Xii Maths Assignment CBSE 2014-2015 BY R.SAIKISHORE

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Contents

1. Matrices and Determinantsi. Important Results………………………… (4 )ii. Assignments……………………………………. (5 )

2. Relations and Functionsi. Assignments…………………………………… (17 )

3. Inverse Trigonometric Functionsi. Assignments………………………………….. (19 )

4. Probabilityi. Assignments………………………………….. (21 )

5. Vector Algebrai. Important Results………………………….. (32)ii. Assignments………………………………….. (34 )

6. Three dimensional Geometry i. Important Results…..…….……………….. (36 )ii. Assignments………………………………….. (44 )

7. Continuity i. Assignments………………………………….. (56 )

8. Differentiabilityi. Assignments………………………………….. ( 59)

9. Applications of Derivativesi. Assignments…………..……………………… ( 62)

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10. Integralsi. Indefinite Integrals……………………….. (66)ii. Substitution………………………………….. (67)iii. Partial Fraction……………………………… (68)iv. Integration by Parts………………………. (69)v. Cancellation Property……………………. (70)vi. Definite Integrals…………………………. (71)

11. Application of Integralsi. Area under the curve…..………………. (74)

12. Differential Equationsi. Forming Differential Equations……… (75)ii. Variable separable………………………… (76)iii. Homogenous………………………………… (77)iv. First Order Linear Differential Equation... (79)

13. Linear Programming……………. (80)

Extra Problems – Unsolved 1. Equivalence Relations………………………………..(87)2. Binary operations……………………………………….(88) 3. Inverse of a function………………..…………………(90)4. Rate of change of quantities……………………….(92)5. Probability…………………………………………………..(93)

ANSWERS………………...………………………………………. (96)

TYPES OF PROBLEMS…………………………………………… (121)3 | P a g e

MATRICES AND DETERMINANTS

IMPORTANT RESULTS

1. |A|=|A '|

2. If A is a square matrix of order ‘n’ then |kA|=k n|A|

3. A is a singular matrix if |A|=0

4. If A is a skew-symmetric matrix of odd order then |A|=0

5. If A is a non-singular matrix of order ‘n’ |adjA|=|A|n−1

6. If A is an invertible matrix of order ‘n’ then det(A-1)=1/det(A)

7. A(AdjA)=(AdjA)A=|A|I

8.(A-1)-1=A

9. (AB)-1=B-1A-1

10. (AT)-1=(A-1)T

11. adj(AB)=(adjA)(adjB)

12. adj(AT)=(adjA)T

13. A+A’ is always symmetric, A-A’ is always skew-symmetric.

14. If A and B are symmetric matrices of same order then AB+BA is always symmetric and AB-BA is always skew-symmetric matrices.

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ASSIGNMENT 1

MATRICES AND DETERMINANTS:

1. Construct a 2x3 matrix whose elements are

(a) aij= (i+ j)2

2(c) aij= i.j

(b) aij= 3i+4 j

2 (d) aij= {i− j , i≥ j

i+ j , i< j

2. If A=( 2 −24 2

−5 1 ) , B=(8 04 −23 6 ), find ‘X’ if 2A+3X=5B.

3. Find X and Y given that 3X-Y=( 1 −1−1 1 ), X-3Y=(0 −1

0 −1)

4. Find x and y, 2(x 57 y−3) + (3 4

1 2 ) = ( 7 1415 14)

5. If A=(2 −33 4 ). Show that A2-6A+17I=0.

6. If A=( 0 3−7 5), find k, if kA2+5A-21I=0.

7. If A=(4 32 5), find x and y such that A2-xA-yI=0.

8. Find x,y,z such that

(x−2 y 3x+4 y 2−1 z+w 2 z−w)=(−3 11 2

−1 2 −5)

9. Find ‘x’ if (1 2 1)(1 2 02 0 11 0 2)(

02x)=0.

10. If f(x) = x2-4x+1. Find f(A) when A=(2 31 2).

11. If A= (−8 52 4) , Show that A2+4A-42I=0.

12. If A=(1 −1 02 3 40 1 2), B=( 2 2 −4

−4 2 −42 −1 5 ), Verify that BA=6I.

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ASSIGNMENT 2


1) If A= ( 0 a b−a 0 c−b −c 0) ,find

12

(A+A’) and 12

(A-A’)

2) If A= (2 34 5) , prove that A-AT is a skew symmetric matrix.

3) Express the matrix A=( 3 1−4 −1) as the sum of symmetric and skew symmetric

matrices.

4) For the following problems prove that (AB)’=B’A’

(a) A = (−1 3 0−7 2 8) B = (−5 0

0 31 −8)

(b) A = (5 3 −12 0 4 ) B = (−3 2

2 1−1 0)

(c) A = ( 1 −23 0

−5 6 ) B = (0 −3 41 2 0)

(d) A = (1 −32 4 ) B = (1 4

2 5)

(e) A = (−123 ) B = (2−1 – 4 )

5) If A = (0 2 y zx y −zx − y z ) and A’A=I ,Find x,y,z.

6) If A = ( cos A sin A−sin A cos A) , using induction prove that

An = ( cosnA sinnA−sinnA cosnA)

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Assignment 3

By using elementary row transformations find the inverse of the matrix if it exists (1-19)

1. A=(1 −12 3 ) 2. A=(1 2

2 −1) 3. A=(4 33 2) 4. A=(1 2

3 7)5. A= (5 2

2 1)6. A=(1 24 9) 7. A= (2 1

7 4 ) 8. A=(4 53 4)

9. A=( 2 −3−1 2 ) 10. A= (2 1

4 2) 11. A= (2 11 1)

12. A= (1 32 7) 13. A=(2 3

5 7) 14. A= (2 51 3) 15. A= (3 1

5 2)

16. A= (2 0 −15 1 00 1 3 ) 17. A=( 1 2 3

−3 5 00 1 1)

18. A=( 1 2 52 3 1

−1 1 1)19. A=( 1 3 −2

−3 0 −52 5 0 )

20. If A=(2 14 2) show that |2 A|=4|A|.

21. Find ‘x’ if |2 45 1|=|2 x 4

6 x|. 22. Find ‘x’ if |2 3

4 5|=| x 32 x 5|.

23. Factorize: |x 1 11 x 11 1 x|. 24. Solve: |15−2 x 11−3 x 7−x

11 17 1410 16 13 |=0.

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ASSIGNMENT 4

1. Find the area of the triangle whose vertices are:

(i) (2, 7), (1, 1), (10, 8).(ii) (-3, 5), (3, 6), (7, 12). (iii)(2, -7), (1, 3), (10, 8).(iv) (-2, 4), (2, -6), (5, 4). ( v) (at1

2, 2at1), (at22, 2at2), (at3

2, 2at3).

2. Show that the following points are collinear:

(i) (-1, -1), (5, 7), (8, 11). (ii) (3, 8), (-4, 2), (10, 14).

3. If (a, 0), (0, b) and (x, y) are collinear, prove that xa

+ yb

= 1.

4. If (3, -2), (x, 2) and (8, 8) are collinear, find ‘x’.

5. Find the equation of the line passing through:

(i) (1, 2), (3, 6). (ii) (3, 1), (9, 3) (iii) (3, 4), (-7, 2).

6. Solve for ‘x’: |a a xx x xb x b|=0.

7. Solve for ‘x’: |x2

0 3x 1 −41 2 0 |=0.

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ASSIGNMENT 5

Solve the following:

1.|x2

0 3x 1 −41 2 0 |=0. 2. |1 4 4

1 −2 11 2 x x2|=0. 3. |x+1 3 5

2 x+2 52 3 x+4|=0.

4. |x−2 2 x−3 3 x−4x−4 2 x−9 3 x−16x−8 2 x−27 3 x−64|=0. 5. | x −6 −1

2 −3 x x−3−3 2 x x+2|=0.

Using properties of determinants prove the following:

6. | a a+b a+b+c2 a 3 a+2b 4 a+3 b+2 c3 a 6 a+3 b 10 a+6 b+3c|=a3.

7. | a2 b2 c2

(a+1)2 (b+1)2 (c+1)2

(a−1)2 (b−1)2 (c−1)2|= -4(a-b)(b-c)(c-a).

8. |(x−2)2 (x−1)2 x2

(x−1)2 x2 (x+1)2

x2 ( x+1)2 (x+2)2|= −8.

9. |1 bc+ad b2c2+a2d2

1 ca+bd c2 a2+b2d2

1 ab+cd a2b2+c2d2|= − (a−b) (b−c) (c−a) (a−d) (b−d) (c−d).

10. If A=(3 22 1), show that A2−4A-I=0 and hence findA−1.

11. If A=( 3 4−2 5), verify that A (adjA) = (adjA) A=|A|I.

12. A= ( 3 7−5 4 ) and B=( 2 3

−1 5), verify that ( AB)−1= B−1 A−1.

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13. If A=(−123 )andB=(1 0 4 ), verify that ( AB)1=B1 A1.


ASSIGNMENT 6

Solve the following system of linear equations using matrix method:

1. 2x+6y=2; 3x-z=-8; 2x-y+z=-3. 2. x-y=3; 2x+3y+4z=1; y+2z=7.3. 2x-y+z=-3; 3x-z=-8; 2x+6y=2. 4. x+2y+z=7; x+3z=11; 2x-3y=1.

5.5x+3y+z=16; 2x+y+3z=19; x+2y+4z=25. 6. x+y+z=1; x-2y+3z=2; x-3y+5z=3.

7. 2x-y-z=7; 3x+y-z=7; x+y-z=3. 8. x-y=5; y-z=2; z-x=-3.9. 2x-y=4; 2y+z=5; z+2x=7. 10. 5x-y=-7; 2x+3z=1; 3y-z=5.

11. FindA−1, if A=(1 1 21 −1 11 1 −1) and hence solve the following system of linear

equations, x+y+z=6; x-y+z=6; 2x+y-z=6.

12. Find A−1, if A=(2 −1 31 3 23 −4 −1) and hence solve 2x-y+3z=13; x+3y+2z=1; 3x-4y-z=8.

13. Find A−1, if A=(1 1 12 2 −32 −1 3 ) and hence solve the following system of linear equations,

x+2y+2z=0; x+2y-z=0; x-3y+3z=-14.

14. Solve the following system of linear equations using matrix method Where x≠0,y≠0,z≠0.

(a). 2x

− 3y+ 3

z = 10;

1x+ 1

y+ 1

z = 10;

3x− 1

y+ 2

z = 13.

(b). 2x+ 3

y+ 10

z = 4;

4x−6

y+5

z = 1;

6x+ 9

y−20

z = 2.

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15. Determine the product (−4 4 4−7 1 35 −3 −1)(

1 −1 11 −2 −22 1 3 ) and use to solve the system

of equations: x−y+z=4; x−2y−2z=9; 2x+y+3z=1.

16. If A= (1 −1 02 3 40 1 2) and B= ( 2 2 −4

−4 2 −42 −1 5 ) are two square matrices, Find AB

and hence solve the system of linear equations: x−y=3; 2x+3y+4z=17; y+2z=7.


ASSIGNMENT 7

1. If A=( 4 2−1 1), prove that (A−2I)(A−3I)=0.

2. If (1 2 x )(1 0 20 2 12 1 0)(

111) = 0. Find ‘x’.

3. If ( x 4 1 )(2 1 01 0 20 2 −4)( x

4−1) = 0. Find ‘x’.

4. If (1 1 −x )(0 1 −12 1 31 1 1 )(011) = 0. Find ‘x’.

5. If A=(1 10 1), prove that An=(1 n

0 1). nЄN.

6. If A=(a b0 1), prove that An=(an b(an−1)

a−10 1 ). nЄN.

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7. If A=(1 1 10 1 10 0 1), prove that An=(1 n

n(n+1)2

0 1 n0 0 1

). nЄN

8. Find the inverse of the matrix A= (a b

c1+bc

a ) and show that aA−1=(a2+bc+1)I-aA.


ASSIGNMENT 8

Using the properties of determinants prove the following:

1.

|p2+2 p 2 p+1 12 p+1 p+2 1

3 3 1|=( p−1 )3

2.

|y+z x yz+x z xx+ y y z

|= (x+ y+z ) ( x−z )2

3.

|x x2 1+ px3

y y2 1+ py 3

z z2 1+ pz 3

|= (1+ pxyz ) ( x− y ) ( y−z ) (z−x )(OR )|x x2 1+x3

y y2 1+ y 3

z z2 1+z3

|=(1+xyz ) ( x− y ) ( y−z ) ( z−x )

4.

|a b−c c−b

a−c b c−aa−b b−a c

|=( a+b−c ) (b+c−a ) (c+a−b )5.

|a2 b2 c2

(a+1 )2 (b+1 )2 (c+1 )2

( a−1 )2 (b−1 )2 (c−1 )2|=−4

(a-b)(b-c)(c-a)

6. If a,b,c are positive numbers and unequal prove that the determinant value of the following determinant is always is negative.

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ASSIGNMENT 11

RELATIONS AND FUNCTIONS

1) Prove that the function f:N→N ,defined by f(x)=2x is one-one ,N is the set of all natural numbers.

2) Prove that the function f:X→Y defined by f(x)=5x where X={1,2,3} and Y={5,10,15} is both one-one and onto.

3) Prove that the function g:A→B defined by g(x)=x2 where A = {1,-3,3} and B = {1,9} is onto but not one-one.

4) Let f:N→N be defined by f(n) ={n+12

if n i odd

n2

if n is even∀ n ∈N .Is it bijective?

5) Consider f:R→[4,∞) given by f(x) = x2+4 , shot that f is invertible with the inverse f−1

Of f given by f−1(y)=√ y−4 ,where R is the set of all non negative real numbers.

6) Let f:W→W be defined as f(n) = {n−1 if n is oddn+1 if n is even

show that f is invertible,find f−1 and

prove that f=f−1.

7) Let f:R→R be defined by f(x) = 2 x−1

3 . prove that f is a bijective function and hence

find the inverse of f.

8) Show that f:R-{1} →R-{1} defined by f(x) = x

x+1 is invertible. Also find f−1.

9) Let f(x) = 2x3 -3 and g(x) = ( x+32

)13 , x∈R ,verify that f and g are inverses of each

other.

10)F:R-{0}→R-{0} given that f(x) = 3x

is invertible and it’s the inverse of itself.

11)Find the inverse of the function f(x) = 43√ x−1 x ∈ R.12)If f:R→R ,g:R→R are two functions ,defined by f(x)=3x-2 and g(x)=x2+1 .Find gof(2).

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13)Let f:Z→Z be defined as f(n) =3n ∀ n ∈ Z, g:Z→Z be defined as g(n)=

{ n3

if n is amultiple of 3

0 if n is not a multiple of 3 show that gof=Iz ,fog ≠Iz

14)Show that the function f:R→R defined by f(x)=3x3+5 ∀ x ∈ R is a bijection.15)Prove that the function f:Q→Q given by f(x)=2x-3 ∀ x ∈ Q is a bijection.16)Prove that f:R→R , given by f(x)=2x is one-one and onto.17)Show that the function f:R→R , defined by f(x)=x2 is neither one-one nor onto.18)Show that f:R→R , defined by f(x)=x3 ,is a bijection.Show that f:N→N defined by

f(x) = {n+12

, if n is odd

n2

, if n is even .Is f a many-one onto function?

19)Show that the function f:N→N given by f(n) = n-(−1)n∀ n ∈ N is a bijection.

ASSIGNMENT-12

RELATIONS AND FUNCTIONS

1) If f: R→R , is defined by f(x)=x

x2+1 . Find f(f(2)).

2) If f(x)=x2-3x+2 be a real valued function of the real variable .Find fof.

3) Let f:R→R be defined by f(x)=3x-2,g:R→R be defined by g(x)=x+2

3 .Show that

fog=gof=IR.4) If f(x)=x2+1 and g(x)=3x-1 find fog,gof,fof,gog.5) Let f(x)=x2-2x+3 ∀ x ϵR ,find f(f(x)).

6) If f(x)=(a−xn)1n ,Prove that f(f(x))=x.

7) If f(x) = ax−bbx−a

, Show that f(f(x))=x.

8) A={1,2,3,5} f={(1,5),(2,1),(3,3),(5,2)} and g={1,3),(2,1),(3,2),(5,5)}. Find fog,gof,fof,gog.9) If f(x)=3-4x ∀ x ϵ R , find g:R→R such that gof=IR.10) Let f(x)=x+1 , g(x)=3x2-2.Find fog,gof,fof,gog.11) IF f(x)=sin x, g(x)=3x .Find fog,gof.12) If f(x)=2x+3 , g(x)=x2+1 .Find fog,gof,fof,ff. Show that fof≠ff.13) f(x)=x3 ,g(x)=cos3x. Find fog,gof,fof,gog.14) f,g:R→R,f(x)=2x2-x+7 and g(x)=x-4 , ∀ x ∈ R . Find gof,fog,gog,fof.15) Find gof,fog if

a) f(x)=|x| g(x)=|5x-2|b)f(x)=8x3 g(x)=x13

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16) If f: R→R defined by f(x)=x2-3x+2.Find f(f(x)). Prove that the function f:N→N , defined by f(x)=x2+x+1 is one-one but not onto.

17.Show that the function f:R-{3}→R-{3} given by f(x) =x−2x−3

is a bijection.

18.Let f:N→N be defined by f(x) = { n+1 , if n is oddn−1 , if n is even

.Show that f is a bijection.

19.Let A={ x∈ R ; -1 ≤ x ≤ 1 }=B .Show that f:A→B given by f(x) = x|x| is a bijection.

20.If f(x) = x−1x+1

, x≠∓1. Show that fof−1 is an identity function

ASSIGNMENT-13

INVERSE TRIGONOMETRIC FUNCTIONS

1. Find the principle value ofcos−1(√32 ).

2. Evaluate : 2 tan−1(12 )+ tan−1( 1

7 )=tan−1( 3117 )

3. Using principal value, evaluate, cos−1(cos2 π3 )+sin−1(sin

2 π3 )

4. Prove the following: tan−1( 13 )+¿ tan−1( 1

5 )+¿ tan−1( 17 )+¿ tan−1( 1

8 )=π4

¿¿¿

5. Solve: tan−1( x−1x−2 )+ tan−1( x+1

x+2 )=π4

6. Evaluate: sin [ π3−sin−1(−1

2 )]7. Evaluate: tan−11+sin−1(−1

2 )+¿cos−1(−12 )¿

8. Evaluate: cos−1( 12 )+2 sin−1( 1

2 )9. Evaluate: tan−1 √3−sec−1 (−2 )+cosec−1( 2

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10. Evaluate: sin−1(sinπ3 )

11. Evaluate: cos−1(cos ( 2 π3 ))

12. Evaluate: cos−1(cos7 π6 )

13. Evaluate: tan−1( tan3 π4 )

14. Prove the following: sin−1 817

+sin−1 35=sin−1 77

85

15. Prove the following: cos−1 1213

+sin−1 35=sin−1 56

65

16. Prove the following: cos−1( 45 )+cos−1( 12

13 )=cos−1( 3365 )

17. Prove the following: sin−1( 35 )−sin−1( 8

17 )=cos−1( 8485 )

18. Evaluate: tan{2tan-1¿) –( π4 )}

ASSIGNMENT-14

INVERSE TRIGONOMETRIC FUNCTIONS

1. Prove that tan−1( √1+cos x+√1−cos x

√1+cos x−√1−cos x )=π4− x

2; π≺x≺3 π

2

2. Prove that cot−1( √1+sin x+√1−sin x

√1+sin x−√1−sin x )=π4− x

2;

π2≺x≺π

3. Simplify: cos−1( 3

5cos x+ 4

5sin x )

4. Simplify:sin−1 (sin x+cos x

√2 ) ,π4≺x≺5π

4

5. Simplify: sin−1 ( 5sin x+12cos x

13 ) ,π4≺x≺5 π

4

6 Write in the simplified form: sin−1 (x √1−x−√x √1−x2)21 | P a g e

7. Write in the simplified form:

sin {tan−1( 1−x2

2 x )+cos−1(1−x2

1+ x2 )}

8. If

sin−1 ( 2a1+a2 )−cos−1( 1−b2

1+b2 )=tan−1( 2 x1−x2 ) prove that x= a−b

1+ab

9. Prove that tan−1 2

3=1

2tan−112

5 10. Solve: tan−1 x+ tan−13 x= π

4

11. Solve: tan−1( x+1)+ tan−1( x−1)= tan−1 8

31

12. Solve: tan−1 ( x−1)

( x+1)+tan−1 (2 x−1 )

(2 x+1 )=tan−123

36

13. Solve: tan−1( x+1)+ tan−1( x )+ tan−1 ( x−1)= tan−13 x

14. Solve:cot−1 x−cot−1 ( x+2 )=π /12 15. Solve: 2tan-1(cosx)=tan-1(2cosecx)

16. Prove that sin [cot−1 (cos (tan−1 x) )]=√ x2+1

x2+2

17. Simplify:cos ec [ tan−1 {cos (cot−1 (sec (sin−1 a )) )}]

ASSIGNMENT – 15

PROBABILITY

CONDITIONAL PROBABILITY

1. If P(A)=0.3, P(B)=0.7 and P(B/A)=0.5, find P(A/B) and P(AUB).

2. A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

3. 12 cards, numbered 1 to 12 are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

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4. A die is rolled, if the outcome is an odd number, what is the probability that it is a prime number?

5. A pair of dice is thrown. Find the probability of getting a sum of 10 or more if 5 appears on the first die.

6. In a certain school 20% of the students failed in English, 15% failed in Mathematics and 10% in both English and Mathematics. A student is selected at random. If he failed in English, what is the probability that he also failed in Mathematics?

7. For two events A and B, it is being given that P(A)=0.47, P(B)=0.63 and P(AUB)=0.8. Find the conditional probabilities P(A/B), P(B/A).

8. Let A and B are two events P(A)=3/5, P(B)=2/5 and P(A/B)=5/8, find P(AUB).

9. If P(not A)=0.7, P(B)=0.7 and P(B/A)=0.5, then find P(A/B) and P(AUB).

10. A die is thrown twice and sum of the numbers appearing is observed to be 6. What is the conditional probability that 4 have appeared at least once?

11. Three coins are tossed. Find the probability that all coins show head, if atleast one of the coins show head.

12. Two numbers are selected at random from the integers 1 through 13. If the sum is even, find the probability that both are odd.

ASSIGNMENT – 16

PROBABILITY

INDEPENDENT EVENTS

1. The probability of A hitting the target is 4/5 and that of B hitting it is 2/3. They both fire at the target. Find the probability that

(i) At least one of them will hit the target?(ii) Only one of them hit the target.

2. 2 balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that,

(i) Both balls are red.(ii) The first ball is black and second is red.

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(iii) One of them black and other red.3. X is taking up subjects Mathematics, Physics and Chemistry in the examination. His

probability of getting grade A in these subjects is 0.2, 0.3 a d 0.5 respectively. Find the probability that X gets,

(i)Grade A in all subjects.(ii) Grade A in no subjects.(iii) Grade A in 2 subjects.

4. A cin hit a target 4 times out of 5 times. B can hit a target 3 times out of 4 times and C can hit a target 2 times out of 3 times. They fire simultaneous. Find the probability that,

(i) any 2 out of A, B, C will hit the target.(ii) none of them will hit the target.

5. The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university

(i)None will graduate(ii) Only one will graduate(iii) All will graduate

6. An urn contains 4 red and 7 blue balls. 2 balls are drawn at random with replacement. Find the probability of getting,

(i)2 red balls(ii) 2 blue balls(iii) 1 red and 1 blue ball

7. A and B appear in an interview for 2 vacancies in same post. The probability of A’s selection is 1/6 and that of B’s selection is ¼. Find the probability that,

(i)both of them are selected.(ii) only one of them is selected.(iii) One of them selected(iv) At least one of them is selected.

8. If A and B are 2 independent events such that P(AUB)=0.5 and P(A)=0.2. Find P(B).9. There are 3 urns A, B and C. Urn A contains 4 white balls and 5 blue balls. Urn B contains 3

white balls and 4 blue balls. Urn C contains 3 white balls and 6 blue balls. One ball is drawn from each of 3 urns. What is the probability that out of these 3 balls drawn, 2 are white balls and on is a blue ball?

10. The probability that a student A can solve a question 6/7 and that another student B solving the question is ¾. Assuming that the 2 events, A can solve the question and B can solve the question are independent, find the probability that only one of them solves the question.

11. If A and B are independent events such that P(AUB)=0.7, and P(A)=0.3. Find P(B).

12. The probability that a randomly selected voter will vote for party A is 0.2 and the probability that he will vote for party B is 0.5, otherwise he will vote for independent parties. What is the probability that out of 6 voters 3 or more will vote for party B?

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13. Ganesh appears for an interview for 2 posts A and B, selection for which is independent. The probability of selection for post A is 1/5 and for post B is 1/8. What is the probability that Ganesh will get selected for atleast one of the posts?

14. A problem in Mathematics is given to 3 students whose chances of solving it are 1/3, 1/5, 1/6 respectively. Find the probability that one of them is able to solve the problem correctly.

15. A speaks truth in 70% of the cases and B in 80% of the cases. In what percentage of the cases are they likely to contradict each other in stating the same fact?

16. An article manufactured by a company consists of 2 parts X and Y. In the process of manufacture of part X, 8 out of 100 parts may be defective. Similarly, 5 out of 100 parts of Y may be defective. Calculate the probability that the assembled product will not be defective.

ASSIGNMENT – 17

PROBABILITY

TRANSFERRING OF BALLS

1. Bag A contains 6 red and 5 blue balls another bag B contains 5 red and 8 blue balls. A ball is drawn from the bag A and without noticing the colour it is put into the bag B. Then, a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour.

2. Bag A contains 4 white and 5 black balls and another bag B contains 6 white and 7 black balls. A ball is drawn from the bag A and without noticing the colour it is put into the bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is white in color.

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3. Bag A contains 4 yellow and 5 red balls another bag B contains 6 yellow and 3 red balls. A ball is drawn from the bag A and without noticing the colour it is put into the bag B. Then, a ball is drawn from the bag B at random. Find the probability that the ball drawn is yellow in colour.

4. Bag A contains 5 white and 6 black balls and another bag B contains 7 white and 3 black balls. A ball is drawn from the bag A and without noticing the colour it is put in bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour.

5. Bag A contains 7 red and 4 blue balls and another bag B contains 5 red and 6 blue balls. 2 balls are drawn from the bag A and without noticing the colour it is put in bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour.

6. Bag A contains 6 pink and 7 blue balls and another bag B contains 8 pink and 8 blue balls. A ball is drawn from the bag A and without noticing the colour it is put in bag B. Then two balls are drawn from bag B at random. Find the probability that the ball drawn is blue in colour.

7. Bag A contains 7 black and 5 pink balls and another bag B contains 8 black and 8 pink balls. A ball is drawn from the bag A and without noticing the colour it is put in bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour.

ASSIGNMENT – 18

PROBABILITY

PROBABILITY DISTRIBUTION

1. 2 cards are drawn successively with the replacement from a well shuffled deck of 52 cards. Find the probability distribution of the number of Aces.

2. 3 cards are drawn one by one with replacement. Find the probability distribution of number of queens.

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3. From a lot of 30 bulbs which include 6 defective bulbs, a sample of 3 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

4. 2 cards are drawn successively with the replacement from a well shuffled deck of 52 cards. Find the probability distribution of number of Jacks. Find the mean and variance.

5. A pair of dice is tossed twice. If the random variable X defined as the number of doublets. Find the probability distribution of X, also find the mean and variance.

6. Find the probability distribution of number of success of a die, where a success is defined as a number less than 3, also find the mean and the variance.

7. Find the mean and variance for the following probability distribution:

x i❑ 0 1 2 3P(x) 1/6 ½ 3/10 1/30


x i❑ 0 1 2 3P(x) 1/8 3/8 3/8 3/8


x i❑ 0 1 2 3P(x) 1/64 18/64 18/64 27/64

10. A pair of dice is thrown four times. If getting a doublet is considered a success, find the probability distribution of number of success.

11. Find the probability distribution of number of heads when 3 coins are tossed.

12. A box contains 13 bulbs. Out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement from the box. Find the probability distribution of number of defective bulbs.

ASSIGNMENT- 19

PROBABILITY

(PROBLEMS WITH RESPECT TO CARDS )

1. A card drawn at random from a pack of 52 cards. Find the probability that is neither an ace nor a king.

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2. A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that it is neither a king nor a heart.

3. Find the probability of drawing a diamond card in each of 2 consecutive draws from a well shuffled pack of 52 cards.

(i) If the card drawn is not replaced after the first draw.

(ii) If the card drawn is replaced after the first draw.

4. 2 cards are drawn from a well shuffled pack of 52 cards one after the other without replacement. Find the probability that one of these is a queen and the other is a king of opposite order.

5. 2 cards are drawn without replacement from a well shuffled pack of 52 cards. What is the probability that is queen of red color and the other is a king of black.

6. 2 cards are drawn from a well shuffled pack of 52 cards one after the after the other without replacement. Find the probability that neither a jack nor a card of spades is drawn.

7. Two cards are drawn one by one without replacement from a well shuffled pack of 52 cards. What is the probability that they are same suit?

8. Two cards are drawn one by one without replacement from a replacement from a well shuffled pack of 52 cards. What is the probability that one is the red card and other is a black card.

9. Four cards are drawn successively one after the other from a well-shuffled pack of 52 cards. If the cards are not replaced, find the probability that all of them are kings.

10. Four cards are drawn successively one after the other from a well-shuffled pack of 52 cards. If the cards are not replaced, find the probability that all of them are of same colour.

ASSIGNMENT – 20

PROBABILITY

(BINOMIAL THEOREM)

1. A pair of dice is thrown 7 times. If getting a total of 7 considering a success. Find the probability of,

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(i)No success(ii) At least 6 success

2. The mean and variance of a binomial distribution and 4 and 4/3 respectively. Find the distribution and P(X≥1).

3. Six coins are tossed simultaneously. Find the probability of getting, (i)3 heads (ii)no heads (iii)at least one head

4. A coin is tossed 4 times. Find the mean and variance of the probability distribution of number of heads.

5. The mean and variance of a binomial distribution and 9 and 6 respectively. Find the distribution.

6. The probability of hitting a target by A is 1/5. If he fires 5 times find the probability that he will hit atleast 2 times.

7. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of getting,

(i)at least 2 successes (ii)exactly 2 successes

8. 5 dice are thrown simultaneously. If the occurrence of an even number in a single dice is success, find the probability of at most 3 successes.

9. There 6% defective items in a large bulk of items. Find the probability that a sample of 8 items will include not mere than one defective item.

10. A pair of dice is thrown 3 times. If getting a total of 10 is considered a success, find the probability of

(i)exactly 2 success

(ii)at least 2 successes

11. A pair of dice is thrown 3 times. If getting a total of 7 is considered a success, find the probability distribution of number of successes.

12. An unbiased die is tossed 3 times. Getting 3 or 5 is considered a success. Find the probability of at least 2 successes

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13. If getting a ‘5’ or a ‘6’ in a thrown of an unbiased die is a success and the random variable ‘x’ denotes th number of successes in 6 throws of the die, find P(X≤4).

14. 5 cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that,

(i)all the 5 cards are spades (ii)only 3 cards are spades (iii) none is spade

15. The probability that a bulb produced by a factory will fuse after 100 days of use is 0.05. Find the probability that out of 5 such bulbs,

(i)none (ii)not more than 1 (iii)more than one (iv)at least one will fuse after 100 days of use

16. The probability of a man hitting a target is ¼. If he fires 7 times what is the probability of his hitting the target at least twice.

17. A die is thrown 7 times. If getting an “even number” is success, find the probability of getting at least 6 successes.

ASSIGNMENT – 21

PROBABILITY

(BALLS, NUMBER, TICKET PROBLEMS)

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1. From a bag containing 20 tickets, numbered from 1 to 20, 2 tickets are drawn at random, Find the probability that,

(i)both the tickets have prime number on them. (ii)on one there is prime number and on the other there is a multiple of four.

2. A bag contains 8red, 3white and 9blue balls. If 3 balls are drawn at random determine that,

(i) all the 3 balls are blue balls. (ii) all the balls are of different colours.

3. A box contains 100 bolts and 50 nuts. It is given that 50% of bolts and 50% of nuts are rusted. 2 objects are selected from the box at random. Find the probability that both are bolts or both are rusted.

4. A bag contains 5 red, 6 white and black balls. 2 balls are drawn at random. What is the probability that both are red or both are black?

5. 2 balls are drawn at random with the replacement form a box containing 10 black and 8 red balls. Find the probability that,

(i)both balls are red (ii)first ball is black while the second is red (iii)one of them is black and other is red

6. A class consists of 10 boys and 8 girls. 3 students are selected at random. Find the probability that the selected group has,

(i) all boys (ii) all girls (iii) 2 boys and 1 girl

7. In a group of 9 students, there are 5 boys and 4 girls. A team of 4 students is to be selected for a quiz competition. Find the probability that there will be 2 boys and 2girls in each team

8. An urn contains 6 red and 5 blue balls. 2 balls are drawn at random with replacement. Find the probability of getting

(i)2 red balls (ii)2 blue balls (iii) 1 red and 1 blue ball9. Bag A contains 6 red and 5 blue balls and bag B contains 5 red and 8 blue balls, bag

C contains 6 blue and 5 red balls. One ball is drawn at random from each bag. Find the probability that all the 3 balls are of the same colour.

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10. A bag contains 30 tickets, numbered from 1 to 30. 5 tickets are drawn at random and arranged in ascending order. Find the probability that the third ticket is 20.

11. 3 bags contains 5 white, 8 red, 7 white, 6 red and 6 white and 5 red balls respectively. One ball is drawn at random. Find the probability that all the three balls drawn are of the same color.

12. 2 balls are drawn at random from a bag containing 3 white, 3 red, 4 green, and 4 black balls, one by one without replacement. Find the probability that both the balls are of different colors.

13. A bag containing 5 white, 7 red and 3 black balls. If 3 balls are drawn one by one without replacement, find the probability that none is red.

14. In a group of students, there are 3 boys and 3 girls. 4 students are to be selected at random from the group. Find the probability that either 1 girl and 3 girls and 1 boy are selected.

15. 4 digit numbers are formed by using the digits 1, 2, 3, 4 and 5 without repeating thedigits. Find the probability that a number chosen at random is an odd number.

16. Cards are numbered 1 to 25. 2 cards are drawn at random. Find the probability that the number on one card is a multiple of 7 and on the other, it is a multiple of 11.

17. A bag contains 25 tickets numbered from 1 to 25. A ticket is drawn and without replacement another ticket is drawn. Find the probability that both tickets will show even numbers.

18. There are 4 letters and 4 addressed envelope. Find the probability that all the letters are not dispatched in right envelopes.

ASSIGNMENT – 22

PROBABILITY

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(INFINITE GP PROBLELMS)

1. A and B can throw a die alternately till one of them gets a number greater than 4 and

Wins the game. Find their respective probabilities of winning, if A starts the game.

2. A and B toss a coin alternately till one of them gets a head and wins the game. If A starts first Find the probability that B will win the game.

3. A and B throw 2 dice simultaneously turn by turn. A will win the game if he throws a total of 5, B will win if he throws a doublet. Find the probability that B will win the game, though A started the game.

4. A and B throw a die alternately till one of them gets a number 6 and wins the game. Find their respective probabilities of winning, if A starts the game.

5. 3 persons A,B,C throw a dice in succession till one gets a six and wins. Find their respective probabilities of winning, if A starts the game followed by B and C.

6. A, B, C play a game and chances of their winning are1/2 , ¼ , 2/3 respectively. Find their respective probabilities of winning, if A starts the game followed by B and C.

7. A,B,C in order toss a coin. The first one to throw a head wins. Find their respective probabilities of winning, if A starts the game followed by B and C.

8. A and B take turns to throw a pair of dice. A wins if he throws 6 before B throws 7, and B wins if he throws 7 before A throws a 6. Show that if A has the first throw, their chances of winning are in the ratio 30:31.

9. A and B take turns in throwing 2 dice, the first to throw 9 being awarded the prize.. Show that if A has the first throw, their chances of winning are in the ratio 9:8.

10. A and B throw a die alternately till one of them gets a doublet and wins the game. Find their respective probabilities of winning, if A starts the game.

VECTOR ALGEBRA

IMPORTANT RESULTS

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Vector is a physical quantity that has got both magnitude and direction. According to triangle law of addition,

AB=OB−OA

1. UNIT VECTOR: If the modulus of given vector is equal to one we call it as unit vector.

a= a|a|

2. DIRECTION COSINES: The cosine of angle made by vector OBwith positive x, y and z axis i.e., cos α , cos β ,cos γ are called as direction cosines. Direction cosine can be represented by l, m and n.

l=cos α= x

√x2+ y2+ z2

m=cos β= y

√x2+ y2+ z2

n=cos γ= z

√ x2+ y2+z2

l2+m2+n2=1

3. DIRECTION RATIOS: Direction ratios are nos. proportional to direction cosines. Direction ratios of a line joining 2 pts. (x1,y1,z1) and (x2,y2,z2) is

(x2-x1 , y2-y1 , z2-z1)

4. Condition for 2 vectors a1 i+b1 j+c1 k and a2 i+b2 j+c2 k to be collinear and parallel :a1

a2

=b1

b2

=c1

c2

5. Section formula:

Internal division :OP= l b+m al+m

External division :OP= l b−m al−m

6. Midpoint formula: OP= a+ b2

Scalar/Dot product:

7. a . b=|a||b|cosθ

8. cosθ= a . b

|a|.|b|

9. i . i= j . j=k . k=1

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10.i . j= j . k= k . i=0

11.a . b=b . a

12. Condition for two vectors to be perpendicular : a . b=0

13. Projection of a on b : a . b

|b|Vector/Cross product:

14. a x b=|a||b|sin θ n,wheren is the unit vector acting ⊥ r to both a&b.

15.sin θ=|a x b||a|.|b|

16.i x i= j x j=k x k=0

17.i x j= k ; j x k=i ; k x i= j

18.a x b≠ b x a ; a x b=− b x a

19. Condition for 2 vectors to be parallel: a x b=0. 20. Area of a triangle whose adj. sides are a&b :

A=12|a x b|sq .units

21. Area of a parallelogram whose adj. sides are a&b :

A=|a x b|sq .units

22. Area of quadrilateral whose diagonal are d1&d2 :

A=12|d1 x d2|sq .units

23. Unit vector acting ⊥ r to both a&b :

n=±a x b

|a x b|

ASSIGNMENT – 23

Vector algebra

1. Find the magnitude of the vector a= 3i - 2 j + 6k .2. Find the unit vector in the direction of 3i- 6 j + 2k .3. Show that the three points A(-2,3,5) , B(1,2,3) and C(7,0,-1) are collinear.

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4. If a = 3i - 2 j + k and b = 2i - 4 j - 3kfind |a−2 b|.5. If a = i + 2 j + 3k and b = 2i + 4 j - 5k epresent two adjacent sides of a parallelogram,

find unit vectors parallel to the diagonals of the parallelogram.6. Find a.b ifa = 2i + 2 j - k and b = 6i - 3 j + 2k .7. Find the value of λ so that the vectors a = 2i + λ j + k and b = i - 2 j + 3k are

perpendicular to each other.8. Find the projection of 7i + j - 4k on 2i + 6 j + 3k .9. Find the angle between the vectors 5i + 3 j + 4k and 6i - 8 j - k .10. If a and b are the unit vectors inclined at an angle θ, then prove that,

a. Sinθ/2 = 12|a−b| (b) Cos θ/2 =

12|a+ b|(c)Tan θ/2 =

12|a−b||a+ b|

11. Let a , b , c be three vectors of magnitude 3,4,5 respectively. If each one is

perpendicular to the sum of the other two vectors, prove that |a+b+ c| = 5√2

12. If a , b , c are three mutually perpendicular unit vectors, then prove that|a+b+ c| = √3 .

13. If the vertices of a triangle are A(1,2,3), B(-1,0,0) and C(0,1,2) respectively, what is the magnitude of ∠ ABCand ∠BCA

14. Find a × b, if a = 2i + k and b = i+ j + k .15. Find a unit vector perpendicular to both the vectors a = i - 2 j + 3k and b = i + 2 j - k .16. Find a vector of magnitude 9, which is perpendicular to both the vectors

a = 4i - j + 3k and b = -2i + j - 2k .

17. Given |a| = 10, |b| = 2 and a.b = 12, find |a× b|.18. Find the area of the triangle whose vertices are (3,-1,2), (1,-1,-3), (4,-3,1).19. Find the area of the parallelogram determined by the vectors i + 2 j + 3k and 3i - 2 j+ k .20. Show that the vectors a = 3i - 2 j + k , b= i - 3 j + 5k and c = 2i + j - 4k form a right

angle triangle.

ASSIGNMENT – 24

Vector algebra

1. Prove that (a¿×b)2=|a|2|b|

2−( a .b )2¿

2. Find the angle between a+b and a− b if a = 2i - j + 3 k and b= 3 i+ j-2k

3. Find the value of λ so that the vectors 3i -6 j-3 k and i -2 j + λ k are (i) parallel (ii) perpendicular.

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4. The dot product of a vector with the vectors i + j -3k , i +3 j -2k , 2 i + j +4k are 0,5,8 respectively. Find the vector.

5. Show that the vectors a = 3i - 2 j + k , b= i - 3 j + 5k and c = 2i + j - 4k form a right angle triangle.

6. If a = 3i+2 j+ 9 k , b= i+λj + 3k , find the value of λ so that a+ b and a−b are perpendicular to each other.

7. Express the vector a = 5i-2 j + 5 k as sum of 2 vectors so that one is parallel and other is perpendicular to b=3i+ k ,

8. Find the projection of b+ con a if a = i+2 j+ k , b= i+ 3 j + k and c = i + k .

9. Find the projection of b+ c on a if a = 2 i-2 j+ k , b= i+2 j-2k andc =2i- j+ 4 k.

10. Find the projection of b+ c on a if a = 7 i- j +8k , b= i+2 j+3k and c =2i 4 k .

11. Find the angle between a = i- j +k , b= i+ j-k

12. Find the value of λ so that the vectors 2i+λ j+k and i -2 j + 3 k are perpendicular to each other.

13. If a = i+ j +k , b= j-k , find a vector c suchthat a x c=b∧a . c=3

14. If a+ b+c=0, |a| = 3, |b| = 5and |c|=7. Show that the angle between a∧b is 60o

3-D GEOMETRY

IMPORTANT RESULTS

I. Equatio of a Straight line passing through a (a point) and parallel to a vector

b∈the vector form : r = a + λ b ¿ the cartesian form :x−x1

l = y− y1

m = z−z1

n

Same formula is used for:

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Eqn. of a line passing through a pt. and parallel to a vector joining two pts. (x1,y1,z1) ; (x2,y2,z2)

b = (x2-x1)i + (y2-y1)j +(z2-z1)k Eqn. Of a line passing through a pt. and parallel to a line

x−x1

l = y− y1

m = z−z1

nb = liᶺ + mjᶺ + nkᶺ

Eqn. of a line passing through a pt. and perpendicular to plane ax + by + cz + d = 0b = aiᶺ + bjᶺ + ckᶺ

II. Eqn. of a St.line passing through 2 pts. A & B ⟹ r = a + λ (b−a)

III. Angle between 2 lines r1 = a1 + λ b1

r2 = a2 + μ b2

cosθ = | b1 . b2

|b1|−|b2|| If b1.b2 = 0, θ = 90o⟹ l1⊥ l2 ; If they are proportional, then l1∥ l2

IV. Shortest distance b/w Skew lines

S.D = |(a2−a1 ) .(b1 X b2)

|b1 X b2| | r=a1+λ b1 r=a2+λ b2

Parallel lines

S.D = |(a2−a1 ) X b

|b| | Condition for 2 lines to intersect

⟹S .D=0

V. To prove 3 pts. are collinear (A , B , C) First find AB∧BC. They will be proportional ( AB= λ BC ¿ .

This indicates that they are either parallel or collinear. In the end, write

Since B is a common pt. , AB∧BC cannot be parallel.

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Therefore they are collinear.

VI. To prove 2 lines are intersecting and to find the pt. of intersection :x−x1

a1

=y− y1

b1

=z−z1

c1

.

First, prove S.D = 0 ; x−x2

a2

=y− y2

b2

=z−z2

c2

.

The equate the lines with λ∧μ .

x−x1

a1

=y− y1

b1

=z−z1

c1

= λ

x−x2

a2

=y− y2

b2

=z−z2

c2

=μ

x=a1 λ+ x1 x=a2 μ+x2

y=b1 λ+ y1 y=b2 μ+ y2

z=c1 λ+z1 z=c2 μ+z2

Equate the values of x,y,z & find value of λ∧μ

Then substitute and find the values of (x,y,z) i.e., the pt. of intersection.

VII. To find the image of the pt. (x1,y1,z1) on the line x−x2

a1

=y− y2

b1

=z−z2

c1

Let B a pt. on the line. B can be found by equating the line with λ.

x−x2

a1

=y− y2

b1

=z−z2

c1

= λ

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x=a1 λ+ x2 y=b1 λ+ y2 z=c1 λ+z2

Therefore (x , y , z) = pt. B Then find the dr of AB , ( B− A)

= (x-x1 ; y-y1 ; z-z1) Dr of CD : (a1 , b1 , c1) Then AB .CD=0 (since they are ⊥) Substitute in the pt. B therefore we get the foot of the perpendicular. Using mid-pt. formula find the pt. E (image)

VIII. Eqn. of a line thro’ a pt. (x1 , y1 , z1) and ⊥r to two lines. First find the cross product of two lines, the vector formed will be parallel

vector (b) and we already have the pt. (a). Then apply the normal formula, r = a+ λ b.

IX. To find the co-ordinates of a pt. on the line x−x1

a1

=y− y1

b1

=z−z1

c1 which are at

dist. of ‘k’ units from (p , q , r) First find the pt. B by equating the line eqn. withλ.

x−x1

a1

=y− y1

b1

=z−z1

c1

= λ

x=a1 λ+ x1

y=b1 λ+ y1

z=c1 λ+z1∴ Pt . B⟹(x , y , z)

Use the distance formula,k = √¿¿

Now square both sides and find the values of λ (both+ve∧−ve). Then substitute and find pt. B

B⟶ is the req . pt .

X. To prove that 2 lines are skew lines. Just show that S . D ≠ 0.

PLANES

I. Eqn. of a plane whose ⊥ r distance from the origin ‘p’ and unit normal vector nᶺ

r .nᶺ = p

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General eqn. of a plane :ax + by + cz + d = 0 where (a , b , c) are dr’s acting ⊥ r to plane.

II. Eqn. of a plane whose ⊥ r distance ‘p’ and normal n :

r .n

|n|=p

III. Eqn. of a plane passing thro’ a pt. a and ⊥ r to n : n=a . n : V.E ( r−a ) . n=0

l(x-x1) + m(y-y1) + n(z-z1) = 0 : C.F l,m,n are dr’sIV. Eqn. of a plane passing through 3 pts. :

| x−x1 y− y1 z−z1

x2−x1 y2− y1 z2−z1

x3−x1 y3− y1 z3−z1|=0

V. Eqn. of a plane passing through 2 pts. and ⊥ r to a plane :

| x−x1 y− y1 z−z1

x2−x1 y2− y1 z2−z1

l m n |= 0

l , m , n ⟶ dr’s of the plane

VI. Eqn. of plane passing through 2 pts. and parallel to line x−x1

l = y− y1

m =

z−z1

n :

| x−x1 y− y1 z−z1

x3−x2 y3− y2 z3−z2

l m n |= 0

2 pts.: (x2 , y2 , z2) ; (x3 , y3 , z3)

VII. Eqn. of a plane passing through a pt. (x1 , y1 , z1) and ⊥ r to 2 planes :

|x−x1 y− y1 z−z1

l1 m1 n1

l2 m2 n2|=0

VIII. Eqn. of line passing through a pt. (x1 , y1 , z1) and ∥ to 2 lines :

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|x−x1 y− y1 z−z1

l1 m1 n1

l2 m2 n2|=0

(l1 , m1 , n1) = dr. of 1st line (l2 , m2 , n2) = dr. of 2nd line

IX. Eqn. of plane passing through pt. , ⊥ r to a plane and ∥ to a line :

|x−x1 y− y1 z−z1

l1 m1 n1

l2 m2 n2|=0

(l1 , m1 , n1) = dr. of the plane (l2 , m2 , n2) = dr. of the line

X. Intercept form :

xa+ y

b+ z

c=1

XI. Angle b/w 2 planes : a1x + b1y + c1z + d1 = 0 a2x + b2y + c2z + d2 = 0

cosθ=¿

If the planes are ⊥ r : a1a2 + b1b2 + c1c2 = 0

∥el :a1

a2

=b1

b2

=c1

c2

XII. Angle between a line and a plane :

x−x1

a1

=y− y1

b1

=z−z1

c1

a2x + b2y + c2z + d2 = 0

sin θ=a1 a2+b1 b2+c1 c2

√a12+b1

2+¿c12 .√a2

2+b22+c2

2 ¿

XIII. ⊥ r distance from a pt. (x1 , y1 , z1) to a plane ax + by + cz + d =0 :

p=|a x1+b y1+c z1+d

√a2+b2+c2 |XIV. ⊥ r distance from the origin to the plane 3x + 4y + 5z – 3 = 0 p=| d

√a2+b2+c2|42 | P a g e

XV. Distance b/w 2 parallel planes : ax + by + cz + d1 = 0 ax + by + cz + d2 = 0

p=| d1−d2

√a2+b2+c2|XVI. Eqn. of a plane through intersection of 2 planes P1& P2 :

P1+λ P2=0

A) If the pt. is given, substitute (x , y , z) in the eqn. and find λ. Thereafter substitute λ and find the eqn. of plane.

B) If it is ⊥ r to another plane P1+λ P2⟹P

= P ⊥ r P3

= P . P3 = 0⟹a1 a2+b1b2+c1 c2=0

C) If it is parallel to line x−x1

a2

=y− y1

b2

=z−z1

c2 :

P⟹P1+λ P2 P ∥ l

a1a2+b1b2+c1c2=0

(a1 , b1 , c1) = dr. of P (a2 , b2 , c2) = dr. of line

XVII. Image of a pt. (p , q , r) on the plane ax + by + cz + d = 0 Dr of line AB : (a , b , c) Now the eqn. of AB can be found because we have a pt. A and the dr’s also.

Eqn. = x−p

a= y−q

b= z−r

c

Equate the above equation with λ, And express x , y , z in terms of λ = Pt. B

Since the pt. B lies on the plane substitute (x , y , z) in the plane eqn. and find the value of λ.

Sub the value of λ in (x , y , z) ⇒ this is the foot of the ⊥ r. Using the mid-pt. formula we can find the coordinates of the pt. E.

43 | P a g e

Using the distance formula the length of the ⊥ r can also be found.

XVIII. To prove lines are co-planar and to find the eqn. of the plane containing them.

First prove S.D = 0 Then eqn. of plane

⟹|x−x1 y− y1 z−z1

l1 m1 n1

l2 m2 n2|=0

(x-x1) , (y-y1) , (z-z1) = can be from any of lines (l1 , m1 , n1) = dr. of 1st line

(l2 , m2 , n2) = dr. of 2nd line

XIX. Distance of a pt. (p , q , r) from the plane a1x + b1y + c1z + d = 0 and

measured parallel to the line x−x1

a2

=y− y1

b2

=z−z1

c2

Dr of AB : (a2 , b2 , c2) Now find the eqn. of AB (the pt. and dr’s are given). Eqn. of AB:

x−pa2

= y−qb2

= z−rc2

Now equate the above eqn with λ and express x , y , z. This is the pt. B. Sub the values of (x , y , z) in the plane eqn. and find the value of λ. Sub the value of λ in (x , y , z) and hence find the pt. B. Using distance formula, find the length of AB.

XX. Distance of the pt. (p , q , r) from the line x−x1

a1

=y− y1

b1

=z−z1

c1and

measured parallel to the plane ax + by + cz + d = 0. Equate the line eqn. with λ and express x , y , z in terms of λ, this is pt.B.

44 | P a g e

Dr. of AB : Pt.B Pt.A Dr. of EF : (a , b , c) AB ⊥ r EF∴ AB . EF=0

Hence find the value of λ.

Sub in x , y , z and find the pt.B. Using distance formula, we can find the distance b/w A and B.

ASSIGNMENT – 25

Three dimensional geometry

1. Find the equations of the line passing through the points,a. (3,2,5) and (-1,0,3)b. (4,-6,1) and (0,3,-1)c. (a,a,a) and (a,0,a)

2. Find the equation of the line parallel to the line x−1

5 =

3− y2

= z+1

4and passing

through the point (3, 0,-4).

3. Find the equation of the line passing through the point (1, 2,-4) and perpendicular to

the lines x−8

3 =

y+19−16

= z−10

7 and

x−153

= y−29

8 =

z−5−5

.

45 | P a g e

4. Find the vector equation of the line through the points (3, 1, 2) and perpendicular to the plane whose vector equation is r .(2i - j +k) -2 = 0.

5. Find the vector equation of the line passing through the points 2i - 3 j - 5k and perpendicular to the plane whose vector equation is r .(6i - 3 j - 5k) + 2 = 0.

6. Find the vector equation of the line through the points 2i - j + k and parallel to the line joining the 2 points whose vectors are -i + 4 j + k and i + 2 j + 2k . Also find the Cartesian equation.

7. The Cartesian equations of a line are 3x + 1 = 6y - 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its direction cosines.

8. The Cartesian equations of a line are 6x – 2 = 3y + 1 = 2z – 2. Find ,a. The direction ratio of line.b. Cartesian and vector equation of line parallel to this line passing through the point

(2, -1, -1).

9. Find the vector equation of the line through the point whose position vector is i - j + 2k and perpendicular to the plane whose vector equation is r .(2i - j +3 k) = 5.

10. Find the vector equation of the line through the points 2i - j + k and parallel to the line joining the 2 points whose vectors are - i + 2 j + 4k and i - 2 j - 2k . Also find the Cartesian equation.

11. Find the vector equation of the line through the pointsi - 3 j - 2k and parallel to the line joining the 2 points whose vectors are i - j + 4k and 2i + j + 2k . Also find the Cartesian equation.

12. A line passes through the point with position vectors 2i - j + 4k and in the direction i+ j - 2k . Find the equation of line in the vector and the Cartesian form.

13. Find the coordinates of the point where the line joining (3, 4, 1) and (6, 1, 6) crosses the xy plane.

14. Find the vector equation of the line through the origin and perpendicular to the plane whose vector equation is r .(i + 2 j +k) = 3

15. Find the vector and Cartesian equations of the line passing through (3, 0, 1) and parallel to the planes x+2y = 0 and 3y-z = 0.

16. Find the vector and Cartesian equations of the line passing through (2, -1, 1) and parallel to the line joining the points (-1, 4, 1) and (1, 2, 2).

46 | P a g e

17. Find the vector and Cartesian equations of the line passing through (2, -3, 7) and parallel to the line joining the points (-2, -2, 0) and (1,-2,-3)

18. Find the vector and Cartesian equations of the line passing through (3,-1, 2) and parallel to the planes x+y+2z-4=0 and 2x-3y+z+3 = 0.

19. Find the vector equation of the line through the origin and perpendicular to the plane whose vector equation is r .(-3i - 4 j + 2 k) = 31

20. Find the equation of the line passing through the point (-1, 3, -4) and perpendicular

to the lines x−2

1 =

y−12−3

=z+4

7 and

x−15−3

= y−9

3 =

z+5−5

.

21. Find the vector & Cartesian forms of the equation of the plane passing through the

point(1,2,3) and perpendicular to the lines x−3

7 =

1− y2

= z−3

7

22. Find the equation of a line passing through (1,2,3) and parallel to the planes x-y+2z=5

and 3x+y+z=6

23.Find the vector & Cartesian forms of the equation of the line passing through the point

(1,3,-2) and parallel to the planes 2x+4y+5z=1 and x-3y+2z=3

ASSIGNMENT – 26

Three dimensional geometry

(SHORTEST DISTANCE BETWEEN 2 LINES)

1)Determine whether or not the following pairs of lines intersect.If these intersect then

find the point of intersection,otherwise the shortest distance between them.

a) x+1

3 =

y+35

= z+5

7 and

x−21

= y−4

3 =

z−65

b)x−1

3 =

y−5−2

= z−7

1 and

x+17

= y+1−6

= z+1

1

c)x−3

1 =

y−5−2

= z−7

1 and

x+17

= y+1−6

= z+1

1

47 | P a g e

d)x−3

3 =

y−8−1

= z−3

1 and

x+3−3

= y+7

2 =

z−64

e)r = (î-7ĵ-2k) +μ(î+3ĵ+2k) and r=(3-λ)î-(4+2λ)ĵ+(λ-2)k

f) r=(-î+5ĵ)+λ(-î+ĵ+k) and r=(-î+3ĵ+2k)+μ(3î+2ĵ+k)

g) r=(î-2ĵ+3k)+λ(-î+ĵ-2k) and r=(î-ĵ-3k)+μ(î+2ĵ-2k)

h) r=(λ-1)î+(1+λ)ĵ+(λ-1)k and r=(1-μ)î-(1-μ)ĵ+(1-μ)k

i) r=(î+2ĵ+3k)+λ(î-3ĵ+2k) and r=(4î+5ĵ+6k)+μ(2î+3ĵ+k)

j) r=(1-t)î+(t-2)ĵ+(3-2t)k and r=(s+1)î+(2s-1)ĵ-(2s+1)k

2) Show that the lines x+1

3 =

y+35

=z+5

7 and

x−21

= y−4

3 =

z−65

intersect each other.

Also find the point of intersection.

3) Determine whether or not the following pairs of lines intersect.If these intersect then

find the point of intersection,otherwise the shortest distance between them.

r=(î+ĵ-k)+λ(3î-ĵ) and r=(4î-k)+μ(2î+3k)

ASSIGNMENT – 27

3-D GEOMETRY

(IMAGE, FOOT OF THE PERPENDICULAR, LENGTH OF THE PERPENDICULAR FROM A POINT TO A PLANE)

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

2. Find the coordinates of the image of the point (1,2,3) in the plane x+2y+4z=38.

3. Find the coordinates of the image of the point (1,3,4) in the plane x-y+z-5=0.

48 | P a g e

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

5. Find the coordinates of the image of the point (2,7,5) in the plane 4x-7y+3z-11=0.

6. From the point (1,2,4), a perpendicular is drawn on the plane 2x+y-2z+3=0. Find the equation, the length and the coordinates of the foot of the perpendicular.

7. Find the length and the coordinates of the foot of the perpendicular from the point (1,1,2) to the plane 2x-2y+4x+5=0

8. Find the distance of the point (3,3,3) from the plane 5x+2y-7z+9=0.

9. Find the reflection of the point (-1,-1,3) on the plane 2x+3y-4z-10=0

10. Find the distance of the point (7,14,5) to the plane 2x+4y-z=2

ASSIGNMENT – 28

3-D GEOMETRY

(IMAGE, FOOT OF THE PERPENDICULAR, LENGTH OF THE PERPENDICULAR FROM A POINT TO A LINE)

1. Find the coordinates of the foot of the perpendicular drawn from the point A(1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1).

49 | P a g e

2. Find the length of the perpendicular drawn from the point (2,-1,5) to the line x−11

10 =

y+2−4

= z+8−11

3. Find the foot of the perpendicular drawn from the point (1,6,3) to the line x1

= y−1

2 =

z−23

4. Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining the points B(14,6) and C(5, 4,4).

5. A(0,6,-9), B(-3,-6,3) and C(7,4,-1) are 3 points. Find the equation of the line AB. If D is the foot of the perpendicular drawn from C to the line AB, find the coordinates of D.

6. A(-3,-2,-1`), B(3,4,2) and C(4,1,-) are 3 points. Find the equation of the line AB. If D is the foot of the perpendicular drawn from C to the line AB, find the coordinates of D.

7. Find the foot of the perpendicular drawn from the point (0,2,7) to the line x+2−1

= y−1

3 =

z−3−4

8. Find the image of the points (1,6,3) on the line x1

= y−1

2 =

z−23

9. Find the foot of the perpendicular drawn from the point (0,2,3) to the line x+3

5 =

y−12

=

z+43

10. Find the equation of perpendicular from the point (3,-1,11) to the line x2

= y−2

3 =

z−34

11. Find the length of the perpendicular drawn from the point (-1,3,9) to the line

x−135

= y+8−8

= z−31

1

ASSIGNMENT-29

50 | P a g e

3-D GEOMETRY

(EQUATION OF A PLANE PASSING THROUGH 2 POINTS AND PERPENDICULAR TO A PLANE/ PARALLEL TO LINE)

1. Find the equation of a line passing through the points (1,-1,2) & (2,-2,2), perpendicular to the plane 6x-2y+2z=9.

2. Find the equation of the plane passing through the points (3,4,1) & (0,1,0) & parallel to

a line x+3

2 =

y−37

= z−2

5

3. Find the equation of a plane passing through the points :i) (1,1,0),(1,6,5) perpendicular to the plane 5x+3y-17z=10.ii) (2,-3,1),(5,2,1) perpendicular to the plane x-4y+5z+2=0.iii) (2,1,-1),(-1,3,4) perpendicular to the plane x-2y+4z=10.iv) (2,2,-1),(3,4,2) parallel to the line whose direction ratios are 7:0:6v) (1,-1,1),(0,3,3) perpendicular to x+2y+2z=5.vi) (3,4,2),(7,0,6) perpendicular to 2x-5y=15.vii) (1,0,-1),(3,2,2) parallel to the line r=î+ĵ+λ(î-2 ĵ+3k)

viii) (1,2,-3),(3,0,5) & parallel to the line x3

= y

−2 = z

4. Find the equation of the PLANE passing through the points :

i) (0,-1,0),(1,1,1),(3,3,0)ii) (0,-1,-1),(4,5,1),(3,9,4)

5. Find the equation of the plane throughthe following points

i) (1,1,1),(1,-1,1),(-7,3,-5)ii) (1,3,4),(2,1,-1),(1,4,3)iii) (1,0,-1),(2,1,3),(3,2,1)iv) (-1,0,1),(-1,4,2),(2,4,1)

6. Show that the following points are coplanar

i) (0,-1,-1),(4,5,1),(3,9,4),(-4,4,4)ii) (1,0,1),(2,4,0),(3,1,2),(0,-4,2)iii) (3,1,2),(0,4,0),(1,1,1),(-1,1,0)

ASSIGNMENT-3051 | P a g e

3D-GEOMETRY

(ANGLE BETWEEN 2 LINES/ 2PLANES/ LINE AND PLANE)

1) Find the angle between the 2 lines r =2î-5ĵ+k+λ(3î+2ĵ+6k) ; r=7î-6k+μ(î+2ĵ+2k)2) Find the angle between the line r= (î-ĵ+k) + λ(2î-ĵ+3k) & the plane r.(2î+ĵ-k)=4

3) Find the angle between the line and the plane in the following problems.

i)r=(5î-ĵ-4k)+λ(2î-ĵ+3k) , r.(3î+4ĵ+k)+5=0 .

ii¿ r=(2î-ĵ+3k)+λ(3î-ĵ+2k) , r.(î+ĵ+k)=3

iii) x−2

3 =

y+1−1

= z−3

2, 3x+4y+z+5=0

iv) x−3

2 =

y−11

= z+4−2

, x+y+4=0

v) x2

= y1

= z

−2 ; x+y+4=0

vi) x+1

3 =

y+12

= z−2−1

; x-2y-z+13=0

4. Find the angle between the following planes:

i) 2x-3y+4z=1; -x+y=4

ii) x+2y+3z+1=0 ; 3x-y+5=0

iii) x+2y+2z-3=0 ; 3x+4y+5z+1=0

iv) 2x+y-z+3=0 ; x+3y+5z+1=0

5. Find the angle between the following lines:

i) x = y = z0

; x3

= y

−4 =

z5

3D-GEOMETRY

52 | P a g e

ASSIGNMENT-31

PLANE PASSING THROUGH THE LINE OF INTERSECTION OF 2 PLANES

1) Find the equation of the plane passing through the line of intersection of the planes x-2y+z=1 , 2x+y+z=8 and parallel to the line with the direction ratio 1,2,3. Also find the perpendicular distance of the point (3,1,2) from this plane .

2) Find the Cartesian and vector equation of the plane passing through the intersection of the planes r .(2î+6ĵ)+12=0 and r.(3î-ĵ+4k)=0 which is at a unit distance from the origin

3) Find the Cartesian and vector equation of the plane passing through the intersection of the planes r.(2î+ĵ+3k)=7 and r.(2î+5ĵ+3k)=9 passing through the point (3,2,-1)

4) Find the Cartesian and vector equation of the plane passing through the intersection of the planes x+y+z=6 ,2x+3y+4z+5=0 and passing through the point (0,1,2)

5) Find 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

6) Find the Cartesian and vector equation of the plane passing through the intersection of the planes r.(î+ĵ+k)=6 and r.(2î+3ĵ+4k)=-5 and passing through the point (1,1,1)

7) Find the Cartesian and vector equation of the plane passing through the intersection of the planes 2x-3y+z-4=0 , x-y+z+1=0 and is perpendicular to the plane x+2y-3z+6=0

8) Find the Cartesian and vector equation of the plane passing through the intersection of the planes 2x-7y+4z=3 , r.(3î-5ĵ+4k)=-11 and passing through the point (-2,1,3)

9) Find the equation of the plane through the origin and the line of intersection of the two planes , 5x-3y+2z+5=0,3x-5y-2z-7=0

10)Find the Cartesian and vector equation of the plane passing through the intersection of the planes x+3y-z+1=0,r.(3î-ĵ+5k)+3=0 and at a distance of 2/3 from the origin.

11)Find the Cartesian and vector equation of the plane passing through the intersection of

the planes 2x+y-z = 3 , 5x-3y+4z+9=0 and is parallel to a line x−1

2 =

y−34

= z−5

5

12)Find the Cartesian and vector equation of the plane passing through the intersection of

the planes x-2y+3z+1=0 , 3x+y-2z+2=0 and is parallel to a line x−3

2 =

y+1−1

= z+4

3

13)Show that the line r = (ĵ+î)+λ(2î+ j +4k ) is a parallel to a plane r.(-2î+k )=5 find the distance between the line and the plane.

14)Show that the line r = (î+ĵ)+λ(2î+ĵ+4k ) is parallel to a plane r.(î+2ĵ-k )=3 find the distance between the line and the plane. Also state whether the line lies in the plane.

15)Show that the line r = (ĵ+k +λ(4î+2ĵ+k ) is parallel to a plane r.(-î+ĵ+2k )=3 find the distance between the line and the plane

53 | P a g e

3D-GEOMETRY

ASSIGNMENT-32

(POINT OF INTERSECTION OF A LINE AND A PLANE)

1) Find the coordinates of a point where the line x+1

2 =

y+23

= z+3

4 meets the plane

x+y+4z=6

2) Find the point on the line x+2

3 =

y+12

= z−3

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

3) Find the distance between the point position vector –î-5ĵ-10k and the point of

intersection of the line x−2

3 =

y+14

= z−212

with the plane x-y+z=5

4) Find the point R, where the line joining P(1,3,4) and Q(-3,5,2) cuts the planer.(2î-ĵ+k)+3=0

5) Find the point of intersection of the line x−1

1 =

y+23

= z−1−1

and the plane 2x+y+z-9=0

6) Find the point of intersection of the line joining (1,3,-2) and(3,4,1) and the plane x-2y-4z-11=0

7) Find the distance of the point (3,4,5) from the point of intersection of x−3

1 =

y−42

=

z−52

with the plane x+y+z=2

54 | P a g e

ASSIGNMENT-33

3-D GEOMETRY

1)Show that the following lines are coplanar. Also find the equation of the plane containing them.

i)x+1−3

= y−3

2 =

z+21

& x = y−7−3

= z+7

2

ii) x+1

3 =

y+35

= z+5

7&

x−21

= y−4

4 =

z−6−7

2) Find the equation of the PLANE containing the lines r=(î+ĵ)+λ(î+2ĵ-k) & r = (î+ĵ)+μ(-î+ĵ-2k)

3)Prove that the line joining (0,2,-4) & (-1,1,-2) & the line joining (-2,3,3) & (-3,-2,1) are coplanar. Find the point of intersection.

4) Show that the following lines are coplanar. Find the point of intersection and the equation of the plane in which they lie.

i)x−5

4 =

y−74

= z−5−5

; x+8

7 =

y−41

= z−5

3

ii)x+1−3

= y−4

1=

z−53

; x+3−4

= y+1

7 =

z−41

iii)x+1

2 =

y+13

= z+1

4 ;

x−15

= y−2

8 =

z−3−7

iv) x = y−2−2

= z+1

1 ;

x−1−2

= y−2

3 =

z+2−1

v)x−3

2 =

y−2−5

= z−1

3 ;

x−1−4

= y+2

1 =

z−62

55 | P a g e

3-D GEOMETRY

ASSIGNMENT-34

(EQUATION OF A PLANE PASSING THROUGH A POINT AND PERPENDICULAR TO 2 PLANES/ PARALLEL TO 2 LINES)

1) Find the equation of the plane passing through the pt(1,1,1) & perpendicular to each of the planes x+2y+3z=7 , 2x-3y+4z=0

2) Find the equation of the plane passing through the point(-1,-1,2) perpendicular to the planes 2x+3y-3z=2 , 5x-4y+z=6

3) Find the vector & Cartesian forms of the equation of the plane passing through (1,2,-4) & parallel to the line r =î+ĵ-4k+λ(2î+3ĵ+6k) ,r=î-3ĵ+5k+μ(î+ĵ-k)

4) Find the vector & Cartesian forms of the equation of the plane passing through (-1,3,2) perpendicular to x+2y+3z=5 ; 3x+3y+z=9

5) Find the vector & Cartesian forms of the equation of the plane passing through (1,-3,5) perpendicular to 3x-y+5z=1 ; x+3y-2z=7

6) Find the equation of the plane through (3,-1,2) & parallel to each of the lines

x−12

= y+1−1

= z−3

3 and

x4

= y3

= z1

(Distance of a point from a plane/line measured parallel to line/plane)

7) Find the distance of a point(2,3,4) from the plane 3x+2y+2z+5=0 & measured parallel to the

line x+3

3 =

y−26

= z2

8) Find the distance of a point (1,-2,3) from the plane x-y+z=5 & measured parallel to the line

x2

= y3

= z

−6

9. Find the distance of the point (3,-4,3) from the plane 2x+5y-6z=16 & measured parallel to the

line x−1

2 =

y1

= z+3−2

10.Find the distance of the point (-2,3,-4) from the line x+2

3 =

2 y+34

= 3 z+4

5& measured

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

56 | P a g e

3D GEOMETRY

ASSIGNMENT-35

(EQUATION OF THE PLANE THROUGH A POINT AND PARALLEL TO A PLANE/ PERPENDICULAR TO A LINE)

1) Find the equation of the plane through the point (1,3,2) and parallel to plane 3x-2y+2z+33=0. Find the perpendicular distance of the point (3,3,2) from this plane.

2) Find the equation of the plane passing through the point (1,-2,1) and perpendicular to the line joining the points (3,2,1) and (1,4,2). Find also the perpendicular distance of the origin from this plane

3) Find the equation of the plane passing through the point (2,0,1) and perpendicular ro the line joining the points (1,2,3) and (3,-1,6)

4) Find the equation of the plane through (2,3,-1) and parallel to the plane 3x-4y+7z=0

5) Find the equation of the plane passing through the linex−3

2 =

y+29

= z−4−1

and the point

(-6,3,2)

6) Find the equation of the plane containing the line x

−2 =

y−13

= 1−z

1 and passing

through the point (-1,0,2)

7) Find the equation of the plane containing the line x+1

2 =

y+1−1

= z−3

4 and perpendicular

to the plane x+2y+z-12=08) The foot of the perpendicular drawn from the origin o the plane is (2,5,7). Find the

equation of the plane.9) Find the equation of the plane through the origin and the line of intersection of the 2

planes 5x-3y+2z+5=0 , 3x-5y-2z-7=0.10)Find the Cartesian and vector equation of the plane passing through the line of

intersection of the planes x+3y-z+1=0 ,r.(3î-ĵ+5k)+3=0 and at distance of 2/3 from the origin.

11)Find the Cartesian and vector equation of the plane passing through the line of intersection of the planes 2x+y-z=3 , 5x-3y+4z+9=0 and is parallel to a line x−1

2 =

y+1−1

= z+4

3

12)Find the Cartesian and vector equation of the plane passing through the line of intersection of the planes x-2y+3z+1=0,3x+y-2z+2=0 and is parallel to a line x−3

2 =

y+1−1

= z+4

3

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13)Show that the line r=(î+ĵ)+λ(2î+ĵ+4k) is parallel to a plane r.(-2î+k)=5 find the distance between the line and the plane.

14)Show the line r=(î+ĵ)+λ(2î+ĵ+4k) is parallel to a plane r.(î+2ĵ-k)=3 Find the distance between the line and the plane.Also state whether the line lies in the plane.

15)Show that the line r=(ĵ+k)+λ(4î+2j+k) is parallel to a plane r.(-î+ĵ+2k)=3 Find the distance between the line and the plane.

Assignment 36

CONTINUITY

1. Show that f(x) is continuous at x=1. f ( x )={ 5 x−4 ,0<¿ x≤ 14 x3−3 x , 1<¿ x<2

2. Given that f ( x )={1−cos 4 x

x2 if x<0

aif x=0√x

√16+√ x−4if x>0

if f(x) is continuous at x=0, find a.

3. If f ( x )={x2−25x−5

when x ≠5

k when x=5 is continuous at x=5, find ‘k’.

4. Find the value of ‘k’ if the function f ( x )={k x2 ,∧x≥ 14 ,∧x<1

is continuous at x=1.

5. Find the value of ‘k’ for which the function f ( x )={ kx , if ∧x≤ 2x−1 , if ∧x>2

is continuous at x=2.

6. If f(x), defined by the following, is continuous at x=0, find the values of a, b, c.

f ( x )={sin (a+1 ) x+sin x

xif x<0

c if x=0

√ x+b x2−√x

b x32

if x>0

7. For what value of ‘k’ is the function continuous at x=2?f ( x )={2 x+1if x<2k if x=2

3 x−1 if x>2

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8. If f ( x )={x2−9x−3

when x ≠3

k when x=3is continuous at x=3, find ‘k’.

9. If f ( x )={x2−2 x−3x+1

when x≠−1

k when x=−1 is continuous at x=-1, find ‘k’?

Assignment 37

CONTINUITY

1. For what value of ‘k’ is the function continuous at x=2?

f ( x )={2 x−1 if x<2k if x=2

x+1 if x>2

2. Find ‘k’ if the function is continuous at x=0?f ( x )={sin 2 x5 x

when x ≠ 0

k when x=0

3. Find ‘k’ if the function is continuous at x=π2

? f ( x )={k cos xπ−2 x

when x≠π2

3when x=π2

4. Find ‘k’ if the function is continuous at x=0? f ( x )={1−coskxx sin x

when x≠ 0

12

when x=0

5. Find ‘k’ if the function is continuous at x=0? f ( x )={sin 2 xx

when x ≠ 0

k when x=0

6. Find ‘k’ if the function is continuous at x=0?

f ( x )={1−cos 4 x

8 x2when x≠ 0

k when x=0

7. Find ‘k’ if the function is continuous at x=0?

f ( x )={1−cos 2kx

x2when x ≠ 0

8 when x=0

8. For what values of ‘a’ and ‘b’ is the function continuous at x=3 and 5?

f ( x )={ 1 if x≤ 3ax+b if 3<x<5

7 if x>5

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9. Show that the function is continuous at x=1?

f ( x )={ x2

2when0 ≤ x≤ 1

2 x2−3 x+ 32

when1<x ≤ 2

10. Find ‘p’ if f(x) is continuous at x=0?

f ( x )={√1+ px−√1−pxx

when−1 ≤ x<0

2 x+1x−2

when 0 ≤ x≤ 1

Assignment 38

CONTINUITY

1. If f(x) is continuous at x=0, find ‘a’?

f ( x )={a sinπ2

( x+1 ) when x ≤ 0

tan x−sin xx3 when x>0

2. For what value of ‘a’ and ‘b’ is the function continuous at x=3 and 5.

f ( x )={ 2 if x≤ 3ax+b if 3<x<5

9 if x≥ 5

3. For what value of ‘a’ and ‘b’ is the function continuous at x=-1 and 0.

f ( x )={ 4 if x ≤−1a x2+b if −1<x<0

cos x if x ≥ 0

4. For what value of ‘a’ and ‘b’ is the function continuous on [0,∞),

f ( x )={ x2

aif 0≤ x<1

aif 1≤ x<√22b2−4b

x2 if √2≤ x<∞

5. If f(x), defined by the following, is continuous at x=0, find the values of ‘a’?

f ( x )={1−cos 4 x

x2if x<0

a if x=02 x

√ x+4−2if x>0

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6. Find if ‘f’ is continuous at x=0. f ( x )={x sin1x

when x≠ 0

0 when x=0

7. Discuss the continuity at x=1/2?

f ( x )={12−x if 0≤ x<1/2

1 if x=1/232−x if

12< x≤ 1

8. If f ( x )={2 x+3 when x ≤2x+k when x>2

is continuous at x=2, find ‘k’?

Assignment 39

DIFFERENTIATION

1. If y=√ 1−x1+x

prove that (1-x2) y +y=0.

2. If y =√ x+ 1

√ x show that 2x

dydx

+ y=2√x .

3. If y=log √ 1−cos x1+cos x

show that dydx

=cosec x .

4. If y=log √ 1+sin2 x1−tan x

finddydx

.

5. If (cos x) y= (sin y) x, finddydx

.

6. If y=(sin x)log x+ x2−1x2+7 x+9

finddydx

.

7. If y=esin x+¿¿ finddydx

.

8. If y=xcot x+(sin x )x finddydx

.

9. If x p yq=(x+ y)p+q, prove that dydx

= yx

, d2 ydx2 =0.

10. If y=cos xx+xcos x finddydx

.

11. If x16 y9=(x2+ y )17, prove that xdydx

=2y.

12. If ¿ xsin x+sin−1 √x , find dydx

.

13. If ¿5log ¿¿ ¿ , find dydx

.

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14. Finddydx

, y=sin3 √7 x2+2

15. If sin y=x sin (a+ y ), prove that dydx

=sin2(a+ y )

sin a

16. If xy+ y2=tan x+ y finddydx

.

17. If y √x2+1=log(√ x2+1−x) show that ( x2+1 ) dydx

+xy+1=0

18. If y=(log (x+√x2+1))2 find

dydx

.

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Assignment 40

DIFFERENTIATION

(INVERSE TRIGONOMETRIC FUNCTIONS)

1. If y=tan−1(√1+x2−√1−x2

√1+x2+√1−x2 ), find dydx

2. If y=cot−1( √1+sin x+√1−sin x√1+sin x−√1−sin x ), show that

dydx

is independent of ‘x’.

3. If ¿ tan−1 √ 1+sin x1−sin x

, find dydx

.

4. Differentiate sin−1( 1−x2

1+x2 ) with respect to sin−1( 2 x

1+x2 ).5. If y=tan−1( 1−cos x

sin x ), find dydx

.

6. If y=tan−1(√1+x−√1−x√1+x+√1−x ), find

dydx

.

7. If y=sin−1( 1

√1+x2 ), find dydx

.

8. If y=tan−1(√1+x2−1x ), find

dydx

.

9. If y=tan−1 (√1+x2+x ), find dydx

.

10. If y=cos−1 (4 x3−3 x ), find dydx

.

11. If y=cos−1(√1+x−√1−x2 ), find

dydx

.

12. If log ( x2+ y2 )=2 tan−1 ( y /x ), then show that dydx

= x+ yx− y

.

13. If y=cos−1( 2x+1

1+4x ), find dydx

.

14. Differentiate the following with respect to ‘x’:f ( x )=tan−1( 1−x1+x )−tan−1( x+2

1−2 x )

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Assignment 41

DIFFERENTIATION

(SECOND DERIVATIVES)

1. If x=a {(cos t+ log tan( t2 ))}, y=asint , find

d2 ydx2 at x=

π4 .

2. If y=sec x−tan x, show that cos xd2 ydx2 = y2

.

3. Prove that cos2 xd2 ydx2 −2 y+2 x=0.

4. If y=sin−1 x , prove that d2 ydx2 = x

(1−x2 )32 .

5. If y=ex (sin x+cos x), then show thatd2 ydx2 −2

dydx

+2 y=0.

6. If y=sin ( msin−1 x ), show that

(1-x2)d2 ydx2 −x

dydx

+m2 y=0.

7. If x=sin θ, y=sin pθ, show that

(1-x2)d2 ydx2 −x

dydx

+ p2 y=0.

8. If x = a cos3t and y = a sin3t find d2 ydx2 at t=π

4.

9. If y=¿¿ show that

( x2+1 )2 d2 ydx2 +2 x ( x2+1 ) d y

dx=2.

10.If x = a(cos t + t sin t), y = a(sin t – t cos t), find d2 ydx2 ∧dy

dxat= π

4.

11.If x = a(θ+sin θ) and y = a(1-cos θ) find d2 ydx2 at θ = π/2.

12.If y = a cos(log x) + b sin(log x) then

prove that x2 d2 ydx2 +x

dydx

+ y=0.

13.If y = log(1+cos x) prove that d3 ydx3 + d2 y

dx2 + dydx

=0.

14.If y = x + cot x, prove that sin2 xd2 ydx2 +2 x−2 y=0.

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Assignment 42

APPLICATIONS OF DIFFERENTIATION

Rolles theorem:

1. Verify Rolles theorem for the functiona. f(x) = x2+x-6 in the interval [-3,2]

b. f(x) = sin x + cos x -1 on [0,π2

¿

c. f(x) = x2-2x-3 on [-1,3]d. f(x) = sin x – sin 2x on [0,π]

2. Using Rolles theorem find the point on the parabola y = x2 – 3x + 2 where the tangent is parallel to the x-axis.

Legranges Mean Value Theorem [LMVT]

3. Verify LMVT for the following functions in the specified intervalsa. f(x) = x(x-2) on [1,3]b. f(x) = (x-3)(x-6)(x-9) on [3,5] c. f(x) = x(x-1)(x-2) on [0,1/2]

4. Using LMVT find the point on the parabola y = x2 – 3x where the tangent is parallel to the chord joining (1,-2) and (2,-2).

Approximations5. Find the approximate value of the following:

a. 3√0.009

b. (256)18

c. 5√1025

d. √0.037

e. √0.0037

6. Find the approximate value of f (2.01) where f(x) = 4x3 + 5x2 +2.7. Find the approximate value of f (1.02) where f(x) = 2x3-2x2 – 5.8. Find the approximate value of f (2.03) where f(x) = 3x3 – 4x2 + 6.

Increasing and decreasing functions :

1. Find the intervals on which the following functions are increasing or decreasing:1. f(x)=2x3+9x2+12x+202. f(x)=x4-2x2

3. f(x)=5x3-15x2-120x+34. f(x)=2x3-9x2+12x-55.f(x)=x3-12x2+36x+176.f(x)=15-9x+6x2-x3

7.f(x)=2x3-6x2-48x+178.f(x)=8+36x+3x2-2x3

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9.f(x)=sinx+cosx , 0≤x≤π2. Find whether f(x)=x2-6x+3 is increasing or decreasing on [4,6].

3. Show that f(x)=x/2+sinx is always increasing in the interval -π3 ≤ x ≤

π3 .

4. Find the intervals on which the functions f(x)=x

1+ x2 is increasing or decreasing.

5. Find the intervals on which the function f(x)=sinx-cosx, 0 ≤ x ≤ π is strictly increasing or strictly decreasing.

Maxima and Minima:1. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere

of radius 20cm is 40

√3 cm.

2. Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius 5√3 cm is 500 π cm3.

3. Find the volume of the largest cone that can be inscribed in a sphere of radius ‘a’ cm.

4. The lengths of the sides of an isosceles triangle are 9+x2, 9+x2and 18-2x2 units. Calculate the area of the triangle in terms of x and find the value of x which makes the area maximum.

5. The sum of the perimeter of a circle and a square is K, where K is a constant. Prove that the sum of their areas in least when the side of the square is double the radius of the circle.

6. Find all the points of local maxima and minima and the maximum and minimum value of

the function, f(x)= -34

x4+2x3+92

x2+100.

7. Show that a closed right circular cylinder of given T.S.A and maximum volume is such that its height is equal to the diameter of its base.

8. Show that of all the rectangles inscribed in a given fixed circle square has the maximum area.

9. Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube.

10. Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse with its vertex coinciding with one extremity of the major axis.

11. Show that the volume of the greatest cylinder that can be inscribed in a cone of height ‘h’

and semi vertical angle 30o is 4

81πh3

12. 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 60o.

13. A manufacturer can sell ‘X’ items at a price of Rs.(5 - x

100 ) each. The cost price of ‘X’ items

is Rs(x5

+ 500). Find the number of items he should sell to earn a maximum profit.

14. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius ‘r’ is 2r/√3.

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15. Show that the height of a right circular cylinder of maximum volume that can be inscribed in a given height is h/3.

16. A window is in the form of a rectangle surmounted by a semi- circle. If the perimeter is 30m, find the dimensions so that the greatest amount of light may be admitted.

17. Show the semi vertical angle of a right circular cone of given T.S.A and the maximum volume is sin-1(1/3).

18. An open box with a square base is to be made out of given quantity of metal sheet of area c2.Show that the maximum volume of the box is c3/6√3.

19. An open box with a square base is to be made out of a given iron sheet of area 27 sq.m. Show that the maximum volume of the box is 13.5 cu.m.

20. A rectangular window is surmounted by an equilateral triangle. Given that the perimeter of the window is 16m,find the width of the window so that maximum amount of light may enter.

21. An open tank with 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 the depth of the tank is half of its width.

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Assignment 43

APPLICATIONS OF DIFFERENTIATION

(Tangents and Normals)

1. Find the point on the curve y=3x2 – 12x +6 at which the tangent is parallel to the x-axis. 2. If the tangent to the curve y=x3 + ax + b at (1,-6) is parallel to the line x-y+5=0, find the

values of ‘a’ and ‘b’.3. Find the point on the curve y=x3 – 3x where the tangent is parallel to the chord joining

(1,-2) & (2, 2).4. Find the equation of the tangent to the curve y= -5x2 + 6x + 7 at the point (1/2, 35/4)5. Prove that the curves x = y2 and xy = k cut at right angles if 8k2=1.6. Find the equation of the tangent to the curve x2 + 3y – 3 = 0 which is parallel to the line y

= 4x - 5.7. Find the equation of the tangent to the curve y = √4 x−2 which is parallel to the line 4x –

2y + 5 = 0.8. Show that the curves xy=a2 and x2 + y2 = 2a2 touch each other. 9. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-

coordinate of the point.10. Find the equations of the normals to the curve y = x3 + 2x + 6 which are parallel to the

line x + 14y + 4 = 0.

11. Find the equation of the tangent to the curve ¿x−7

( x−2 )(x−3) , at the point, where it cuts the

x-axis.12. Find the equation of the tangent to the curve y=√3 x−2 which is parallel to the line 4x-

2y+5=0.13. At what points will the tangent to the curve y=2x3 – 15x2 + 36x - 21 be parallel to the x-

axis? Also, find the equations of tangents to the curve at those points.

14. Find the equations of tangent to the curve x = sin 3t, y = cos 2t, at t=π4

.

15. Find the equation to the tangent to the curve x = a sin3 t, y = b cos3 t, at t=π4

.

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Assignment 44

INDEFINITE INTEGRALS

(COMPLETION OF SQUARES)

1.∫ 1

√5−4 x−2 x2dx

2.∫√x2+8 x−6 dx

3.∫ x2

x2+6 x−3dx

4.∫ 1

√x2−2 x+4dx

5.∫ 1

√ x (1−2 x )dx

6.∫ x+1

√x2+2 x−1dx

7.∫ 1

√4 x−3 x2+5dx

8.∫ 1

√16−6 x−x2dx

9.∫ ex

√5−4 ex−e2xdx

10.∫ 1

√5+4 x−x2dx

11.∫ 6 x+7

√ ( x−5 ) ( x−4 )dx

12∫ x+3

x2−2 x−5dx

13.∫ x

x4−x2+1dx

14.∫ 1

√8+4 x−2 x2dx

15.∫ x2

x2+6 x+12dx

16.∫ 2 x

√1−x2−x4dx

17.∫ cos x

√sin2 x−2sin x−3dx

18.∫ 2 x+1

2 x2+4 x−3dx

19.∫ 2 x−3

x2+3 x−18dx

20.∫ (3x−2)

√ x2+4 x+5dx

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Assignment 45

SUBSTITUTION

1. ∫Sec2 √ x√ x

dx

2. ∫ (2 x+1 ) √x2+x+1 dx

3. ∫ tanx sec 2 x1−tan2 x

dx

4. ∫ sin(2 tan7 x )1+x2 dx

5. ∫ Sinxsin ( x−a )

dx

6. ∫ 2 x tan−1(x2)1+x4 dx

7. ∫ x3+xx4−9

dx

8. ∫ e2 x−e−2 x

e2 x+e−2 x dx

9. ∫ x √x 4−1dx

10.∫cos4 x dx

11.∫ ex

ex−1dx

12.∫ √ x2+a2

xdx

13.∫ dx√ x (1+√x )

14.∫ sin 2 x

(a+bcosx )2dx

15.∫ cos5 xsinx

dx

16.∫√16+¿¿¿¿¿

17.∫ cos ( x+a )sin ( x+b )

dx

18.∫ sin ( x−α )sin ( x+α )

dx

19.∫ x2

1+x3 dx

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Assignment 46

PARTIAL FRACTIONS

1.∫ 1

(2−x ) ( x2+3 )dx

2.∫ dxx ( x−4 ) ( x−2 )

3.∫ 3 x+1

( x−2 )2 ( x+2 )dx

4.∫ sin x¿¿ ¿

5.∫ 2 x

(1+x2 )(3+x2)dx

6.∫ sin 2 x¿¿ ¿

7.∫ x2+1(x+1)2 dx

8.∫ dx

x3+x2+ x+1

9.∫ x( x+1 )(x2+1)

dx

10.∫ cos x¿¿ ¿

11.∫ x4

(x−1 ) ( x2+1 )dx

12.∫ sec x tan x¿¿ ¿

13.∫ x( x+2 )(3−2x )

dx

14.∫ 2 x−1( x−1 ) ( x+2 ) ( x−3 )

dx

15.∫ x2+x+1( x+2 ) (x+1 )2

dx

16.∫ sin x¿¿ ¿

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17.∫ xdx( x−1 )(x2+4)

Assignment 47

Integration by parts

1.∫ x sin−1 x dx

2.∫ x tan−1 x dx

3.∫ x3 log 2 xdx

4.∫ x2cos−1 xdx

5.∫ e3 x sin 2 xdx

6.∫ tan−1 x(1+ x )2

dx

7.∫ x2 cot−1 x dx

Special integrals

1.∫ dx

32−2 x2

2.∫ x2−11+x4 dx

3.∫ 1

1+x4dx

4.∫√ a−xa+x

dx

5.∫√ tan θ dθ

6.∫ 1+x2

1+x4 dx

Assignment 48

72 | P a g e

73 | P a g eCancellation property

1.∫ ex sec x¿¿¿

2∫ ex ( 1−sin x1−cos x )dx

3.∫ ¿¿

4.∫ex ( x2+1( x+1)2 )dx

5.∫ 2+sin2 x1+cos 2 x

e x dx

6.∫ x2+1(x+1)2 e x dx

7.∫ ex¿¿¿

8∫ex ( sin xcos x−1

sin2 x )dx

Different types of sums

1.∫ sin 3 x sin 5 x dx

2.∫ 2cos x3sin 2 x

dx

3.∫( 1+2sin x

cos2 x )dx

Assignment 49

73 | P a g e

74 | P a g e Definite Integrals

1. ∫0

12

1

√1−x2dx

2. ∫0

πx

a2cos2 x+b2 sin2 xdx

3. ∫−π

2

π2

(2 sin|x|+sin|x|¿)dx ¿

4. ∫0

π2

sin3 xsin3 x+cos3 x

dx

5. ∫0

5

|x−3|dx

6.∫0

2

( 3 x2−4 ) dx (As limit of sums)

7. ∫0

ax4

√a2−x2dx

8.∫0

2

(x2+3)dx (As limit of sums)

9. Sketch the graph: y=|x+1| Evaluate:∫−4

2

|x+1|dx

10. ∫−π

4

π4

|sin x|dx

11. ∫0

πx

1+sinxdx

12. ∫−2

2

|2 x+3|dx

13. ∫0

π2

11+ tan3 x

dx

14. ∫1

2

x2 dx (Limit as sum)

15. ∫0

2

(1+x2 ) dx (Limit as sum)

74 | P a g e

75 | P a g e16. ∫2

4x

x2+1dx

17. ∫0

π /21

3+2 cosxdx

18. ∫1

4 √x√5−x+√ x

dx

19. ∫0

π2

cosx1+sin2 x

dx

20. ∫0

π2

11+√cotx

dx

21. ∫1

4

(x2¿−x)dx¿ (Limit as sum)

22. ∫0

π2

cos xsin x+cos x

dx

23. Evaluate ∫1

4

f ( x )dx wheref ( x )={2 x+8 ,∧1≤ x ≤ 26 x ,∧2 ≤ x ≤ 4

24. ∫0

11

2 x−3dx

25. ∫0

3

( x+4 ) dx (As the limit of sum)

26. ∫0

1

√ 1−x1+x

dx = π2−1

27. ∫0

π /4

√1−sin 2x dx = √2 – 1

28. ∫0

π /2sin3 x

sin2 x+cos3 xdx

29. ∫1

2

¿¿¿

30. ∫0

2

( x2+x+2 ) dx (As limit of sums)

31. ∫1

4

( x2−x ) dx (Limit as sum)

32. ∫3

6

|x−3|+|x−4|+|x−5|dx

33. ∫0

πx

1+sin xdx

34. ∫0

11

√2 x+3dx

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76 | P a g e35. ∫1

4

|x−1|+|x−2|+|x−4|dx

36. ∫−5

5

|x+2|dx

37. ∫0

π4

log¿¿¿

38. ∫0

π /21

1+ tan3 xdx

39. ∫1

6 √7−x√ x+√7−x

dx

40. ∫0

π2

sin 2 xsin x+cos x

dx

41. ∫0

πx sin x

1+cos2 xdx

42. ∫0

π2

xsin x+cos x

dx

43. ∫0

π /2

(√ tan x+√cot x ) dx=√2 π

44. ∫−5

0

|x|+|x+3|+|x+6|dx

45. ∫π /6

π /31

1+√ tan xdx

46. ∫0

π1

5+4 cos xdx

47. ∫0

π4

sin 2 x sin3 x dx

48. ∫0

3

( 2 x2+3 x+5 ) dx (Limit of a sum)

49. ∫0

πx tan x

sec x cosec xdx =

π 2

450. ∫0

12 x

1+x2 dx

51. ∫0

πx tan x

sec x+ tan xdx

52. ∫0

π2

log sin x dx

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77 | P a g e53. ∫0

1

x ( tan¿¿−1 x)2 dx¿

54. ∫– a

a

√ a−xa+x

dx

55. ∫0

1√2

1

√1−x2dx

56. ∫1

4

( 3 x2+2 x) dx

57. ∫0

3

( x2−3 x+4 ) dx

58. ∫0

πx

1+sin xdx

59. ∫0

11

√2 x+3dx

60. ∫1

4

|x−1|+|x−2|+|x−4|dx

61. ∫−5

5

|x+2|dx

62. ∫0

π4

log¿¿¿

63. ∫0

π /21

1+ tan3 xdx

64. ∫1

6 √7−x√ x+√7−x

dx

65. ∫0

π2

sin 2 xsin x+cos x

dx

66. ∫0

πx sin x

1+cos2 xdx

67. ∫0

π2

xsin x+cos x

dx

68. ∫0

π /2

(√ tan x+√cot x ) dx=√2 π

Prove that

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1. ∫0

1

log( 1x¿¿−1)dx ¿¿=0

2. ∫0

2

( x2+1 ) dx using limit as sum (14/3)

3. ∫0

2 a

f (x )dx = ∫0

2 a

f (2 a−x ) dx

4. ∫1

2

( x2−1 ) dx (As the limit of sums) (4/3)

5. ∫0

9

f ( x )dx = ∫0

a

f (a− x ) dx

Hence, evaluate ∫0

21

1+ tan xdx (

π4

)

Assignment 50

Area under the curve

1. Find the area of the region bounded by the curves i. y2=4ax and x2=4ay

ii. x2=8y and y2=8x2. Find the area of triangle: A (2, 5) B (4, 7) C (6, 2).3. Find the area of the following region :

{( x , y ): x2+ y2≤ 2 ax , y2 ≥ ax , x , y≥ 0}

4. Sketch the region {( x , y ): 4 x2+9 y2=36} and find its area using integration.

5. Find the area of the triangle: A (1, 0) B (2, 2) C (3, 1).6. Using integration, find the area of the region bounded by the line 2y=8-x, x-axis and

the lines x=2 and x=4.7. Find the area of the region bounded by the curve x2=4 y and the line x=4y-2.8. Find the area bounded by the circle x2+ y2=16and the line y=x in the 1st quadrant.9. Find area: A (1, 6), B (2, 8), C (3, 4).

10. Find area: {( x , y ): y2 ≤ 4 x ;4 x2+4 y2≤ 9}11. Find area: {( x , y ):25 x2+9 y2≤ 225 ,5x+3 y ≥15 }12. Sketch the region enclosed between the circles x2+y2=1 and x2+( y−1)2=1. Also, find the

area of the region using integration.13. Find the area of the region bounded by the curve ¿√1−x2 , line y=x and the positive x-

axis.14. Find the area bounded by the lines y=4x+5, y=5-x and 4y=x+5.

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79 | P a g e15. Find the area: (2, 0) (4, 5) (6, 3).16. Find the area of the circle x2+y2=16 which is exterior to the parabola y2=6x, by using

integration.17. Find area:

i. x2+ y2=1, (x-1)2+y2=1.ii. y=x2, x=y2.

18. Find the area bounded by the lines y=4x+5, y=5-x and 4y=5+x.

19. Find the area of the region {( x , y ): x2+ y2≤ 1 ≤ x+ y }20. Find the area bounded by the curve: x2=4y & the line x=4y-2.21. Find the area of the circle 4x2+4y2=9 which is the interior to the parabola y2=4x.22. Area of triangle whose vertices are (2, 0) (4, 5) (6, 3)

ASSIGNMENT-51

Differential Equations

(Forming differential equations)

1. Form a differential equation of a family if circles in the second

quadrant and touching the co-ordinate axis.

2. Form a differential equation of a family of curves by x2+y2 = 2ax.

3. Form a differential equation of a family of curves Y=(a+bx)e3X a and b

are arbitrary constants.

4. Find the differential equation rep. the family of ellipses having foci

on x-axis and centre at the origin.

5. Form a differential equation representing the family of curves give by

(x-a)2 +2y2=a2, where a is an arbitrary constant.

6. Form the differential equation representing the family of parabolas

having vertex at the origin and axis along positive direction of x-axis.

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7. Form the differential equation representing the family of circles

touching the x-axis at the origin.

Assignment-52


(Variable and separable)

1. solve: x(x2-1)dydx =1. Y=0,x=2.

2. For the differential equation xy dydx =(x+2).(y+2),find the solution

curve passing through (1,-1);

3. solve: (x2-yx2)dy+(y2+xy2)dx=0.

4. dydx =y.sin2x given that y(0)=1.

5. log(dydx )=ax+by.

6. (x3+x2+x+1)dydx =2x2+x .

7. dydx = ytanx, y=1,x=0.

8. 3extany dx+(2-ex)sec2ydy=0 ,x=0,y=π/4.

9. dydx

=¿1+x2+y2+x2y2,give that y=1, when x=0.80 | P a g e

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10. (x+1)dydx =2e-y-1, y=0 when x=0

11. ex √1− y2dx + yx dy=0 when x=0,y=1.

Assignment-53

Differential equations

Homogeneous equations

1. Solve the following differential equation (x3+y3)dy-x2ydx=0.

2. (y+x)dydx =y-x.

3. (x2-y2)dx+2xydy=0 given that y=1,x=1.

4. ye(x/y)dx=(xe(x/y)+y)dy

5. xdy-ydx=√ x2+ y2dx

6. ydx+xlog( yx )dy=2xdy

7. (x cos( yx )+ysin( y

x¿)y-( y sin( y

x )+xcos( yx )xdy

dx =0.

8. Find the particular solution of the differential equation satisfying

the given conditions x2dy+(xy+y2)dx=0 ,y=1, x=1.

9. Solve:dydx - y

x +cosec( yx )=0;y=0;when x=1

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10. x2 dydx =y2+2xy ;given that y=1;when x=1;

11. dydx + 2ytanx=sinx.

12. xdydx =y-tan( y

x )

13. dydx - y

x +cosec( yx )=0

14. 2x2 dydx

−¿2xy+y2=0

15. (x-y)(dx+dy)=dx-dy

16. 2ye yx dx+(y-2xe y

x¿dy=0given that x=0, y=1.

17. Show that the differential equation (x-y) dydx =x+2y, is

homogeneous and solve it.

18. xdydx

= y+√ x2+ y2

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Assignment-54


First order differential equations

1. Find the particular solution of the differential equation

dydx

+ycotx=4xcosecx x≠0,given y=0,whenx=π/2 .

2. Solve the differential equation given that y=1 when x=2

xdydx +y=x3 .

3.(1+x2)dydx +y=tan-1x.

4. (X2-1)dydx +2xy=1/(x2-1) .

5.(x2+1)dydx +2xy=√ x2+4

6. Xlogxdydx +y=2logx/x

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7. (3x2+y)dydx =x;x>0 when x=1,y=1 .

8. (1+x2)dy+2xydx=cotx dx (x≠0) .

9. Cos2x dydx +y=tanx .

10.dydx

+y=cosx-sinx .

11. Secx dydx –y=sinx .

12.dydx +secx y=tanx.

LINEAR PROGRAMMING

1. A diet for a sick person must contain at least 4000 units of vitamins, 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 respectively. One unit of the food contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the 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 should be used to have least cost, but it must satisfy the requirements of the sick person. Form the question as L.P.P and solve it graphically.

2. A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:

Machine Area occupied

Labour force

Daily output (in units)

A

1000 m2 12 men 60

B 1200 m2 8 men 40

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3. A man has Rs.1,500 for purchasing rice and wheat. A bag of rice and a bag of wheat cost Rs.180 and Rs.120 respectively. He has the storage capacity of at most 10 bags. He earns a profit of Rs.11 and Rs.9 per bag of rice and wheat respectively. Formulate the above problem as an L.P.P to maximize the profit and solve it graphically.

4. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5,760 to invest and has a space for at most 20 items. A fan costs him Rs.360 and a sewing machine Rs.240. His expectation is that he can selll a fan at a profit of Rs.22 and a sewing machine at a profit of Rs.18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise the profit? Formulate this as a linear programming problem and solve it graphically.

5. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g 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 graphically.

6. A man has Rs.1,500 for purchasing wheat and rice. A bag of rice and a bag of wheat cost Rs.180 and Rs.120 respectively. He has a storage capacity of only 10 bags. He earns a profit of Rs.11 and Rs.9 per bag of rice and wheat respectively. Formulate the problem as an L.P.P to find the number of bags of each type he should buy for getting maximum profit and solve it graphically.

7. One kind of cake requires 300g of flour and 15g of fat, another kind of cake requires 150g of flour and 30g of fat. Find the maximum number of

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cakes which can be made from 7.5kg of flour and 600 g of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an LPP and solve it graphically.

8. A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs.300 and that on a chain is Rs.190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

9. A factory makes two types of items A and B, made of plywood. One piece of item A requires 5 minutes for cutting and 10 minutes for assembling. One piece of item B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours for assembling. The profit on one piece of item A is Rs.5 and that on item B is Rs.6. How many pieces of each type should the factory make so as to maximize profit? Make it as an L.P.P and solve it graphically.

10. A library has to accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weigh 1 kg and 1 1/2kg each respectively. The shelf is 96 cm long and at most can support a weight of 21 kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an L.P.P and solve it graphically.

11.A merchant plans to sell two types of personal computers - a desktop model and a portable model that will cost Rs.25,000 and Rs.40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and his profit on the desktop model is Rs.4,500 and on the portable model is Rs.5,000. Make

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an L.P.P and solve it graphically.

12.A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P and solve graphically.

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

14.A dietician wishes to mix two types of foods 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 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs.5 per kg to purchase Food I and Rs.7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a L.P.P and solve it graphically.

15.A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs.17.50 per package on nuts and Rs.7 per package of bolts. How many packages of each should be produced each day so as to maximize his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.

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16.A company produces soft drinks that have a contract which requires that a minimum of 80 units of the chemical A and 60 units of the Chemical B go into each bottle of the drink. The chemicals are available in prepared mix packets from two different suppliers. Supplier S had a packet of mix of 4 units of A and 2 units of B that costs Rs.10. The supplier T has a packet of mix of 1 unit of A and 1 unit of B that cost Rs.4. How many packets of mixes from S and T should the company purchase to honour the contract requirement and yet minimize cost? Make a L.P.P and solve graphically.

17.A producer has 30 units and 17 units of labour and capital respectively to produce two types of items A and B. To produce one unit of A, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of item B.If A and B yield a profit of Rs.100 and Rs.120 per unit respectively, how many items of each type should he produce so as to maximize the total profit? Make the above problem as an L.P.P and solve it graphically.

18.A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts while it takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs.2.50 per package of nuts and Rs.1.00 per package of bolts. How many packages of each type should he produce each day so as to maximise his profit, if he operates his machines for at most 12 hours a day? Formulate this problem as a linear programming problem and solve it graphically.

19.A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at

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Rs.100 and Rs.120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an L.P.P and solve graphically.Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

20.An aeroplane can carry a maximum of 200 passengers. A profit of Rs.500 is made on each executive class ticket out of which 20% will go to the welfare fund of the employees. Similarly a profit of Rs.400 is made on each economy ticket out of which 25% will go for the improvement of facilities provided to economy class passengers. In both cases, the remaining profit goes to the airline's fund. The airline reserves at least 20 seats for executive class. However at least four 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 net profit of the airline. Make the above as an L.P.P and solve graphically. Do you think more passengers would prefer to travel by such an airline than by others?

21.A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs.10,500 and Rs.9,000 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 maximise the total profit? Form an L.P.P 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?

22.A cottage industry manufactures pedestal lamps and wooden shades,

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each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on sprayer to manufacture a pedestal lamp. It takes 1 hour on grinding/cutting machine and 2 hours on sprayer to manufacture a shade. To keep the noise and dust pollution level under prescribed limits, a sprayer can be used for a maximum of 20 hours per day while cutting and grinding machine can be used for a maximum of 12 hours per day. The profit from the sale of a lamp is Rs.5 and that from sale of a shade is Rs.3. Assuming that the manufacturer can sell all the pedestal lamps and shades that are made, how should he schedule his daily production to maximize profit. Make it as an L.P.P and solve it graphically. Which social value is indicated in the above question?

23. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a cutting/grinding machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on sprayer to manufacture a pedestal lamp. It takes 1 hour on grinding/cutting machine and 2 hours on sprayer to manufacture a shade. On any day, to keep the environment pollution under minimum level, sprayer can be

used for at the most 20 hours while grinding/cutting machine can be used for at the most 12 hours. The profit from selling a pedestal lamp is Rs.5 and for selling a shade is Rs.3. Assuming that it can sell all that it produces, how should it schedule its daily production to maximize its profits? Make it as an L.P.P and solve it graphically. Which value is described in this question?

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Extra Problems

Assignment - 55

Relations and functions

Equivalence relation

1.Show that the relation R on the set R of al real nos defined by R={(a,b): a≤b2}

is not transitive .

2. R= {(a,b): |a-b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.

3. Show that relation R in the set Z of integers given by R={(a,b): 2 divides (a-b)}

is an equivalence relation .

4.Let R be a relation defined on A={1,2,3} by R={(1,1), (2,2),(3,3),(1,2),(2,3),(3,2),(2,1)} .Is R (i) symmetric (ii) transitive ?

5. Show that the relation S defined on the set NxN by (a,b) S (c,d) implying ad=bc is an equivalence relation.

6.Let f: x R y be a function. Define a relation R on X given by R ={(a,b): f(a)=f(b)}.

Show that R is an equivalence relation on X.

7.Show that the relation S in the set A={x € Z: 0≤x≤12} given by S={(a,b): a,b € Z,

|a-b| is divisible by 4} is an equivalence relation .Find the set of all elements related to 1.

8.Show that the relation R defined by (a,b) R (c,d) => a+d=b+c on the set NxN is an equivalence relation.

9.Let N denote the set of all natural nos and R be the relation NxN defined by (a,b) R(c,d) if and only if ad(b+c) = bc (a+d). Prove that R is an equivalence relation.

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Assignment-56

Binary Operations

1. A binary operation * on Q is defined as a*b=ab/4. Find the identity element on Q.

2.Let A=ZxZ and * be the binary operation on A def by (a,b)*(c,d)=(a+c,bd).

Find {(4,2)*(-2,5)*(3,1)}.

3.Let * be a binary operation defined on Q such that a*b=ab/12. Check whether * is

(i) commutative (ii) associative (iii) identity element of the binary operation * in Q,if it exists.(iv)Invertible elements in Q for the binary operation *,if there are any.

4.Is a*b=|a-b| defined on Z+ a binary operation ? Justify .

5.Let R be the set of all real nos and a binary * is defined on R as a*b=a+b-5 ,¥ a,b€R, then find the identity element for * on R.

6. Let * be a binary operation on N given by a*b=LCM (a,b), ¥ a,b € N. Find (16*20)*40.

7.Consider the binary operation * :R x R -> R and 0: R x R -> R defined by a*b= |a-b|

and a o b = 0, ¥ a,b € R. Show that * is commutative but not associative . o is associative but not commutative .

8.Let * be a binary operation defined by a*b=2a+b-3. Find 3*4.

9.Define a binary operation * on the set {0,1,2,3,4,5} as

a*b= a+b if a+b ≤ 6

a+b-6 if a+b ≥ 6

Show that zero is the identity element and each element a≠0 of the set is invertible with 6-a being the inverse of ‘a’.

10. Let * be a binary operation on A defined by a*b =a2 +ab. If 3*b=21 , find b .

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11. The binary operation * on Z defined by a*b=a+b+2; a,b € Z. Find the identity element .

12. Let A=NƲ{0} × NƲ{0} and let * be a binary operation on A defined by

(a, b) * (c, d) = (a+c, b+d) for all (a b) € A. Show that

(i) * is commutative on A (ii) * is associative on A (iii) Find the identity element on A.

15.Let * be a binary operation on N given by a*b=HCF(a,b) ¥ a,b € N .Find 22*4.

16.If the binary operation * on the set of integers Z ,is defined by a*b=a+3b2,then find the value of 2*4 .

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Assignment -57

Inverse of functions

1.Let f: N →R be a function defined by f(x)=4x2+12x+15. Show that f: N→S where S is the range of f is invertible . Find the inverse of f .

2.Show that the function f : N→N defined by f(x)=2x-1is not onto .

3.Show that the function f : [- ½ ,∞) →[ ¾ ,∞) defined by f(x)=x2 +x+1 is invertible . Find f ˉ1 (7).

4.If f : P→R be function defined by f(x)=4x3 -7. Show that function f is a bijective function.

5. Let f(x)=an

x+1 ,x≠ -1. If fof =x,find a.

6.f :R→R be defined by f(x)= (3-x3)⅓ ,then find fof (x).

7.Consider the functions f&g : f(x)= sinx, g(x)=cosx. Show that f is one to one ,g is one to one ,but f+g is not one to one.

8.Let f : N→R be a relation defined as f(x)=4x2 +12x+15 .Show that f : N→S (where S is the range of f ) is invertible. Find the inverse of f .

9.Show that f :R →R defined by

f(x)= { 1 x>00 x=0

−1 x<0 is neither one to one nor onto.

10.If f(x) = x−1x+1 , (x≠ -1,1),show that fof-1 is an identity function.

11.If f(x) is an invertible function ,find the inverse of f(x) = 3 x−2

5 .

12.Let f : R→{-3/5 }→R be a function defined as f(x) =2 x

5 x+3 ,find f-1.

13.If f : A→B is defined by f(x) =5 x+67 x−9 ,where A = R – {9/7} and B = R- {5/7} ,show

that f is one to one and onto function. Hence find f-1 .

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14. Show that the function f : R→{x € R: -1<x<1} defined by f(x) = x

1+¿ x∨¿¿ ,x € R is

one to one and onto.

15.Consider the function f : R+→x2 +4. Show that f is invertible with inverse given by f-1(y) =√ y−4 , where R+ is the set of non-negative real nos.

16.What is the number of bijective functions from {-1,2,3} to {0,3,5} ?

95 | P a g e

96 | P a g e

Assignment – 58

RATE OF CHANGE OF QUANTITIES

1. The volume of a spherical balloon is increasing at the rate of 25c m3/sec. Find the rate of change of its surface area at the instant when its radius is 5cm.

2. A stone is dropped in a quiet lake and waves move in circles at a speed of 3.5 cm/sec. At the instant when the radius of the circular wave is 7.5 cm, how fast is the enclosed area increasing?

3. The surface area of a balloon being inflated changes at a constant rate. If initially its radius is 3cm and after 2 sec it is 5 units, find the radius after‘t’ seconds.

4. The length ‘x’ of a rectangle is decreasing at the rate if 5cm/min and the width ‘y’ is increasing at the rate of 4cm/min. When x=8cm and y=6cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.

5. A water tank has the shape of an inverted cone with its axis vertical and vertex lower most. Its semi-vertical angle is tan-1

(0.5). Water is poured into it at a constant rate of 5m3/hr. Find the rate at which the level of water is rising at the instant when the depth of water in the tank is 4m.

6. A particle moves along the curve 6y= x3+2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as its x-coordinate.

7. An inverted cone has depth 10cm and a base radius of 5cm. Water is poured into it at the rate of1.5 cc/min. Find the rate at which the level of water in the cone is rising when the depth is 4cm.

8. At what points in the ellipse 16x2+9y2=400, does the ordinate decreases at the same rate at which the abscissa increases?

96 | P a g e

97 | P a g e

PROBABILITY (BAYE’S THEOREM)

1. In a bulb factory, machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

2. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck 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.

3. In a factory which manufactures bolts, machines A, B and C, manufacture respectively 35%, 25% and 40% of the bolts. Of their output 5, 4 and 4 percent are respectively defective bolts. A bolt is drawn at random from the total production and is found to be defectiver. Find the probability that it is manufactured by the machine B.

4. An insurance company insured 3000 scooter drivers, 5000 car drivers and 7000 truck drivers. The probability of their meeting with an accident are 0.02, 0.03 and 0.12 respectively. One of the insured persons meets with an accident . What is the probability that he is a truck driver?

5. A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4. Find the probability that it is actually a number greater than 4.

6. A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white?

7. In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/5 be the probability that he knows the answer and 2/5 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/3, what is the probability that the student knows the answer, given that he answered it correctly?

8. In a class, 5% of the boys and 10% of the girls have an IQ of more than 150 . In the class 60% of the students are boys and rest girls. If a student is selected at random and found to have an IQ of more than 150, find the probability that the student is a boy.

97 | P a g e

98 | P a g e

9. Given three identical boxes I, II and III each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

10. A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that is is actually six.

11. Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II at random. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

12. There are three coins. One is a two-tailed coin (having tail on both faces), another is a biased coin that comes up heads 60% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, and it shows tails. What is the probability that is was a two-tailed coin?

13. Suppose a girl thrws a die. If she gets a 5 or 6, she tosses a coin 3 times and notes the number of heads. If she gets 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2 3 or 4 with the die?

14. A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she threw 1, 2 3 or 4 with the die?

15. In a certain college, 4% of boys and 1% of girls are taller than 1.75 metres. Furthermore, 60% of the students in the college are girls. A student is selected at random from the college and is found to be taller than 1.75% metres. Find the probability that the selected student is a girl.

16. A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both hearts. Find the probability of the missing card to be a heart.

98 | P a g e

99 | P a g e

17. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers, the probabilities of their meeting an accident respectively are 0.01, 0.03 and 0.15. One of the insured persons meets with an accident. What is the probability that he is a car driver?

18. In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternatively to decide that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

19. Often it is taken that a truthful person commands more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is actually a six. Find the probability that it is actually a six. Do you also agree that the value truthfulness leads to more respect in the society?

20. 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 chance by 25%. At a time a patient can choose 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 of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

21. In a test, an examinee either knows the answer or guesses or copies the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/6 and the probability that he copies the answer is 1/9. The probability that his answer is correct, given that he copied, is 1/8. Find the probability that he knew the answer to the question, given that he correctly answered it. Does the result of this question indicate that most of the students believe in the value of passing an examination with honesty and self knowledge?

22. Assume that the chances of a patient having a heart attack are 50%. Assume that a meditation and yoga course reduces the risk of heart attack by 30% and the prescription of certain drugs and certain restrictions reduces the risk by 25%. At a time a patient chooses only one of the two options with equal probabilities. 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 of meditation and yoga. What values are reflected in this question?

99 | P a g e

100 | P a g eANSWERS

ASSIGNMENT-1

1. (i) ( 2 9 /2 89/2 8 25/2)

(ii) (7 /2 11 /2 15/25 7 9 )

(iii) (1 2 32 4 6)

(iv)(0 3 41 0 5)

2. ( 12 4 /34 −14 /3

25 /3 28 /3 )3. x = ( 3 /8 −1 /4

−3 /8 1/2 ) ; y = ( 1/8 1/4−1 /8 1 /2 )

4. x = 2 ; y = 9

5. proving sum.

6. K = 1

7. x = 9 ; y = -14

8. x = 1 ; y = 2 ; z = -1 ; w = 3

9. x = -1

10. (0 00 0)

ASSIGNMENT – 2

1. (i) 0

(ii) ( 0 a b−a 0 c−b −c 0)

2. Proving sum.

100 | P a g e

101 | P a g e

3. Symmetric ----( 3 −3/2−3 /2 −1 ) Skew symmetric ---( 0 5/2

−5 /2 0 )4. Proving sum.5. x = ±1/√2 ; y =± 1/√6 ; z = ±1/√36. Proving sum.

ASSIGNMENT - 31. A−1= ( 3 /5 1/5−2 /5 1/5)

2. A−1= (1/5 2/52 /5 −1/5)

3. A−1= (−2 −33 −4 )

4. A−1= ( 7 −2−3 1 )

5. A−1= ( 1 −2−2 5 )

6. A−1= ( 9 −2−4 1 )

7. A−1= ( 4 −1−7 2 )

8. A−1= ( 4 −5−3 4 )

9. A−1= (2 31 2)10. A−1does not exist.11. A−1=( 1 −1

−1 2 )12. A−1= ( 7 −3

−2 1 )13. A−1= (−7 3

5 −2)14. A−1= ( 3 −5

−1 2 )15. A−1= ( 2 −1

−5 3 )16. A−1= ( 3 −1 1

−15 6 −55 −2 2 )

17. A−1= ( 5 /2 1/2 −15/23 /2 1/2 −9/2

−3 /2 −1/2 11/2 )

101 | P a g e

102 | P a g e

18. A−1= ( 2 /21 1/7 −13/21−1 /7 2/7 3/75 /21 −1/7 −1/21 )

19. ( 1 −2/5 −3/5−2 /5 4/5 11 /25−3 /5 1/25 9/25 )

20.Proving sum.

21.X = ±√3

22. X = 2

23.(1−x)2 (x + 2)

24. x= 4

ASSIGNMENT – 41. (i) 47/2 sq. units

(ii) 16 sq. units(iii) 95/2 sq. units(iv) 35 sq. units

(v) a2 (t 12 (t 2-t 3) - t 2

2 (t 3- t 1) + t 32 (t 1 - t 2 ))

2. Proving sum.

3. Proving sum. 4. (i) x=5 (ii) x = 57

5.(i) y = 2x (ii) x-3y = 0 (iii) x-5y+17=0

6. x = -a (or) x = -b, x=0 7. x = −3 ±√33

8

ASSIGNMENT – 5

1. x = −3 ±√3

8 2. x = 2; -1 3. x = -1,9,0 4. x = 4 5. x = 1,2,-3n 10. (−1 2

2 −3 ) ASSIGNMENT – 6

1. x = -2; y = 1; z = 2

2. x = 2; y = -1; z = 4

3. x = -2; y = 1; z = 2

4. x = 2; y = 1; z = 3

5. x =1; y = 2; z = 5

102 | P a g e

103 | P a g e6. x = ½; y =0; z = ½

7. x = 2; y = -1; z = -2

8. No soln.

9. x = 3; y = 2; z = 1 10. x = -1; y = 2; z = 1

11. x = 4; y = 0; z = 2 12. x = 1; y = -2; z = 3

13. x = -28/5 ; y = 24/5 ; z = 0 4. (a) x = ½; y = 1/3 ; z = 1/5 (b) x = 2; y = 3; z = 5

15.(8 0 00 8 00 0 8) ; x = 3 ; y = -2; z = -1 16.(6 0 0

0 6 00 0 6) ; x = 2 ; y = -1 ; z = 4

ASSIGNMENT – 71. Proving sum. 2. X = -33. X = -2±√24. X = 2

ASSIGNMENT – 14

(1) π /4 +X/2

(2) X/2

(3) cos−1( 3

5 ) ,sin−1( 45 )

( 4) x+π /4

(5) x+sin−1 (12

13 )(6) sin−1 x+sin−1√x

(7) 1

(10) x=−2+√7

3

(11) x=1/4 , x¿−8

(12) x=4/3 , -3/8 not possible

(13) x=0, x=1/2, x¿ -1/2

103 | P a g e

104 | P a g e

(14) x=-1±√4+2√3

(15) x=0, x=π /4

(17) √3−a2

ASSIGNMENT – 15

1. P(A/B) = 3/14

P(AUB) = 0.85

2. P(E/F) = 2/5

3. P(E/F) = 5/9

4. 2/3

5. 1/3

6. 1/2

7. 10/21; 30/47

8. ¾

9. 3/14 ; 0.85

10. 2/5

11. 1/7

12. P(both odd) = 7 C2

7 C2+6C2

ASSIGNMENT – 16

1. 14/15, 2/5

2. 16/81, 20/81, 40/81

3.0.03, 0.28, 0.22

4. 13/30;1/60

5. 0.216; 0.432; 0.064

6. 16/121; 49/121; 56/121

7. 1/24; 1/3; 1/3; 3/8

8. 3/8104 | P a g e

105 | P a g e9.55/189

10. 9/28

11. 4/7

12. 42(0.5)6

13. 3/10

14. 19/45

15. 38%

16. 437 /500 (or) 87.4%

ASSIGNMENT - 17

1. 93/154

2. 29/63

3.29/45

4. 39/121

5. 360/715

6. 105/442

7. 103/204.

ASSIGNMENT 18

1.

Xi 0 1 2Pi 144/169 24/169 1/169

2.

Xi 0 1 2 3Pi 1724/2197 432/2197 36/2197 1/2197

3.

Xi 0 1 2 3

Pi 64/125 48/125 12/125 1/1254.

105 | P a g e

106 | P a g eXi 0 1 2Pi 144/169 24/169 1/169

5.

Xi 0 1 2Pi 25/36 10/36 1/36

Mean=1/3; variance=5/18.

6..

Xi 0 1 2Pi 4/9 4/9 1/9

Mean=2/3 Variance=4/9

7. Mean = 6/5 Variance= 14/25

8. Mean= 9/4 Variance= 3/16

9. Mean= 135/64 Variance= 99/32

10.

Xi 0 1 2 3 4

Pi 625/1296 500/1296 150/1296 20/1296 1/129611 Same as Qn.8

12.

Xi 0 1 2 3

Pi 5/143 40/143 70/143 28/143.

ASSIGNMENT 19

1. 11/13 2. 36/52 3. (i) 13C2

52C2 (ii) ¼ x ¼ 4. ( 2C1 X 2C1

52C2)2

5. ( 2C1 X 2C1

52C2) 6. ( 36 C2

52 C2) 7.

36 C2

52C2

X 4 8. ( 26 C1 X 26 C1

52 C2) 9.

4C4

52C4 10.

26 C4

52C4

X 25

ASSIGNMENT 20

1. ( 56 )

7

( 16 )

5

2. 1−( 1

3 )6

(q+p)n= ( 13+ 2

3 )6

3. 1. 5/16 2. 1/64 3. 63/64

106 | P a g e

107 | P a g e

4. mean=2 variance =1 5. ( 2

3+ 1

3 )27

6. 821/3125 7. 1. 19/144 2. 25/216.

8. 13 /16. 9. .

7150 (47

50 )7

10. 1. 11/576 2. 17/64

11. 124/216, 75/216, 15/216, 1/216

12. 7/27 13. P=1/5, (0.8+0.2)5

14. ( 56+ 1

6 )36

15. 716/729

16. ( 1

4+ 3

4 )12

17. (1/4)5 , 90(1/4)5 , (3/4)5

18. i. (19/20)5 ii.( 6/5) (19/20)4 iii. 1-( 6/5) (19/20)4 4. 1-(19/20)5 19. 4547/8192

20. 1/16

ASSIGNMENT 21

1. (i) 8C2

20C2 (ii)

8C1 X 5 C1

20C2

2. (i) 9C3

18C3 (ii)

6 C1 X3 C1 X 9C1

18 C3 3.

100 C2+75C2−50C2

150 C2 4.

5C2+6 C2+4 C2

15 C2

5. (i)

1681 (ii)

2081 (iii)

4081

6. (i)

10 C3

18 C3 (ii)

8C3

18C3 (ii)

10 C2 X 8C1

18 C3

7.

5C2 X 4C2

9C4 8. (i) 36/121 (ii) 25/121 (iii) 60/121

9.

611

× 513

× 511

+ 511

× 813

× 611 10.

19 C2 X 10 C2

20 C5 11.

513

× 713

× 611

+ 813

× 613

× 511

107 | P a g e

108 | P a g e

12. 1-( 3C2+3C2+4C2+4 C2

14 C2) 13.

8C3

15 C3 14. 2( 3C3 X 3 C1

6 C4) 15. 1/10

16. 1/5

17.

12C2

25C2 18.

2324 (1− 1

4 !)

ASSIGNMENT 22

1. 3/5, 2/5 (2) 2/3 , 1/3 (3) 3/7, 4/7 (4)6/11, 5/11 (5) 36/91, 30/91, 25/91 (6) 4/7, 2/7, 1/7. 8. 30/61 (9) 9/17, 8/17 (10)6/11, 5/11

ASSIGNMENT 23

1. 7. 2.

3 i−6 j+2 k7 3.dr’s proportional 4.

√86 5. (i)

3 i +6 j−2 k7 (ii)

i +2 j−8 k√69

6. 4 7. λ=5/2 8. 8/7 9. cos−1( 2

√5050 )13.

cos−1( −4

√18 ) 14. − i− j+2 k

15. ± 1

√3( i− j−k )

16. ±3(− i+2 j+2 k ) 17. 16 18. √165

/2 19. 8√3

20. (√35 )2=(√14 )2+(√21 )2

ASSIGNMENT 24

2. 90o 3. λ=−1 , 5 (7) λ=±2√21 8. 10/√6 9. 2 10. 68 /√114 11. cos−1(−1

3 )

12. λ=5/2 13.

5 i +2 j+2 k3

Asignment 25

108 | P a g e

109 | P a g e

1. a. x−3

2 =y-2=z-5 b. x−4−4 =

y+69 =

z−1−2 c.

x−a0 =

y−a−a =

z−a0

2.x−3

5 =y−0

2 =z+4

4 3. x−1

2 =y−2

3 =z+4

6 4.x−3

2 =y−1−1 =

z−21

5.x−2

6=

y+3−3

=z+5−5

6.x−2

2=

y+2−2

=z−1

17. (

−13

,13

,1)

8.x−2

16

=y+1

13

=z+1

12

9. x−1

2=

y+1−1

=z−2

3 10.

x−22

=y+4−4

=z+6−6

11. x−1

1 = y+32 = z+2

−2 12. x−21 = y+1

1 = z−4−2 13. (12/5, 23/5, 0)

14. x−0

1=

y−02

=z−0

1 15.

x−3−2

=y1

=z−1−3

; r→

=3i+ j+2k+λ(-2i+ j−3 k)

16. x−2

2 =y+1−2 =

z−11 17.

x−23 =

y+30 =

z−7−3 18.

x−37

= y+113

= z−2−5 ; r

→=3i- j+2k+λ(7i+3

j−5 k)

19.x

−3=

y−4

=z

−2 20.

x+1−6

= y−316

= z+4−6

21.7 ( x−1 )−2 ( y−2 )+7 ( z−3 )=0

22. x−1−3

= y−25

= z−34

23. x−123

= y−31

= z+24−10

Asignment 26

1. a. (12

,−12

,−32 )

b. (−79

2,

321

,−13

2 )

c. √116 unitsd. 3√30

e.5

√59

f. 18

√42

g.14

√29

h. (-1, 1, -1)

109 | P a g e

110 | P a g e

i. 3

√19

j. 8

√29

2. (12

,−12

,−32 )

3. (4, 0, -1)

Asignment 27

(Image, Foot of the perpendicular, Length of a perpendicular from a point to a plane)

1. (-3, 5, 2)2. (3, 6, 11)3. (3, 1, 6)

4. (53

,53

−43

¿

5. (6, 0, 8)

6. eqn : x−1

2= y−2

1= z−4

2 ; co-ordinates (11/9, 19/9, 34/9) ; length 1/3 unit

7. (-1/12, 25/24, -1/12)

8. 3√7826

units

10. 7929

,13329

,−129

29

11. 17 units

Assignment 28Three Dimensional Geometry

(Image, Foot of the perpendicular, Length of a perpendicular from a point to a line)

1. (-5/3, 2/3, 19/3)2. √14units3. (1 , 3 ,5 )

4. (3, 4, 5)5. (-1,2, -5)6. (1, 2, 1)

7. (−3726

,−7126

,−2213

¿

110 | P a g e

111 | P a g e

8. (1, 0, 7)

9. (2, 3, -1)foot = (2, 5, 7) ; eqn : x−2

1= y−5

−6= z−7

4 ; length = √53units

10. 21 units

Assignment 29

Three Dimensional Geometry

Equation of a plane Passing through 2 points and perpendicular to the plane / parallel to the line

1. 2x+2y-4z+8=02. 8x-13y+15z+61=03. (i)4x-y+z-3=0

(ii)25x-15y-17z-78=0(iii) 18x+13y+2z-47=0(iv) 12x+15y-14z-68=0(v) 2x+2y-3z+3=0(vi)5x+2y-3z-17=0(vii) 4x-y-2z-6=0(viii) 7x-11y+z+18=0

4 (i)4x-3y+2z-3=0

(ii)5x-7y+11z+4=0

5. (i) 3x-4z+1=0

(ii)33x-y+7z-58=0(iii) x-y-1=0(iv) 4x-3y+12z-8=0

111 | P a g e

112 | P a g e

Assignment – 30

1. θ = cos−1 1921

2. θ = sin−10 = 00, 1800

3. (i) θ = sin−1 514

(ii) θ = sin−1 4

√42

(iii) θ = sin−1 72√91

(iv) θ = π4

,3 π4

(v) θ = π4

,3 π4

(vi)θ = 0 , π

4. (i) θ =cos−1 −5

√58

(ii) θ = π3

,−π3

(iii) θ = cos−1 75√2

(iv) θ = π2

,3 π2

5.θ = cos−1 −110

Assignment – 31

1. eqn of the plane : x-2y+z-1 = 0

Perpendicular dis : 2

√6

2. Cartesian Eqns : 2x+y+2z+3 = 0

112 | P a g e

113 | P a g e x-2y+2z-3 = 0

Vector eqns :r.(2i + j + 2k ) + 3 = 0

r.(i - 2 j + 2k ) - 3 = 0

3. Cartesian eqn: 6x + 7y + 9z -23 = 0

Vector eqn :r.(6i + 7 j + 9k ) - 23 = 0

4. Cartesian eqn : 22x + 25y + 28z - 81 = 0

Vector eqn :r.(22i + 25 j + 28k ) - 81 = 0

5. eqn : 51x + 15y – 50z + 173 = 0

6. Cartesian eqn : 20x + 23y + 26z – 69 = 0

Vector eqn :r.(20i + 23 j + 26k ) - 69 = 0

7. Cartesian eqn : x – 5y - 3z – 23 = 0

Vector eqn :r.(i - 5 j - 3k ) - 23 = 0

8. Cartesian eqn : 15x – 47y + 28z – 7 = 0

Vector eqn :r.(15i - 47 j + 28k ) - 7 = 0

9. Cartesian eqn : 25x – 23y – 2z = 0

10. 4x + 2y + 4z + 4 = 0 (λ = 1)

X + 3y – z + 1 + 35/59 (3x – y + 5z + 3) = 0 (λ =3559

)

11. Cartesian eqn : 7x + 9y - 10z – 27 = 0

Vector eqn :r.(7i + 9 j - 10k ) - 27 = 0

12. Cartesian eqn : 40x + 11y - 23z + 27 = 0

Vector eqn :r.(40i + 11 j - 23k ) + 27 = 0

13. perpendicular distance = 7

√5



Assignment – 32113 | P a g e

114 | P a g e1. Point = (1,1,1)

2. Points = (-2,-1,3) (2617

,4317

,11117

)

3. Distance = 13 units.

4. Point = (-1,4,3)

5. Point = (3,4,-1)

6. Point = (5,5,4)

7. Distance = 6 units.

Assignment – 33

1. (ii) Eqn : x + y + z = 0

2. Eqn : x – y – z =0

3. Eqn : 12x – 4y + 4z +24 = 0

4. (i) 17x – 47y -24z +172 = 0

(ii) Not coplanar

(iii) 53x – 34y – z +18 = 0

(iv) x + y + z = 1

(v) 13x + 16y + 18 z – 89 = 0

Assignment – 34

1. 17x + 2y – 7z – 12 = 0

2. 15x -17y -7z + 12 = 0

3. 9x – 8y + z + 11 = 0; r.(9i - 8 j + k ) + 11 = 0

4. r.(7i - 8 j + 3k ) + 25 = 0 7x-8y+3z+25=0

5. 13x -11y -10z +4 = 0; r.(13i - 11 j - 10k ) + 4 = 0

6. x – y – z – 2 = 0

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115 | P a g e7. 7 units

8. 1 unit

9. 6√2

10. 17/2

Assignment – 35

1. Eqn : 3x + 2y +2z -13 = 0

Distance : 3√22 / 11 units

2. Eqn : 2x – 2y – z – 5 = 0

Distance : 5/3 units

3. Eqn : 2x – 3y + 3z -1 = 0

4. Eqn : 3x – 4y + 7z +13 = 0

Assignment 36

Continuity

2. a =8. 3. K=10. 4. K=4. 5. K=126. A=

−32 , b≠1, c=

127. K=5. 8. K=6.

9. k=-4 10. K=1.

Assignment 37

Continuity

1. K=3. 2. K=253. K=6. 4. K=±1 5. K=2. 6. K=1. 7. K=±2

8. a=3, b=-8.10. p=−12

Assignment 38

Continuity

1. A= 122. A=

72 , b=

−172 3. A=3, b=1. 4. A=±1,b=1,1±√2. 5. K=8. 8. K=5.

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Assignment 39Differentiation

4. ½¿ - sec 2 x1−tan2 x

]

5.log ( siny )+ ytanxlog (cosx )−xcoty

6. sinxlogx ¿

7. cosx.esinx+(tanx)x (xsecxcosecx+log(tanx)).

8. xcotx((cotx/x)-cosec2xlogx)+sinxx(xcotx+logsinx)

10. (cos x )x (log (cos x )−x tan x )+xcos x(cos x

x−sin x log x )

12. xsin x(sin x

x+cos x log x )+ 1

√1−x1

2√x ) 13. cot x log 55logsin x+ (sin x )x ( xcot x+logsin x )

14.21x sin2 √7 x2+2

√7 x2+2

16. sec 2 x− yx+2 y−1

.18. 2 log (x+√ x2+1)

√ x2+1

Assignment 40

Differentiation: Inverse trigonometric functions:

1.x

√1−x4.2.

12 3.

124. -1.5.

1

2√1−x2 6.½. 7.−1

1+ x2 8. 1

2(1+x2)

9. 1

2(1+x2).10. −3

√1−x2.11. −1

2√1−x2 .13. -21

1+(2X)2 -2x log2 14.−2

1+ x2

Assignment 41

Differentiation: Second Order Derivatives:

1. 2√2a

8. 4 √23 a

. 10. 8√2aπ

.11. 1/a.

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Assignment 42Applications of derivatives: (Rollesthm, LMVT)

1. (a) c=-1/2 (b) π4 (c) 1 (d)

cos−1( 1±√338 )

(2) (3/2, -1/4)

3. (a) c=2, (2,0) (b) 23.6/6 (c) 0.24

4. c=3/2 (3/2, -9/4)

5. (a) 0.2083 (b) 2.000976 (c) 4.000781

(d) 0.1925 (e)0.06083

6. 54.68 (7) -4.96 8. 14.6

Assignment 43

Tangents and Normals

1. (2,-6) 2. a=-2, b=-5 3.(√21

3,−2√21

9 ) 4. 4x-4y+33=0 6. 4x-7y=35

7.4x-2y=1 9. (0,0) (3,27) 10.X+14Y-254=0, X+14Y+86=0 11.X-20Y=7 12. 48X-24Y=23 13.(2,7)

(3,6) Y=7, Y=6 14. 2√2x−3 y=2

15.4bx+4ay=2√2ab

Assignment 44

Completion of Squares

1. 1√2

sin−1(√14 ( x+1 )7 )+C

2. ( x+42 )√x2+8 x−6−11log|x+4+√x2+8 x−6|+C

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3. x−3 [ log|x2+6 x−3|− 72√12

log( x+3−√12x+3+√12 )]+C

4. log|x+1+√ x2−2x+4|+C

5. √22

sin−1 (4 x−1 )+C

6. √ x2+2 x−1+C

7.1

√3sin−1( 3 x−2

√19 )+C

8. sin−1( x+35 )+C

9. sin−1( ex+23 )+C

10. sin−1( x−23 )+C

11. 6√ x2−9 x+20+34 log|x−92+√ x2−9 x+20|+C

12.12

log|x2−2 x−5|+ √63

log|x−1−√6x−1+√6 |+C

13. √36

log| x2−√3+12

x2−1−√32

|+C

14.1

√2sin−1( x−1

√5 )+C

15. x−3 log|x2+6 x+12|−2√3 tan−1( x+3

√3 )+C

16. sin−1( x2+12

√5 )+C

17. log|sin x−1+√sin2 x−2sin x−3|+C

18.12

log|2 x2+4 x−3|−√510

log|x+1−√52

x+1+ √52

|+C

19. log|x2+3 x−18|−23

log|x+ 32+√ x2+3 x−18|+C

20.3√ x2+4 x+5−8 log|x+2+√ x2+4 x+5|

Assignment 45

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3.5

16log¿ x−2∨¿− 7

4 ( x−2 )− 5

16log ¿ x+2∨¿+C ¿¿

4. log ¿1+cos x∨¿−3 log¿2+3cosx∨¿+C ¿¿

5. 12

log|1+x2

3+x2|+C

6. −2 log|1+sin x|+4 log|2+sin x|+C

7. x−2 log|x+1|− 2x+1

+C

8.12

log ¿ x+1∨¿−14

log ¿ x2+1∨¿−12

tan−1 x+C ¿¿

9.12

log|x+1|+ 14

log|x2+1|+ 12

tan−1 x+C

10. 15 [ log|(3+4sin x )4

2+sin x |]+C

11. x2

2+x+ 1

2log ¿x−1∨¿−1

4log ¿x2+1∨¿−1

2tan−1 x+C ¿¿

12.13

log|2+sec x5+sec x|+C

13. 17 (log|(3−2 x )3

( x+2 )2 |)+C

14. same asQ .9

15.−16

log|x−1|−13|x+2|+ 1

2log|x−3|+C

16. log|( x+2 )3

( x+1 )2|− 1x+1

+C

17. log|( 2−cos x1−cos x )|+C

Assignment 47

Integration by parts

1. x2

2sin−1 x+ x

4√1−x2−3

2sin−1 x+C

2. x2

2tan−1 x−1

2x+ 1

2tan−1 x+C

3. x4 log 2x4

− x4

16+C

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6. ex tan x+C

7. ex cot x+C

Different Types of Sums

1.sin 2x

4− sin 8 x

16+C

2.13

log|cosecx−cot x|+C

3. tan x+2 sec x+C

Assignment 49

Definite Integrals1. π

6

2. π2

2 ab

3. π4

4. 132

5. 0

6. 263

7. 98. 2−√2

9. π

10. 252

11. 73

12. 143

13. 12

log175

14. 12

log175

15. 32

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123 | P a g e

16. π4

17. π4

18. 47

19. 12

log13

20. 332

21. π2−1

22. Proving sum, √2−1

23. π4

24. 2(( log2 )2−2 log2+1)

25. 263

26. 272

27. 192

28. π

29. √5−√3

30. 232

31. 29

32. π8

log 2

33. π4

34. 52

35. 2−√2 log(√2+1)

36. π 2

4

37. π

2√2log (√2+1 )

38. Proving sum

39. 732

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40. π12

41. Omit42. √2

43. 932

44. π 2

4

45. log 2

46. π 2−2 π2

47. −π2

log 2

48. π2

16+ 1

2log2−π

4

49. aπ

50. π4

51. 78

52. 152

Assignment 50

Area under Curve1. I) 16

3a2 sq. units

ii) 643 sq. units

2. 9 sq. units124

3. π a2

4−2a2

3 sq. units

4. 6π sq. units

5. 32 sq. units

6. 5 sq. units

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125 | P a g e

7. 98 sq. units

8. 2 πsq. units9. 3 sq. units

10. 152 ( π

2−1) sq. units

11. 2 π3

−√32 sq. units

12. π8 sq. units

13. 152 sq. units

14. 3 sq. units

15. 43(8 π−√3) sq. units

16. I) 2 π3

−√32 sq. units

II) 13 sq. units

17. 12 ( π

2−1) sq. units

18.1

3√2+ 9 π

8−9

4sin−1 1

3 sq. units

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TYPES OF PROBLEMS

(Chapterwise)

RELATIONS AND FUNCTIONS: Equivalence relation Find an inverse of a function. Composition of functions. Binary Operations.

INVERSE TRIGNOMETRIC FUNCTIONS: Principle Values. Prove the equality. Solve for X. Simplify to the simplest form.

MATRICES: Construction of a matrix. Transpose of a matrix- (AB)T=BTAT. Symmetric and Skew Symmetric Matrix. Given a matrix A prove an algebraic equation hence fine inv(A). Inverse of a matrix by elementary transformations. Solving the system of equations by matrix method. Singular Matrix. Induction. (AB)-1=B-1A-1. BAC=D, Find A(where B,C,D are square matrices).

DETERMINANTS: Area of a Triangle. Equation of a line. Properties of a Determinant.

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DIFFRENTIAL CALCULUS: Discuss the continuity of the given function. Find the values of ‘a’ and ‘b’ if the function is continuous. Differentiation.

Implicit Function. Inverse Function. Parametric Function. Log Function. Second Derivative.

Rolle’s and LMVT.

APPLICATION OF DERIVATIVES: Rate of change. Tangents and Normals. Increasing and Decreasing Functions. Approximations. Maxima and Minima.

INTEGRATION: Simplify and Integrate. Substitution. Partial Fraction. Parts. Special Integrals. Definite Integrals. Limit of Sum. Properties of Definite Integrals.

APPLICATIONS OF INTEGRALS: Area of standard curves viz Parabola, Ellipse, Circle. Area enclosed between the two curves. Area of the triangle.

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DIFFERENTIAL EQUATIONS: Order and Degree. Formations of differential equations. Verifications of a given equation as solution of a differential equation. Variable Separable. Homogeneous Equations. Linear Equations.

VECTORS: Addition of vectors. Sections Formulae. Dot Product. Cross Product. Extra Problems in Dot Product. Extra Sums in Cross Product. Scalar Triple Product.

PLANES: Passing through three points. Co-Planar Points. A point and dr’s of Normal is given. Passing through a point and parallel to a plane. Two points and perpendicular to a plane. One point and perpendicular to two planes. A point and normal to a line joining two points. Line of intersection of two planes and a point. Line of intersection of two planes and perpendicular to a plane. Line of intersection of two planes and at a distance of some unit from

another point. Distance between the parallel planes. Angle between two planes.

LINES AND PLANES:

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Passing through two points. Point of intersection of a line and a plane. Distance of a point from the point of intersection of a line and a

plane. Distance of a plane from a point measured parallel to a line. Distance of a line from a point measured parallel to a plane. Foot of the perpendicular from a point on a plane. Image of a point. Foot of the perpendicular from a point on a line. Plane passing through intersection of two planes and passing through

a third point. Plane through intersection of two planes and parallel to a line. Equation of a plane through the two points and parallel to a line. Equation of a plane through the one point and parallel to two lines. Equation of a line through a point and perpendicular to two lines. Co Planar Lines. Shortest Distance between two skew lines. Angle between a line and a plane. Equation of a plane containing a line/containing two parallel lines:

A Point. Perpendicular Plane. Parallel to a line.

Angle between two lines.

LINEAR PROGRAMMING PROBLEMS(LPP): Manufacturing problems. Diet problems. Transportation problems.

PROBABILITY: Conditional Probability.

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Independent Events. Baye’s Theorem. Random Variable. Probability Distribution. Mean and Variance. Binomial Distribution.

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Xii Maths Assignment CBSE 2014-2015 BY R.SAIKISHORE

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Transcript of Xii Maths Assignment CBSE 2014-2015 BY R.SAIKISHORE