KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 2016-17

CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)

For Slow Learners:

1- A Relation is said to be Reflexive if……………………………….. every a A where A is non empty set.

2- A Relation is said to be Symmetric if…………………………………… …….a,b, A. 3- A Relation is said to be Transitive if…………………………………………(a,c) R , a,b,c A. 4- Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1, T2):

T1 is congruent to T2} . Show that R is an equivalence relation. 5- Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a-b} is

an equivalence relation. 6- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b =

a+1} is reflexive, symmetric or transitive. 7- Prove that the function f: R R, given by f(x) = 2x, is one – one. 8- State whether the function is one – one, onto or bijective f: R R defined by

f(x) = 1+ x2

9- Find gof f(x) = |x|, g(x) = |5x + 1|.

10- If f : R R be defined as f(x) = , then find f f(x)

11- Let * be the binary operation on H given by a * b = L. C. M of a and b. find

(a) 20 * 16

(b) Is * commutative

(c) Is * associative (d) Find the identity of * in N.

12- Show that f: RR defined by f(x)= x3 + 4 is one-one, onto. Show that f -1 (x)=(x - 4)1/3. For Bright Students:

1- Show that the relation R on N x N defined by (a,b)R(c,d) a+d = b+c is an equivalence relation. 2- Let N denote the set of all natural numbers and R be the relation on N x N defined by (a,b)R(c,d)

ad(b c) bc(a d). Show that R is an equivalence relation on N x N. 3- Let a relation R on the set R of real numbers defined as (x,y)R x2 4xy 3y2 0. Show that R

is reflexive but neither symmetric nor transitive.

4- Show that the relation in the set R of real no. defined R = {(a, b) : a b3 }, is neither reflexive nor symmetric nor transitive.

5- Let A = N N and * be the binary operation on A define by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative.

6- Let f : R [5,) given by f(x) = 9x2+6x-5. Show that f is invertible with

f1(y) −1+ 푦+6 3

1

3 33 x

7- Let L be the set of all lines in xy plane and R be the relation in L define as

R = {(L1, L2): L1 || L2}. Show then R is on equivalence relation. Find the set of all lines related to the line Y=2x+4.

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

a * b =

Show that zero in the identity for this operation & each element of the set is invertible with 6 - a being the inverse of a. CLASS - XII MATHEMATICS : (Inverse Trigonometric Functions)

For Slow Learners:

1- Write the principal value of the following

2- Write the principal value of .

3- Write the following in simplest form :

4- Prove that

5- Prove that .

6- .

For Bright Students:

1. Prove that

2. Prove that

3. Solve

6ba,6ba6baif,ba

23cos.1 1

21sin.2 1

3tan.3 1

21cos.4 1

3π2sinsin

3π2coscos 11

0x,x

1x1tan2

1

3677tan

53sin

178sin 111

4π

81tan

71tan

51tan

31tan 1111

1731tan

71tan

21tan2thatovePr 111

4,0x,

2x

xsin1xsin1xsin1xsin1cot 1

xcos21

4x1x1x1x1tan 11

4/πx3tanx2tan 11

4. Solve

5. Solve

6. Prove that

7. Prove that

8. Prove that

CLASS - XII MATHEMATICS : (Matrices & Determinant)

For Slow Learners:

1. If a matrix has 5 elements, what are the possible orders it can have?

2. Construct a 3 × 2 matrix whose elements are given by aij = |i – 3j |

3. If A = , B = , then find A –2 B.

4. If A = and B = , write the order of AB and BA.

5. For the following matrices A and B, verify (AB)T = BTAT,

where A = , B = 6. Give example of matrices A & B such that AB = O, but BA ≠ O, where O is a zero

matrix and A, B are both non zero matrices.

7. If B is skew symmetric matrix, write whether the matrix (ABAT) is Symmetric or skew symmetric.

8. If A = and I = , find a and b so that A2 + aI = bA

9. Find the adjoint of the matrix A =

10. If A = , find A-1 and hence solve the following system of equations: 2x – 3y + 5z = 11, 3x + 2y – 4z = - 5, x + y – 2z = - 3

11. Using matrices, solve the following system of equations: a. x + 2y - 3z = - 4 2x + 3y + 2z = 2 3x - 3y – 4z = 11

318tan1xtan1xtan 111

4π

2x1xtan

2x1xtan 11

1,

21x,xcos

21

4x1x1x1x1tan 11

1663tan

54cos

1312sin 111

4yxyxtan

yxtan 1

b. 4x + 3y + 2z = 60 x + 2y + 3z = 45 6x + 2y + 3z = 70

3. Find the product AB, where A = , B = and use it to solve the equations x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7 4. Using matrices, solve the following system of equations:

- + = 4

+ - = 0

+ + = 2 5. Using elementary transformations, find the inverse of the matrix

11.

For Bright Students:

1. Using properties of determinants, prove that :

2. Using properties of determinants, prove that :

3. Using properties of determinants, prove that :

4. . Express A = as the sum of a symmetric and a skew-symmetric matrix.

5. Let A = , prove by mathematical induction that : .

6. If A = , find x and y such that A2 + xI = yA. Hence find .

2yxqpbaxzpraczyrqcb

zrcyqbxpa

322

22

22

22

ba1ba1a2b2

a2ba1ab2b2ab2ba1

222

2

2

2

cba11ccbca

bc1babacab1a

542354323

3141

n21nn4n21

An

5713 1A

7. Let A= . Prove that .

8. Solve the following system of equations : x + 2y + z = 7, x + 3z = 11, 2x – 3y = 1.

9. Find the product AB, where A = and use it to solve

the equations x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1.

10. Find the matrix P satisfying the matrix equation .

11. Using properties of determinants, prove the following : 1-

2-

3- = (1 + pxyz)(x - y)(y - z) (z - x)

02

tan2

tan0

1001

Iand

cossinsincos

)AI(AI

312221

111Band

135317444

12

2135

23P

2312

abc4bacc

bacbaacb

322

22

22

22

ba1ba1a2b2

a2ba1ab2b2ab2ba1