin2( xx

dx

38. dtan

39.

dxxx

44 cossin

1

40.

dxxx cossin1

1

41. dx

xcos45

1

42.

dxxx cos3sin

1

43. xdxx 3sin 44. xdxx

2sin

45. dxexx3 46.

dxx 1sin

47. dx

x

xx

4

213

1

sin 48.

dx

x

xe

x

22

2

)1(

)1(

49. dxx

xe

x

cos1

sin1 50.

dx

x

xe

x

2)2(

1

51. dxx

xe

x

2cos1

2sin2 52.

dxxxx 1)23( 2

53. dx

xx

x

)2)(1(

3

54.

dxxx

x

)2)(1(

222

55. ƒ dxxxx

xxx

)6)(5)(4(

)3)(2)(1(

56 dx

xx 1)3(

1

57. dx

x

x

44 58.

dxxx2sec

59. dx

x 4

14

60.

dxx

x

44

4

61. dxx

xx

2sin1

cossin 62.

dxxex

x

x

)1(

1



ASSIGNMENT – 7 (b)

TOPIC : DEFINITE INTEGRALS

Q.1 Evaluate

i) 4

1

)( dxxf where

42,53

21,34)(

xx

xxxf [37]

ii) dxx

4

4

2 [20]

iii) 2

0

2 23 dxxx [ 1 ]

iv) dxxx

x

2

1 3 [½]

v) dxx

3

6 cot1

1

[ 12 ]

vi) 2

0

tanlog

xdx [0]

vii) 2

0

tanlog2sin

xdxx [0]

viii)

2

2

2sin

xdx [ 2 ]

ix)

4

4

43 sin

xdxx [0]

x) dxx

x

4

42cos2

4

36

2

xi) dxx2

0

2cos

[1]

xii)

0sin1

sindx

x

xx [

1

2

]

xiii)

2

0

44 cossin

cossin

xx

xxxdx [ 162 ]

xiv) dxxx

2

2

cossin

[4]

xv) ∫ 𝑥 𝑎𝑛𝑥𝑒𝑐𝑥+ 𝑎𝑛𝑥𝜋0 𝑥 [𝜋 𝜋2 − 1 ]

xvi) ∫ |𝑥 𝑜𝑠𝜋𝑥| 𝑥0 [5𝜋−22𝜋 ]

Q.2 Evaluate the following integrals as limit of a sum.

i) dxxx 4

1

2 ii)

b

a

xdxe iii)

4

0

2dxex

x

2

27 ab

ee 2

15 8e


ASSIGNMENT NO :8


CHAPTER : APPLICATION OF INTEGRALS

Q.1 Find the area enclosed between the curves y = x2 & y = x .

Q.2 Find the area between the x – axis & the curve y = sin x from x = 0 to x = 2.

Q.3 Find the area of the region lying in the first quadrant & bounded by y = 9x2, y = 1 & y = 4

using integration.

Q.4 Find the area bounded by y = x3, the x – axis, x = -2 & x = 1.

Q.5 Find the area bounded by y = 1x +1, x = - 3, x = 3 & y = 0.

Q.6 Make a rough sketch of the function xy 4 & find the area enclosed between the

curve & the coordinate axes.

Q.7 Find the area bounded by y2 = 8x & its latus rectum.

Q.8 Sketch the graph of y = 1x & evaluate dxx

1

1

3

Q.9 Find the area bounded by the curves y = sinx & y = cos x for x0 .

*Q.10 Find the area bounded by the curve )3(4 22 xay & the lines x = 3 & y = 4a.

Q.11 Find the area bounded by the lines x + 2y = 2, y – x = 1 & 2x + y = 7.

Q.12 Find the area of the region bounded by the curve y = 22 x & the line y = x, x = 0 & x

=3.

Q.13 Sketch the region common to the circle x2 + y

2 = 16 & the parabola yx 62 . Also find

the area of the smaller region bounded by the curves.

*Q.14 Compare the areas under the curves y = cos x & y = cos2

x between x = 0 & x = 4

.

*Q.15 Find the area of the region bounded by the curve y2 = 2y – x & the y – axis.



ASSIGNMENT NO :9

TOPIC : DIFFERENTIAL EQUATIONS

Q.1 Show that y = - ( 1 + x ) is a solution of the differential equation

(y –x ) dy – (y2 – x

2) dx = 0

Q.2 Show that y = x Sin3x is a solution of the differential equation 03cos692

2

xydx

yd

Q.3 Form a differential equation of the curve represented by (2x –a)2 - y

2 = a

2, where a is a constant.

Q.4 Solve the differential equation log

dx

dy = 3x – 4y

Q.5 Show that the differential equations that represents all parabolas each of which has a latus

rectum 4a & whose axis is parallel to the x – axis is 2a 0

3

2

2

dx

dy

dx

yd

Q.6 Solve sec2y (1 + x

2) dy + 2x tan y dx = 0 given that

4

y when x = 1.

Q.7 Solve x2 dy + y ( x + y) dx = 0, given that y = 1 when x = 1.

Q.8 Solve dx

dyxy

dx

dyxy 22

, given that when x = 1, y = 1

Q.9 xxxxydx

dycot2cot

2 , given that y(0) = 0

Q.10 Show that the differential equation, of which xy = aex + be

-x + x

2 is a solution, is

0222

2

2

xxydx

dy

dx

yxd

Q.11 Solve 1

121

2

2

xxy

dx

dyx

Q.12 The slope of tangent at any point of a curve is four times the abscissa of the point of contact.

Find the equation of the curve if it passes through the origin.

Q.13 Solve 2

2

dx

yd =

xe

xxx

32

221

Q.14 Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its

radius originally is 3 mm & 1 minute later has been reduced to 2 mm, find an expression for the

radius of the rain drop at any time t.

Q.15 Solve the differential equation √ + 2 + 2 + 2 2 + 𝑑𝑑 =

Q.16 Solve 𝑑𝑑 = + 2 + 2 + 2 2, given that when x=0, y=1.

Q.17 A population grows at the rate of 5% per year. How long will it take for the population to

double? Use differential equation for it.

Q.18 Solve 2/

xedxdy

Q.19 Solve 0)1()1(22 dxeydye

xx, given that when x = 0, y = 1

Q.20 Find the differential equation, of which x

cexy1

tan1tan

is a solution where c is a

constant.



ASSIGNMENT NO: 10

TOPIC : VECTOR ALGEBRA

Q.1 Find the value of so that the two vectors ^^^

32 kji and ^^^

64 kji are

a. Parallel

b. Perpendicular to each other.

Q.2 Show that the area of the parallelogram having diagonals ^^^

23 kji and ^^^

43 kji is 35

sq. units.

Q.3 If

cba ,, are vectors such that

ba . =

ca . ,

caba and

0a , then prove that

cb .

Q.4 Three vectors

cba &, satisfy the condition

0cba . Evaluate the quantity

accbba ... if ,1

a

b = 4 and

c = 2

Q.5 Find a vector of magnitude 19 which is perpendicular to both the vectors ^^

kj & ^^^

84 kji

Q.6 Find the projection of

cb on

a where

a = ^^^

22 kji ,

b = ^^^

22 kji &

c = ^^^

42 kji

Q.7 If

cba ,, are three vectors such that

0cba ,3

a

b = 5,

c = 7, find the angle

between

ba& .

Q.8 If

a = ^^^

kji &

b =^^

kj , find a vector

c such that

ca =

b &

ca . = 3.

Q.9 Find the area of a triangle whose 2 sides are represented by the vector ^^^

43 kji & ^^^

kji

Q.10 Find the relation between & such that

a +

b is perpendicular to

c where

a = ^^^

23 kji ,

b = ^^^

32 kji , &

c = ^^^

2kji .

Q.11 If ,32^^^

kjia

& ,23^^^

kjib

show that

ba is perpendicular to

ba

Q.12 The dot product of a vector with the vectors ,3^^^

kji ^^^

23 kji and ^^^

42 kji are 0, 5, &

8 respectively. Find the vector.

Q.13 If

0cba , show that

accbba

Q.14 Express the vector ^^^

525 kjia

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

vector ^^

33 kib

& the other is perpendicular to

b .

Q.15 If ,

a

b ,

c are three mutually perpendicular vectors of equal magnitude, find the angle

between

a &

a +

b +

c .

Q.16 Find the angle between the vectors

a +

b &

a -

b if ^^^

32 kjia

and ^^^

23 kjib

.

Q.17 Prove that

baba

bababa

..

..

Q.18 Show that the points whose position vectors are

^^^

765 kji ,

^^^

987 kji &

^^^

5203 kji are collinear.

Q.19 Using vector method find the area of the triangle whose vertices are A(1, 1, 1), B (1, 2, 3) & C (2,

3, 1).

Q.20 If

ba& are vectors such that ,2

a 3

b & 4.

ba , find

ba .

Q.21 Find

cba , when

i) ^^^

432 kjia

, ^^^

2 kjib

and ^^^

23 kjic

ii) ^^

32 jia

, ^^^

kjib

and ^^

3 kic

Q.22 Find the volume of the parallelepiped whose co-terminous edges are represented by the

vectors.

i) ^^^

32 kjia

, ^^^

2 kjib

, ^^

kjc

ii) ^

6 ia

, ^

2 jb

, ^

5 ic

Q.23 Find the value of , for which the vectors

a ,

b ,

c are coplanar, where

i) ^^^

2 kjia

, ^^^

32 kjib

and ^^^

53 kjic

ii) ^^^

kjia

, ^^^

2 kjib

and ^^^

kjic

Q.24 Show that the four points with position vectors ^^

76 ji , ^^^

41916 kji , ^^

63 kj and

^^^

1052 kji are coplanar.

Q.25 Find the value of for which the points A (3, 2, 1), B (4, , 5), C (4, 2, -2) and D (6, 5, -1) are

coplanar.

Q.26 If the vectors ^^^

kcjaia , ^^

ki and ^^

kbjcic be coplanar, show that abc 2.

Q.27 For any three vectors

cba ,, show that the vectors ,

ba

cb ,

ac are coplanar.

Q.28 Show that

cbaaccbba 2

Q.29 For any three vectors ,

a

b ,

c prove that 0

cbacba

Q.30 The volume of the parallelepiped whose edges are

^^

12 ki ,

^^

3 kj and

^^^

152 kji

is 546 cubic units. Find the value of .

ANSWERS SHEET

12. ^^^

2 kji

14. ^^

26 ki , ^^^

32 kji

15.

3

1cos

1

16. 2

19. 2

21 𝑞. 𝑛𝑖

20. 5

21. i) -7 ii) 4

22. i) 12 cu. Units ii) 60 cu. Units

23. i) 4 ii) 1

25. = 5

30. = -3


ASSIGNMENT:11


TOPIC: THREE DIMENSIONAL GEOMETRY

1. If the equation of the line AB is 4

5

2

2

1

3

zyx

, Find the direction ratio and

direction cosines of a line parallel to AB.

2. If a line L makes angles ,,, with the four diagonals of a cube, prove that

3

4coscoscoscos 2222 or

3

82222 SinSinSinSin .

3.

4. Find the coordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line

joining the points B( 0, -1, 3) and D (2, -3, -1).

5. Find the vector equation of the line which is parallel to the vector ^^^

32 kji and which

passes through the point (5, -2, 4). Also find its Cartesian equations.

6. Find a point on the line 2

3

2

1

3

2

zyx

at a distance 23 from the point (1, 2, 3)

7. Find the equations of the perpendicular drawn from the point A (2, 4, -1) to the line

9

6

4

3

1

5 zyx

.

8. Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line

11

8

4

2

10

11

zyx

.

9. Find the shortest distance between the lines :

i) )2(^^^^^

kjijir

and )253(2^^^^^^

kjikjir

ii) 1

1

6

1

7

1

zyx

and 1

7

2

5

1

3

zyx

9. Find whether or not the two lines given below intersect:

^^^

1112 kjir

^^^

125523 kujuiur

10. Find the coordinates of the point where the line

4

3

3

2

2

1

zyx meets the plane x + y + 4z = 6.

11. Find the vector equation of the line passing through the point with position vector

^^^

532 kji and perpendicular to the plane 02536.^^^

kjir . Also find the point of

intersection of this line and the plane.

12. Find the equation of the line passing through the point P(4, 6, 2) and the point of

intersection of the line 7

1

23

1

zyx and the plane x + y – z = 8.

13. Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured along a

line parallel to 632

zyx

.

14. Find the distance of a point A (-2, 3, 4) from the line 5

43

4

32

3

2

zyx measured

parallel to the plane 4x + 12y – 3z + 1 = 0.

15. From the point P(1, 2, 4) a perpendicular is drawn on the plane 2x + y – 2z + 3 = 0. Find

the equation, the length and coordinates of the foot of the perpendicular.

16. Find the coordinates of the image of the point (1, 3, 4) in the plane 2x – y + z + 3 = 0.

17. Find the equation of the plane passing through the points (3, 4, 1) and ( 0, 1, 0) and parallel

to the line 5

2

7

3

2

3

zyx

18. Find the Cartesian and vector equations of the plane passing through the points R (2,5,-3),

S (-2, -3, 5) and T (5, 3, -3).

19. Find the equation of the plane passing through the intersection of the planes

^^^

32. kjir = 7 and

^^^

352. kjir = 9 and the point (2, 1, 3)

20. Find the equation of the plane passing through the points (1, -1, 2) , (2, -2, 2) and

perpendicular to the plane 6x – 2y + 2z = 9.

21. Find the equation of the plane passing through the points (-1, -1, 2) and perpendicular to

each of the following planes :

2x + 3y – 3z = 2 and 5x – 4y + z = 6.

22. pFind the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and

which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and

2x + y – z + 5 = 0.

23. Find the Cartesian and the vector equation of the planes through the intersection of the

planes 01262.^^

jir and 043.^^^

kjir which are at a unit distance from the

origin.

24. Show that the lines 7

5

5

3

3

1

zyx and

5

6

3

4

1

2

zyx intersect. Also find

their point of intersection.



ASSIGNMENT -12

TOPIC: LINEAR PROGRAMMING

1. Solve the following linear programming problems graphically:

a) Max and min Z = 60 x + 15 y s.t x + y 50, 3 x + y 90 , x , y 0

b) Max Z = 8x + 7y s.t 3x + y 66, x + y 45, x 20, y = 40, x , y 0

c) Min Z = x – 5y + 20 s.t x – y 0 , -x + 2y 2 , x 3, y 4, x , y 0

d) Min Z = x – 7y + 190 s.t x + y 8, x 5, y 5, x + y 4 , x , y 0

a) Max Z = 4x + 8y s.t 2x + y 30, x + 2y 24, x 3, y 9, y 0.

2. A manufacturer produces two types of steel trunks. He has two machines A and B. The

first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second

type of trunk requires 3 hours on machine A and 2 hours on machine B. Machine A and

B can work almost for 18 hours and 15 hours per day respectively. He earns a profit of

Rs. 30 and Rs. 25 per trunk of the first type and second type respectively. How many

trunks of each type must he make each day to make maximum profit?

3. Two tailors, A and B, charge Rs150 and Rs200 per day respectively. A can stitch 6 shirts

and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each

work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?

4. An aeroplane can carry a maximum 200 passengers. A profit of Rs400 is made on each

first class. However, at least four times as many passengers prefer to travel by second

class than by first class. Determine how many tickets of each type must be sold to

maximize the profit for the airline. Form an L.P.P and solve it graphically.

5. If a young man rides his motor cycle at 25 km per hour, he has to spend Rs 2 per

kilometer on petrol; if he sides at a faster speed of 40 km per hour, the petrol cost

increases to Rs 5 per km. He has Rs100 to spend on petrol and wishes to find the

maximum distance he can travel within one hour. Express this as a L.P.P and solve it

graphically.

6. A factory owner purchases two types of machines, A and B, for his factory. This

requirement and limitations for the machines are as follows.

Machine Area occupied

by the machine

Labour force for

each machine

Daily output

(in Units)

A 100 sq. m 12 men 60

B 200 sq.m 8 men 40

He has an area of 9000 sq. m available and 72 skilled men who can operate the machines.

How many machines of each type should he buy to maximize the daily output?

7. An oil company requires 13,000, 20,000 and 15,000 barrels of high grade medium grade

and low grade oil respectively. Refinery A produces 100, 300 and 200 barrels per day of

high medium and low grade oil respectively whereas the Refinery B produces 200, 400

and 100 barrels per day respectively. If refinery A costs Rs 400 per day and B costs Rs

300 per day to operate, how many days should each be run to minimize the cost of

requirement?

8. A housewife wishes to mix together two kinds of food F1 and F2 in such a way that the

mixture contains at least 10 units of Vitamin A, 12 units of Vitamin B and 8 units of

Vitamin C. The Vitamin contents of one kg of foods F1 and F2 are as follows.

Vitamin A Vitamin B Vitamin C

Food F1 1 2 3

Food F2 2 2 1

One kg of food F1 costs Rs 6 and one kg of food F2 costs Rs 10. Formulate the above

problem as a L.P.P and use corner point method to find the least cost of the mixture

which will produce the diet.

9. A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and

6% phosphoric acid. After testing the soil conditions of the field, it was found that atleast

14 kg of nitrogen and 14 kg of phosphoric acid is required for good crop. The fertilizer

of type A costs Rs 5 per kg and type B costs Rs 3 per kg. How many kg of each type of

fertilizer should be used to meet the requirement at the minimum possible cost? Using

L.P.P, solve the above problem graphically.

10. A firm deals with two kinds of fruit juices – pineapple and orange juice. One tin of A

requires 4 litres of pineapple and 1 litre of orange juice. The form has only 46 litres of

pineapple juice and 24 litres of orange juice. Each tin of A and B are sold at a profit of

Rs 4 and Rs 3 respectively. How many tins of each type should the firm produce to

maximize the profit? Solve the problem graphically.

11. A dealer wishes to purchase a number of fans and radios. He has only Rs 5760 to earnest

and has a space for at most 20 items. A fan costs him Rs 360 and radio Rs 240. His

expectation is he can sell a fan at a profit of Rs 22 and radio at a profit of Rs 18.

Assuring that he can sell at the items he buys, how should he invest his money for

maximum profit? Translate the problem as L.P.P and solve it graphically.

12. A diet for a sick person must contain atleast 400 units of vitamin, 50 units of minerals

and 1400 units of calories. Two foods A and B are available at a cost of Rs 5 and Rs 4

per unit. One unit of food A contains 200 units of vitamin, 1 unit of minerals and 40 unit

of calories, while one unit of food B contains 100 units of vitamins, 2 units of minerals

and 40 units of calories. Find what combination of the foods A and B be used to have

least cost, but it must satisfy the requirements of the sick person. Form the L.P.P and

solve it graphically.

13. A man has Rs 1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost

Rs 180 and Rs 120 respectively. He has a storage capacity of 10 bags only. He earns a

profit of Rs 11 and Rs 9 per bag of rice and wheat respectively. Formulate an L.P.P to

maximize the profit and solve it.


ASSIGNMENT NO 13


Topic: PROBABILITY

1. If P(A) = 0.2, P(B) = p, P (AB) = 0.6 and A, B are given to be independent events,

find the value of p.

2. An Urn contains 4 red and 7 blue balls. Two balls are drawn at random with

replacement. Find the probability of getting

i) 2 red balls ii) 2 blue balls iii) one red and one blue ball.

3. A and B throw a pair of die turn by turn. The first to throw 9 is awarded a prize. If A

starts the game, show that the probability of “A” getting the prize is 17

9.

4. There are two bags I and II. Bag I contains 2 white and 4 red balls and Bag II contains 5

white and 3 red balls. One ball is drawn at random from one of the bags and is found to

be red. Find the probability that it was drawn from bag II.

5. In a class having 70% boys, 20% of boys and 10% of the girls are players. A student is

selected at random from the class and is found to be a player. Find the probability that

the selected student is a girl.

6. A man is know to tell a lie 1 out of 4 times. He throws a die and reports that it is a six.

Find the probability that it is actually a six.

7. For A, B and C the chances of being selected as the manager of a firm are in the ratio 4:

1: 2 respectively. The respective probabilities for them to introduce a radical change in

marketing strategy are 0.3, 0.8 and 0.5. If the change does takes place, find the

probability that it is due to the appointment of B or C.

8. A pair of dice is tossed twice. If the random variable X is defined as the number of

doublets, find the probability distribution of X.

9. Two cards are drawn successively with replacement from a well – shuffled deck of 52

cards. Find the probability distribution of the number of jacks.

10. Four bad oranges are mixed accidentally with 16 good oranges. Find the probability

distribution of the number of bad oranges in a draw of two oranges.

11. Find the mean and variance for the following probability distribution –

X 0 1 2 3

P(X) 6

1

2

1

10

3

30

1

12. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the

mean and standard deviation of the number of kings.

13. An experiment succeeds twice as often as fails. Find the probability that in the next six

trials, there will be atleast four successes.

14. A pair of dice is thrown 6 times. Getting a total of 7 on the two dice is considered a

success. Find the probability of getting: i) atleast 5 successes. ii) exactly 5 successes

iii) atmost 5 successes iv) no success.

15. There are 6% defective items in a large bulk of items. Find the probability that a sample

of 8 items were include not more than one defective item.

16. If the mean and variance of a binomial distribution are respectively 9 and 6, find the

distribution.

17. If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the

distribution.

18. The probability of A solving a problem is 7

3 and that B solving it is

3

1. What is the

probability that i) atleast one of them solves the problem ii) only one of them will solve

the problem.

19. Ramesh appears for the interview for two posts A and B for which selection is

independent. . The probability of his selection for the post A is 6

1 and for post B is

7

1.

Find the probability that Ramesh is selected for atleast one of the post.

20. A problem in statistics is given to three students whose chances of solving it are

2

1,

3

1,

4

1respectively. What is the probability that only one of them solves it correctly?

AHLCON PUBLIC SCHOOL ASSIGNMENT -1 CLASS XII … · 2017-09-11 · AHLCON PUBLIC SCHOOL ASSIGNMENT...

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Transcript of AHLCON PUBLIC SCHOOL ASSIGNMENT -1 CLASS XII … · 2017-09-11 · AHLCON PUBLIC SCHOOL ASSIGNMENT...