VERY SHORT ANSWER QUESTIONS
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Total 131 questions
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1) If A is and m x n matrix and B is n x p matrix does AB exists? If yes, write its order. Yes,.mxp
2) If A= 2 1 4 and B= 3 -1
4 1 5 2 2
1 3 , write the order of AB and BA. 2x2 and 3x3
3) If A= 4 3 and B= - 4
1 2 3
write AB. -7
2
4) If A= 1
2
3 write AA'. 1 2 3
2 4 6
3 6 9
5) Give an example of two non-zero 2 x 2 matrices A and B such that AB= O. A= 2 0 &B= 0 0
3 0 2 -1
6) If A= 2 3
5 7 find A+ A'. 4 8
8 14
7) If A= i 0
0 i write A². -1 0
0 -1
8) If A= cosx sinx
- sinx cosx find x satisfying 0< x < π/2 when A+ A'= I. π/3
9) If A= cosx sinx
- sinx cosx find AA'. 1 0
0 1
10) If A= 1 0 and B= x 0
y 5 1 -2 and the relation A+ 2B= I, where I is 2 x 2 unit matrix. find x and y. 0,-2
11) If A= 1 -1
-1 1 satisfies the matrix equation A²=kA,write the value of k. 2
12) If A= 1 1
1 1 satisfies A⁴= kA, then write the value of k. 8
13) -1 0 0
If A= 0 -1 0
0 0 -1 find A². -A or I₃
14) -1 0 0
If A= 0 -1 0
0. 0 -1 find A³. A
15) If A= -3 0
0 -3 find A⁴. 81 0
0 81
16) If A= x 2 and B= 3
4 and the relation AB= 2 find x. -2
17) If A= [aᵢⱼ] is a 2 x 2 matrix such that aᵢⱼ= i+ 2j, write A. 3 5
4 6
18) write matrix A satisfying A+ B= C where B= 2 3 & C = 3 -6
-1 4 -3 8
1 -9
-2 4
19) if A= [aᵢⱼ] is a square Matrix that aᵢⱼ = i² - j², then write whether A is symmetric or skew symmetric. sk
20) For any square matrix write whether AA' is symmetric or skew-symmetric. symm
21) If A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑aᵢⱼ. 0
22) If A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑ ⱼ∑aᵢⱼ . 0
23) If A and B are symmetric matrices, then write the condition for which AB is also symmetric. AB= BA
24) If B is a skew-symmetric matrix, write whether the matrix AB A' is symmetric or Skew-symmetric. Sk
25) If B is a symmetric matrix, write whether the matrix AB A' is symmetric or Skew-symmetric. sy
26) If A is a skew-symmetric and n ∈ ℕ such that (Aⁿ)ᵀ = K Aⁿ, write the value of K. (-1)ⁿ
27) If A is a symmetric matrix and n∈ ℕ, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two. sy
28) if A is a symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two. sy
29) If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two. sk
30) If A and B are symmetric matrices of the same order, write whether AB - BA is symmetric or Skew-symmetric or neither of these two. sk
31) write a square matrix which is both symmetric as well as skew symmetric. null matrix
32) Find the values of x and y, if 2A+ B= C where A= 1 3 B= y 0
0 x 1 2
And C= 5 6
1 8 3,3
33) If x+ 3 4 = 5 4
y-4 x+ y 3 9 then find x and y. 2,7
34) If A is a singular Matrix, then write the value of |A|. 0
35) For what value of x, the following matrix is singular ?
5 - x x+1
2 4 3
35) write the value of the determinant of 2 3 4
2x 3x 4x
5 6 8 0
36) State whether the matrix 2 3
6 4 is singular or non-singular. ns
37) find the value of the determinant 4200 4201
4202 4203 -2
38) find the value of the determinant 101 102 103
104 105 106
107 108 109 0
39) Write the value of the determinant a 1 b+c
b 1 c+a
c 1 a+b 0
40) If A= 0 i & B= 0 1
i 1 1 0 find the value of |A|+| B|. 0
41) if A= 1 2 & B= 1 0
3 -1 -1 0 find |AB|. 0
42) Evaluate 4785 4787
4789 4791 8
43) if w an imaginary cube root of unity, find the value of 1 w w²
w w² 1
w² 1 w 0
44) If A= 1 2 and B= 1 -4
3 -1 3 -2 find |AB|. -70
45) If A=[aᵢⱼ] is a 3x3 diagonal matrix such that a₁₁=11, a₂₂= 2 and a₃₃ = 3, then find |A|. 6
46) If A=[aᵢⱼ] is a 3x3 scalar Matrix such that a₁₁=2, then write the value of |A|. 8
47) If I₃ denotes Identity matrix of order 3x3, write the value of its determinant. 1
48) A matrix A of order 3x3 has determinant 5. What is the value of |3A| ? 135
49) On spending by first row, the value of the determinants of 3x3 square Matrix A=[aᵢⱼ] is a₁₁ C₁₁+ a₁₂C₁₂+ a₁₃ C₁₃, where Cᵢⱼ is the cofactor of aᵢⱼ in A. write the expression for its value on expanding by the second column. a₁₂ C₁₂+ a₂₂C₂₂+ a₃₂ C₃₂
50) let A= [aᵢⱼ] be a square matrix of order 3x3 and Cᵢⱼ denote cofactor of aᵢⱼ in A. If |A|= 5, write the value of a₃₁C₃₁+ a₃₂C₃₂+ a₃₃Ca₃₃. 5
51) In question 18, write the value of write the value of a₁₁C₂₁+a₁₂C₂₂ + a₁₃ C₂₃ . 0
52) Write the value of
sin 20 - cos 20
sin 70 cos 70 1
53) If a square Matrix satisfying A'. A = I, write the value of |A|. ±1
54) If A and B are square matrices of the same order such that |A| = 3 and AB= I, then write the value |B|. 1/3
55) A is a skew-symmetric of order 3, write the value of |A|. 0
56) If A is a square matrix of order 3 with the determinants 4, then write the value of |-A|. -4
57) if A is square Matrix such that |A|= 2, Write the value of |AA'|. 4
58) find the value of the determinant 243 156 300
81 52 100
-3 0 4 0
59) write the value of the determinant of 2 -3 5
4 -6 10
6 -9 15 0
60) If the matrix 5x 2
-10 1 is singular, find the value of x. -4
61) If A is a square matrix of order n x n such that |A| = K, then write the value of |-A|. (-1)ⁿ
62) find the value of the determinant 2² 2³ 2⁴
2³ 2⁴ 2⁵
2⁴ 2⁵ 2⁶ 0
63) if A and B are nonsingular matrices of the same order, write whether AB is singular or non-singular. ns
64) A metrix of order 3 x 3 has determinant 2. What is the value of |A(3I)|, where I is the identity matrix of order 3 x 3. 54
65) If A and B are square matrices of order 3 such that |A| = -1, |B| = 3, then find the value of |3AB|. -81
66) Write the value of a+ ib c+id
-c+id a+ib a²+b²+c²+d²
67) write the cofactor of a₁₂ in the following metrix 2 -3 5
6 0 4
1 5 -7 46
68) If 2x+5 3
5x+2 9 = 0, find x. -13
69) Write the adjoint of the matrix A= -3 4 -2 7
7 -2 4 -3
70) If A is square matrix such that A(adj A)= 5I, where I denotes the identity matrix of the same order. Then, find the value of |A|. 5
71) If A is a square Matrix of order 3 such that |A|= 5, write the value of |adj A|. 25
72) If A is a square matrix of order 3 such that |adj A|= 64, find |A|. ±8
73) if A is a non-singular Square matrix such that |A|=10, find |A⁻¹|. 1/10
74) If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB= AC => B= C. A must be invertible
75) If A is a non-singular square Matrix such that A⁻¹= 5 3
-2 -1 then find (A')⁻¹. 5 -2
3 -1
76) if adj A= 2 3
4 -1 and adjoint of B= 1 -2
-3 1 find adj of AB. -6 5
-2 -10
77) If A is a symmetric matrix, write whether A' is symmetric or Skew-symmetric. Sy
78) If A is a square matrix of order 3 such that |A| = 2, then write the value of adj(adj A). 2A
79) If A is a square matrix of order 3 such |A|=3, then find the value of |adj (adj A)|. 81
80) If A is a square matrix of order 3 such that adj(2A)= k adj(A), then write the value of k. 4
81) If A is a square matrix, then write the matrix adj(A') - (adj)'. Nm
82) Let A be a 3 x 3 square Matrix such that A(adj A)= 2I, where I is the identity Matrix, write the value of |adj A|. 4
83) if A is a nonsingular symmetric matrix, write whether A⁻¹ is Symmetric or Skew-symmetric. Sy
84) If A= cosx Sinx
- sinx cos x and A(adjoint A)= k 0
0 k then find the value of k. 1
85) If A is an invertible Matrix such that |A⁻¹|= 2, find the value of |A|. 1/2
86) If A is a square Matrix such that A(adj A)= 5 0 0
0 5 0
0 0 5 then write the value of |adj A|. 25
87) If A= 2 3
5 -2 be such that A⁻¹= kA, then find the value of k. 1/19
88) Let A be a square matrix such that A² - A + I= O, then write A⁻¹ in terms of A. A⁻¹=(I - A)
89) using cramer's rule write the solution of the system of equations 3x+4y=7; 7x - y= 6. 1,1
90) find the inverse of the matrix
3 -2. 5 2
-7 5 7 3
91) find the inverse of cosx sinx
-sinx cosx
Cox - sinx
Sinx cosx
92) If A= 1 -3
2 0 write adj A. 0 3
-2 1
93) If A= a b & B = 1 0
c d 0 1 find adj(AB). d - b
- c a
94) If A= 1 0 0 & B= x & C= 1
0 1 0 y -1
0 0 1 z 0 with the relation AB= C, then find x,y, z. 1, -1,0
95)If A= 1 0 0 & B= x & C= 1
0 -1 0 y 0
0 0 -1 z 0 with the relation AB= C, then find x,y and z. 1,0, -1
96)If A= 1 0 0 & B= x & C= 1
0 y 0 -1 0
0 0 1 z 1 with the relation AB= C, then find x,y and z. 1,0,1
97) If A= 3 -4 0 & B= x
9 2 0 y with the relation AB= C, then find x,y. 2/3,-2
98) If A= 1 0 0 & B= x & C= 2
0 0 1 y -1
0 1 0 z 3 with the relation AB= C, then find x,y, z. 2,3, -1
99) If A= 2 4 & B= n & C= 8
4 3 1 11 with the relation AC= B, then find n. 2
100) ) A Matrix has 12 elements. Find the possible orders of the matrix. 1x12 or 12x1 or 2x6 or 6x2 or 3x4 or 4x3.
101) Construct a 2x2 matrix whose (i,j)th element as= {(i+j)²/2}.
1/2 9/2
0 2
102) If 3A= 3 -4
15 5 find 4A. 4 -16/3
20 20/3
103) If A+ 1 3 4 -1
-1 2 = 6 2 find A. 3 -4
7 0
104) If 2(x y) + (1 2)=(3 8), find the values of x and y. 1,3
105) if A= 3 and B= x & C= 3
4. y 11 with the relation 2A + 3B= C, find the value of x and y. -1,1
106) Under what condition the product of two matrices AB of two metrices A and B can be determined ?
107) If A= 1 2 3 & B= 7 8
4 5 6 0 9 find which one of the product AB and BA cannot be determined. AB
108) Find the if A and B are two square matrix order of the matrix B if [3 4 2]B=[2 1 0 1 2] 3x5
109) If A= 1 2 3
& B= 4 5 6 & C= 3
5 6 7 2
6 7 8 1 with the relation ABC then find the order of ABC. 1x1
110) If A= 3 0 & B= a b & C= a b
0 3 c d c d Prove AB= 3C.
111) If A= x 2x & B= 2 & C= 16
2y y 3 21 find the value of x and y. 2,3
112) If A= 1 1
0 1 find A² and A³.
1 2 & 1 3
0 1 0 1
113) If A= -3 2
-6 4 show A²= A
114)If A= cosx -sinx B= cosx sinx
sinx cosx - sinx cosx show that AB= I
115) If A= cosx sinx
- sinx cosx show that
A²= cos2x sin2x
- sin2x cos2x
116) Find the value of AB when
A= a & B= a b c
b
c. a² ab ac
ab b² bc
ac bc c²
117) In the case of two matrices A and B if AB= BA, show that (A+B)(A-B)= A² - B².
118) IF A= 7 1 2 & B= 3 & C= 4
9 2 1 4 2
5 find AB+ 2C. 43
44
119) If A and B are two square matrices of the same order then what condition (A+B)²= A²+2AB+B² ? BA= AB
120) for what value of k, the inverse of the matrix A= 2. k
3 5 does not exist? 10/3
121) find the value of the determinant of the square matrix A= 4 -3
9 7 55
122) If A= sin 70 cos 70
Sin 20 - cos 20
find |A|. -1
123) value of logₓy 1
1 logₓy 0
124) find the number of elements of a determinants of order n. n²
125) find the cofactor of the element a in the determinant
7 -6 5
1 2 a
3 -2 1 -4
126) show that a-b b-c c-a
b-c c-a a-b = 0
c -a a- b b-c
127) Evaluate 91 92 93
94 95 96
97 98 99 0
128) Solve: x+ a b
a x+ b = 0. 0, -(a+b)
129) Solve: 1 1 1
p x p = 0
q q x p,q
130) If A=2 3 &B= 0 4 & C= 2 3
-1 6 4 0 -1 6
Is the relation AB= 4C true?
131) If A= 1 -1
-1 1 show that A²= 2A
MULTIPLE CHOICE QUESTIONS
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Total questions - 80
1) If A= 1 0 0
0 1 0
a b -1 then A² is equal to
A) null matrix B) a unit matrix.
C) - A D) A
2) If A= i 0
0 i, n∈ ℕ, then A⁴ⁿ equal
A) 0 i B) 0 0 C). 1 0 D) 0 i
i 0 0 0 0 1 i 0
3) If A and B are two Matrices such that AB= A and BA= B, then B² is equals to
A) B. B) A C) 1 D) 0
4) If AB= A and BA= B, where A and B are square matrices, then
A) B²= B and A²= A.
B) B²≠ B and A²= A
C) A²≠ A and B²= B
D) A²≠ A and B²≠ B
5) If A and B are two Matrices such that AB= B and BA= A, then A²+ B² is equal to
A) 2AB B) 2BA C) A+B. D) AB
6)
7) if the matrix AB is zero, then
A) it is not necessary that either A= O or B= O.
B) A= O or B= O C) A= O and B= O
D) all the above Statements are wrong.
8)
9) If A and B are square matrices of order 3, A is non-singular and AB= O, then B is a
A) null matrix. B) singular Matrix
C) unit Matrix
D) non singular matrix
10) If A= n 0 0 and B= a₁ a₂ a₃
0 n 0 b₁ b₂ b₃
0 0 n c₁ c₂ c₃ then AB is equals to
A) B B) nB. C) Bⁿ D) A+ B
11) If A= 1 a
0 1 then Aⁿ (where n ∈ ℕ) equals
A) 1 na. B) 1 n²a C) 1 na D) n na
0 1 0 1 0 0 0 n
12) If A= 1 2 x and B= 1 -2 y
0 1 0 0 1 0
0 0 1 0 0 1 and AB= I, then x+ y is
A) 0. B) -1 C) 2 D) none
13) If A= 1 -1 & B=a 1
2 -1 b -1 and (A+ B)² = A²+ B², values of a and b
A) a= 4, b= 1 B) a= 1, b= 4.
C) a= 0, b= 4 D) a= 2, b= 4
14) If A= a b
c -a is such that A²= I, then
A) I+ a² + bc= 0 B) I- a² + bc= 0
C) I- a² - bc= 0. D) I+ a² - bc= 0
15) If S= [Sᵢⱼ] is a scalar matrix such that sᵢⱼ = k and A is a square matrix of the same order, then AS= SA= ?
A) Aᵏ B) k+ A C) kA. D) kS
16) If A is a square Matrix such that A² = A, then (I+A)³ - 7A is equals to
A) A B) I - A C) I. D) 3A
17) if a matrix A is both symmetric and skew symmetric, then
A) A is a diagonal matrix
B) A is zero matrix.
C) A is scalar matrix
D) A is a square matrix
18) the matrix 0 5 -7
-5 0 11
7 -11 0 is
A) a skew-symmetric matrix
B) a symmetric matrix
C) a diagonal matrix.
D) an upper triangular matrix
19) If a square matrix, then AA is a
A) skew symmetric matrix
B) symmetric matrix
C) diagonal Matrix
D) none.
20) If A and B are symmetric matrices, then ABA is
A) symmetric matrix.
B) skew-symmetric matrix
C) diagonal Matrix
D) scalar matrix
21) If A= 5 x
y 0 and A= A', then
A) x= 0, y= 5 B) x+y= 5
C) x= y. D) none
22) If A= 3 x 4 matrix and B is a matrix such that A'B and BA' are both defined. then, B is of the type
A) 3x4. B) 3x3 C) 4x4 D) 4x3
23) If A= [aᵢⱼ] is a square matrix of even order such that aᵢⱼ = i² - j³, then
A) A is a skew-symmetric matrix and |A|= 0
B) A is symmetric metrix and |A| is a square.
C) A is a symmetric Matrix and |A|= 0 D) none.
24) If cosx - sinx
sin x cosx then A'+ A= I, if
A) x=nπ, n ∈ Z
B)x=(2n+1)π/2 , n ∈ Z
C) x= 2nπ +π/3, n ∈ Z. D) none
25) If A= 2 0 -3
4 3 1
-5 7 2 is expressed as the sum of a symmetric and skew symmetric matrix, then the symmetric matrix is
A) 2 2 -4 B) 2 4 -5
2 3 4 0 3 7
-4 4 2. -3 1 2
C) 4 4 -8 C) 1 0 0
4 6 8 0 1 0
-8 8 4 0 0 1
26) Iut of the given matrices, choose that matrix which is a scalar Matrix:
A) 0 0 B) 0 0 0 C) 0 0 D) 0
0 0. 0 0 0 0 0 0
27) the number of all possible matrices of order 3x3 with entry 0 or 1 is
A) 27 B) 18 C) 81 D) 512
28) Which of the given values of x and y make the following pairs of matrices equal ?
3x+7 5 = 0 y-2
y+1 2 - 3x 8 4
A) x= -1/3, y= 7
B) x= 7, y= 2/3
C) x= -1/3, y= -2/5
D) not possible to find.
29) If A= 0 2 and kA= 0 3a
3 -4 2b 24
then the values of k, a, b are respectively
A) -6,-12,-18 B) -6,4,9
C) -6,-4,-9. D) -6,12,18
30) If I= 1 0 and J= 0 1
0 1 -1 0 and B= cosx sinx
-sinx cosx then B equals
A) I cosx + J sinx.
B) I sinx + J cosx
C) I cosx - J sinx
D) - I cosx + J sinx
31) The trace of the matrix
A= 1 -5 7
0 7 9
11 8 9 is
A) 17. B) 25 C) 3 D) 12
32) If A=[aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ = k, for all i, then trace of A equal to
A) nk. B) n+ k C) n/k D) none
33) If A and B are square matrices of order 2, then det(A+B)= 0 is possible only when
A) det(A+B)= 0 or det(B)= 0
B) det(A)= 0 + det(B)= 0
C) det(A)= 0 and det(B)= 0
D) A'+ B= O.
34) which of the following is not correct in a given determinant of A, where A=[aᵢⱼ]₃ ₓ ₃.
A) Order of minor is less than order of the det (A)
B) minor of an element can never be equal to cofactor of the same element.
C) Value of a is determinant is obtained by multiplying elements of a row or column by corresponding cofactors
D) order of minors and cofactors of the elements of A is same
35) Let x 2 x
x² x 6
x x 6
Then the value ofax⁴+bx³+cx²+dx+e is equal to
A) 0 B) -16 C) 16 D) none.
36) the value of the determinant. a² a 1
cos nx cos(n+1)x cos(n+2)x
sin nx sin(n+1)x sin(n+2)x is independent of
A) n. B) a C) x D) none
37) If ∆₁= 1 1 1 & ∆₂= 1 bc a
a b c 1 ca b
a² b² c² 1 ab c then A) ∆₁ +∆₂= 0. B) ∆₁+2∆ = 0 C) ∆₁+ ∆₂ D) none
38) Dₖ = 1 n n
2k n²+n+2 n²+n
2k-1 n² n²+n+2 and ⁿₖ₌₁∑ Dₖ = 48, then n equals
A) 4. B) 6 C) 8 D) none
39) Let x²+ 3x x-1 x+3
x+1 -2x x-4
x-3 x+4 3x
= ax⁴ + bx³+ cx² + dx+ e be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is
A) 4 B) 0. C) 1 D) none
40) using the factor theorem it is found that a+b, b+c and c+a are three factors of the determinant
-2a a+ b a+ c
b+a -2b b+c
c+a c+b -2c The other factor in the value of the determinant is
A) 4. B) 2 C) a+b+c D) none
41) if a,b,c are distinct, then the value of x satisfying
0 x²-a x³- b
x²+a 0 x²+ c = 0 is
x⁴+ b x- c 0
A) c B) a C) b D) 0
42)
43)
44)
45)
46) The value of 5² 5³ 5⁴
5³ 5⁴ 5⁵
5⁴ 5⁵ 5⁶ is
A) 5² B) 0 . C) 5¹³ D) 5⁹
47)
48)
49)
50) If A is an invertible matrix, then which of the following is not true
A) (A²)⁻¹= (A⁻¹)² B) |A⁻¹|= |A⁻¹|.
C) (A')⁻¹= (A⁻¹)' D) |A| ≠ 0
51) If A is an invertible matrix of order 3, then which of the following is not true
A) |adj A|= |A|² B) (A⁻¹)⁻¹= A
C) If BA= CA, then B= C, where B and C are square matrices of order 3.
D) (AB)⁻¹= B⁻¹A⁻¹, where B= [bᵢⱼ]₃ ₓ ₃ and |B| ≠ 0
52) If A= 3 4 & B= -2 -2
2 4 0 -1 then (A+B)⁻¹=
A) is a skew-symmetric matrix
B) A⁻¹+ B⁻¹ C) does not exist D) n.
53) If S= a b
c d then adj A is
A) -d -b B)d -b C) d b D) d c
-c a -c a. c a b a
54) If A is a singular matrix, then adj A is
A) non-singular B) singular.
C) symmetric D) not defined
55) If A, B are two n x n non-singular matrices, then
A) AB is non-singular.
B) AB is singular
C) (AB)⁻¹= A⁻¹B⁻¹
D) (AB)⁻¹ does not exist
56) If A= a 0 0
0 a 0
0 0 a then the value of |adj A| is
A) a²⁷ B) a⁹ C) a⁶. D) a²
57) 1 2 -1
If A= -1 1 2
2 -1 1 then det(adj(adj A)) is
A) 14⁴. B) 14³ C) 14² D) 14
58) If B is a non-singular matrix and A is a square matrix, then det(B⁻¹ AB) is equal to
A) Det (A⁻¹) B) Det (B⁻¹)
C) Det (A). D) Det (B)
59) For any 2 x 2 matrix, if A(adj A) = 10 0
0 10 then |A| is equal to
A) 20 B) 100 C) 10. D) 0
60) If A⁵ = O such that Aⁿ for 1≤n ≤ 4, then (I - A)⁻¹ equals
A) A⁴ B) A³ C) I+ A D) none.
61) If A satisfies the equation x³ - 5x³ + 4x + K= 0, then A⁻¹exists if
A) K≠ 1 B) K≠2 C) K≠-1 D) none.
62) If for the matrix A, A³ = I, then A⁻¹=
A) A². B) A³ C) A D) none
63) If A and B are square matrices such that B= - A⁻¹BA, then (A+ B)²
A) O. B) A²+ B². C) A²+2AB+B² D) A+B
64) 2 0 0
0 2 0
0 0 2 Then A⁵ is
A) 5A B) 10A C) 16A. D) 32A
65) For non-singular square matrix A, B and C of the same order (AB⁻¹C)=
A) A⁻¹BC⁻¹ B) C⁻¹B⁻¹A⁻¹
C) CBA⁻¹ D) C⁻¹BA⁻¹.
66) 5 10 3
the matrix -2 -4 6
-1 -2 b is a singular matrix, if the value of b=
A) -3 B) 3 C) 0 D) non-existent.
67) If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is
A) dⁿ B) dⁿ⁻¹. C) dⁿ⁺¹ D) d
68) if A is a matrix of order 3 and |A| = 8, then |adj A|=
A) 1 B) 2 C) 2³ D) 2⁶.
69) If A² - A + I= 0, then the inverse of A is.
A) A⁻² B) A+ I C) I - A. D) A - I
70) if A and B are invertible matrices, which of the following statement is not a correct.
A) adj A= |A| A⁻¹
B) det(A⁻¹)= (A)⁻¹
C) (A+B)⁻¹ = A⁻¹+ B⁻¹.
D) (AB)⁻¹ = B⁻¹A⁻¹
71) If A is a square Matrix such that A² = I, then A⁻¹ is equal to
A) A+ I B) A. C) 0 D) 2A
72) Let A= 1 2 & B= 1 0
3 -5 0 2 and X be a matrix such that A= BX, then X is equal to
A) 1 2 B) -1 2 C) 2 4
3 -5. 3 5 3 -5 D) n
73) If A= 2 3
5 -2 be such that A⁻¹ = kA, then k equal to
A) 19 B) 1/19. C) -19 D) -1/19
74) If A= 1/3 1/3 2/3
2 1 -2
x 2 y is orthogonal, then x+y=
A) 3 B) 0 C) -3. D) 1
75) if A= 1 0 1
0 0 1
a b 2 then aI + bA + 2A² is
A) A B) -A C) ab A D) none .
76) If A= 1 - tan x & B= 1 tanx
tanx 1 -tanx 1 and C= a - b
b a and relation of A B⁻¹= C then
A) a=1, b= 1
B) a=cos 2x , b= sin 2x.
C) a=sin 2x, b= cos 2x D) n
77) If a matrix A is such that 3A³ + 2A² + 5A + I= 0, then A⁻¹ is
A) -(3A² + 2A + 5)
B) (3A² + 2A + 5)
C) (3A² - 2A - 5) D) none .
78) If A is an invertible Matrix, then det A⁻¹ is equal to
A) det(A) B) 1/det(A). C)1 D) n
79) If A= 2 -1
3 -2 then Aⁿ=
A) A= 1 0
0 1 if n is an even natural number.
B) A= 1 0
0 1 if n is an odd natural number
C) A= -1 0
0 1 if n belongs to N
D) none
80) If ∆= a b
c d then K∆ is equal to
A) Ka Kb B) Ka b
Kc Kd c Kd
C) Ka b D) Ka b
Kc d. c d
RELATION
**************
FUNCTION
***********
INVERSE TRIGONOMETRIC FUNCTION
*************************************
Total questions: 74
1) Write the value of sin⁻¹(-√3/2) + cos⁻¹(-1/2). π/3
2) write the difference between maximum and minimum values of sin⁻¹x for x ∈ [-1,1]. π
3) if sin⁻¹x + sin⁻¹y+ sin⁻¹z= 3π/2, then write the value of x+ y +z. 3
4) if x> 1, then write the value of sin⁻¹{2x/(1+x²)} in terms of tan⁻¹x. π - 2 tan⁻¹x.
5) If x < 0, then write the value of cos⁻¹{(1-x²)/(1+x²) in terms of tan⁻¹x. - 2 tan⁻¹x
6) write the value of tan⁻¹x + tan⁻¹(1/x) for x> 0. π/2
7) write the value of tan⁻¹x + tan⁻¹(1/x) for x< 0. -π/2
8) what is the value of cos⁻¹(cos 2π/3) + sin⁻¹(sin 2π/3)? π
9) if - 1< x <0, then write the value of sin⁻¹{2x/(1+x²)} + cos⁻¹{(1-x²)/(1+x²)}. 0
10) write the value of sin(cot⁻¹x). 1/√(1+x²)
11) Write the value of cos⁻¹(1/2) + 2 sin⁻¹(1/2). 2π/3
12) write the range of tan⁻¹x. (-π/2,π/2)
13) value of cos⁻¹(cos1540). 100°
14) value of sin⁻¹(sin(-600°)). 60°
15) value of cos⁻¹(sin(-600°)). 60°
16) value of sin⁻¹(sin(1550°)). 70°
17) Evaluate sin(1/2 cos⁻¹ 4/5). 1/√10
18) Evaluate sin(tan⁻¹ 3/4). 3/5
19) write the value of cos⁻¹(tan 3π/4). π
20) write the value of cos(2sin⁻¹ 1/2). 1/2
21) write the value of cos⁻¹(cos350) - sin⁻¹(sin(350°)). 20°
22) write the value of cos²(1/2 cos⁻¹3/5. 4/5
23) If tan⁻¹x+ tan⁻¹y=π/4, then write the value of x+ y+ xy. 1
24) write the value of cos⁻¹(cos 6). 2π - 6
25) write the value of sin⁻¹(cos π/9). 7π/18
26) value: sin{π/3 - sin⁻¹(sin (-1/2))}. 1
27) value tan⁻¹{tan(15π/4))}. -π/4
28) value of cos⁻¹1/2 + 2 sin⁻¹ 1/2. 2π/3
29) value of tan⁻¹a/b - tan⁻¹{(a-b)/(a+b)}. π/4
30) value of cos⁻¹(cos 5π/4). 3π/4
31) show: sin⁻¹{2x √(1- x²)}=2sin⁻¹x.
32) value:cos⁻¹(1/√2 secπ/4). π/4
33) value of tan⁻¹(cot330°). -π/3
34) value of sin⁻¹(2sin 150). π/2
35)value of cos⁻¹sin cot⁻¹√3. π/3
36)value of sin(1/2cos⁻¹1/2). 1/2
37) value of sec² cot⁻¹1/√3 + tan² cosec⁻¹√2. 5
38) value: sin(π/2 - sin⁻¹3/5). 4/5
39) State with arguments which of the following functions are not defined. cos⁻¹(3), sec⁻¹(√2), cot⁻¹(1/4), cosec⁻¹(-1/√5).
cos⁻¹(3), cosec⁻¹(-1/√5).
40) Indicate the correct one out of the following two statements:
a) cos⁻¹3 + cod⁻¹2=π/2
b) sin⁻¹(1/2)+ cos⁻¹(1/2)=π/2.
41) show sin(cos⁻¹x)= cos(sin⁻¹x) state with reasons whether the equality is valid for all values of x.
42) value of cos⁻¹(cos 4π/3). 2π/3
43) value of cot⁻¹(-√3)+ tan⁻¹(1/√3). 0.33
44) If cos⁻¹(1/√5)= x, then value of cosec⁻¹(√5). π/2- x
45) If sin⁻¹x= k, find the value of cosec⁻¹(1/√(1-x²). π/2 - k
46) value of tan⁻¹3+ tan⁻¹1/3. π/2
47) value of sin⁻¹3/5 + cosec⁻¹5/4. π/2
48) value of tan⁻¹(-2)+ tan⁻¹(-1/2). -π/2
49) if sin⁻¹x +, sin y=sin⁻¹z, then express z in terms of x and y. z= x√(1-y²)+ y√(1-x²).
50) If tan⁻¹x+ tan⁻¹y=π/2, show that, xy = 1.
51) Prove, cos(2 sin⁻¹(x/√2)= 1- x².
52) value of cos(2cos⁻¹3/4). 1/8
53) value of cos(2 sin⁻¹3/√5). 7/25
54) value of tan(2 tan⁻¹1/2). 4/5
55) value of cos(2 tan⁻¹1/3). 4/5
56) value of sin(2 tan⁻¹2/3). 12/13
57) value of sin(2 sin⁻¹4/5). 24/25
58) prove: sin⁻¹3/√5=1/2 cos⁻¹1/3
59) prove: tan⁻¹1/5=1/2 sin⁻¹5/13
60) value of sin(1/2 cos⁻¹4/5). 1/√10
61) If sin⁻¹x= cos⁻¹x, x is. 1/√2
62) If cot⁻¹x+ cot⁻¹2=π/2, x is. 1/2
63) If sin⁻¹x+2 cos⁻¹x= 2π/3, x is. √3/2
64) If cos(2 sin⁻¹x)=-1, x is. ±1
65) The sum of two acute angles tan⁻¹x and tan⁻¹1/2 is 45°, find the value of x. 1/3
66) Find the value of tan⁻¹sin cos⁻¹ √(2/3). π/6
67) value of sin⁻¹sin(sin 5π/6)ᶜ. 1/2 radian
68) value of cos⁻¹x+ cos⁻¹(-x), when 0< x < 1. π
69) value of tan(cos⁻¹4/5 + tan⁻¹2/3). 17/6
70) value of tan{1/2(tan⁻¹x+ tan⁻¹1/x)}. 1
71) If A+ B+ C=π and A= tan⁻¹2, B= tan⁻¹3, show that C=π/4.
72) If sin⁻¹x + sin⁻¹y= 2π/3, find the value of cos⁻¹x + cos⁻¹y.
73) show tan(1/2 cos⁻¹a)= √{(1-a)/(1+a)}.
74) show cos⁻¹√(3/5)= 1/2 cos⁻¹1/5
MULTIPLE CHOICE QUESTIONS
_____________()______()__________
1) value of cos[π/3+ cos⁻¹(-1/2) is
A) 0 B) -1. C) 1 D) n
2) value of sin{tan⁻¹(7π/6) + cos⁻¹(cos 7π/3) is
A) 0 B) -1 C) 1. D) none
3)
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