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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