MATRICES
1) If A= 1 -2 4 & B= 0 -2 4
2 3 2 1 3 2
3 1 5 -1 1 5 then A+ B is
a) 1 -2 4 b) 1 -2 8
3 3 2 3 3 4
2 1 5 2 1 10
c) 1 -4 8
3 6 4
2 2 10 d) none
2) If A²= 8A + kI where
A= 1 0
-1 7 then k is
a) 7 b) -7 c) 1 d) -1
3) The matrix
λ 7 -2
4 1 3
2 -1 2 is a singular matrix if λ is
a) 2/5 b) 5/2 c) -5 d) none
4) If A= a b
c d then A² is
a) a² b² b) a²+ bc ab+ bd
c² d² ac+ dc bc + d²
c) non-existent d) none
5) If A= α 0 & B= 1 0
1 1 5 1
Such that A² = B then α is
a) 1 b) -1 c) 4 d) none
6) If A= 2 -3 & B= 1 5 μ & C= 2 4 1
1 λ 0 2 -3 1 -1 13 with the relation AB= C then find λ and μ
a) λ= 3, μ= 4
b) λ= 3p4, μ= -3
c) no real value of λ, μ are possible d) none
7) A= coa²θ cosθ sinθ
cosθ sinθ sin²θ
B= cos²φ cosφ sinφ
cosφ sinφ sin²φ
With the relation AB= 0 then|θ - φ| is
a) 0 b) π/2 c) π/4 d) π
8) If A= 0 -4 1
2 λ -3
1 2 -1 then A⁻¹ exist (i.e., A is invertible) if
a) λ ≠ 4 b) λ ≠ 8 c) λ = 4 d) none
9) If A= 1 0 2
0 1 -1
1 2 1
Find the reciprocal of A
a) -3 -4 2
-1 1 -1
1 2 -1
b) 3 4 -2
1 -1 1
-1 -2 1
c) -3 -1 1
-4 1 2
2 -1 -1
d) none
10) If A= 1 -1 1
1 2 0
1 3 0 then the value of|adj A| is equal to
a) 5 b) 0 c) 1 d) none
11) If A= sinα - cosα 0
cosα sinα 0
0 0 1 then A⁻¹ is equal to
a) Aᵀ b) A c) adj A d) none
12) If A= 4 -1 -4
3 0 -4
3 -1 -3 then A² is equal to
a) A B) I c) Aᵀ d) none
13) If f(x)= cosx - sinx 0
sinx cosx 0
0 0 1 then f(x + y) is equal to
a) f(x) + f(y) b) f(x) - f(y) c) f(x) . f(y) d) none
14) If A= 1 ω ω² & B= ω ω² 1 & C= 1
ω ω² 1 ω² 1 ω ω
ω² 1 ω² ω ω² 1 ω² where ω is the complex cube root of 1 then (A + B)C is equal to
a) 0 b) 1 0 0 c) 1 d) 1
0 0 1 0 0 1
0 0 0. 1 1 1
15) If A= 0 c -b & B= a² ab ac
-c 0 a ba b² bc
b -a 0 ca cb c² then AB is equal to
a) 0 b) I c) 2I d) none
16) If the matrices such that AB= C
B= 1 -2 & C= 6 0
1 4 0 6 then Find the matrix A
a) 2 4 b) -1 1 c) 4 2 d) none
1 -1 4 2 -1 1
17) The rank of the matrix
-5 3 2
3 2 -5
4 -1 -3 is
a) 3 b) 2 c) 1 d) none
18) The rank of the matrix
1 2 3
λ 2 4
2. -3 1 is 3 if
a) λ≠ 18/11 b) λ= 18/11 c) λ= - 18/11 d) none
19) The rank of the matrix
4 1 0 0
3 0 1 0
5 0 0 1 is
a) 4 b) 3 c) 2 d)
20) The system of equations x+ y+ z=2, 2x- y+ 3z=5; x- 2y - z= -1 written in Matrix form is
a)
21) If A= 1 x 1 & B= 1 3 2 & C= 1
2 5 1. 2
x
With relation ABC= 0 then x is
a) 2 b) -2 c) 1 d) 4 d) none
22) If A= x+ y y & B= 2 & C= 3
2x x- y -1 2 with the relation AB= C then x. y is equal to
a) -5 b) 5 c) 4 d) 6
****
23) 1 -2 3
2 -1 4
3 4 1 is a
a) rectangular matrix
b) singular matrix
c) square matrix
d) non-singular matrix
24) If A= 3 1 & B= 5 4 6
-1 2 4 1 2
0 6 -5 -1 1 then
a) A+ B exists
b) AB exists
c) BA exists d) none
25) If A= 1 1 1
1 1 1
1 1 1
a) A³= 9A b) A³= 27A c) A+ A= A² d) A⁻¹ does not exist
DETERMINANTS
Sap-2
Evaluate with property:
1) b+ c c+ a a+ b 2a 2b 2c
a+ b b+ c c+a = 2c 2a 2b
c+a a+ b b+c 2b 2c 2a
2) 2ab a² b²
a² b² 2ab= (a³+ b³)²
b² 2ab a²
3) x+ z z y
z z+x x = 4xyz
y x x+ y
4) y+ z x x
y z+ x y = 4xyz
z z x+ y
5) (a²+ b²)/c c c
a (b²+c²)/a a = 4abc
b b (c²+a²)/b
6) a b ax+ by
b c bx + cy
ax+ by bx+ cy. 0
= (b²- ac)(ax²+ 2bxy + c²y)
7) a² bc ac+ c²
a²+ ab b² ac = 4a²b²c²
ab b²+ bc c²
8) b²+ c² ab ac
ab c²+ a² bc = 4a²b²c²
ca cb a²+ b²
9) b²+ c² a² a²
b² c²+a². b²= 4a²b²c²
c² c² a²+b²
10) a b c
a- b b- c c- a = a³+b³+c³-3abc
b+c c+a a+b
11) a- b- c 2a 2a
2b b- c- a 2b =(a+ b+c)³
2c 2c c-a-b
12) a+b+2c a b
c b+c+2a b =. 2(a+b+ c)³
c a c+a+2b
13) 1+a²-b² 2ab -2b
2ab 1-a²+b² 2a= (1+a²+b²)
2b -2a 1-a²-b²
14) a² b² c²
(a+1)² (b+1)² (c+1)²
(a-1)² (b-1)² (c -1)²
=
4| a² b² c²
a b c
1 1 1
15) 1+a₁ a₂ a₃ a₄
a₁ 1+a₂ a₃ a₄
a₁ a₂ 1+a₃ a₄
a₁ a₂ a₃ 1-a₄
= 1+ a₁+ a₂+ a₃ + a₄.
16) 1+ a 1 1 1
1 1+b 1 1
1 1 1+c 1
1 1 1 1+d
= abcd (1+ 1/a+ 1/b + 1/c + 1/d)= s - r if a, b, c are the roots of x⁴+ px³+ qx² + rx + s= 0
17) if α, β, γ≠ 0, then
α+ a₁b₁ a₁b₂ a₁b₃
a₂b₁ β+a₂b₂ a₂b₃
a₃b₁ a₃b₂ γ+a₃b₃
= αβγ[1+ a₁b₁/α + a₂b₂/β + a₃b₃/γ]
18) 1+ x 2 3 4
1 2+ x 3 4
1 2 3+x 4
1 2 3 4+x
= x³(x +10).
19) x a a a
a x a a
a a x a
a a a x
= (x + 3a)(x - a)³
20) ∑a² ∑ab ∑ab
Let ∆= ∑ab ∑a² ∑ab
∑ab ∑ab ∑a²
Show that ∆ is non negative and establish the relation between a, b and c if ∆=0.
21) a²+1 ab ac ad
ab b²+1 bc bd
ac bc c²+1 cd
ad bd cd d²+1 =
a²+1 b² c² d²
a² b²+1 c² d²
a² b² c²+1 d²
a² b² c² d²+1
= 1+ a²+ b²+ c²+ d².
22) a²x ab ac
ab b²+x bc
ac bc c²+x
= x²(a²+ b²+ c²+ x)
23) ax by cz a b c
x² y² z²= x y z
1 1 1 yz zx xy
24) bc bc'+ b'c b'c'
ca ca'+ c'a c'a'
ab ab'+ a'b c'b'
= (bc'+ b'c)(ca' - c'a)(ab' - a'b)
25) 0 a² b² c² 0 aα bβ cγ
a² 0 γ² β²= aα 0 cγ bβ
b² γ² 0 α² bβ cγ 0 aα
c² β² α² 0 cγ bβ aα 0
26) If p+ q+ r= 0 show that
pa qb rc a b c
qc ra pb =pqr| c a b
rb pc qa b c a
27) If a≠p, b ≠q, c≠ r and
p b c
a q c= 0
a b r
Then value of p/(p- a) + q/(q- b) + r/(r- c)
28) a b- y c- z
If a- x b. c - z = 0
a- x b- y c
Then a/x + b/y + c/z= _____
29) x³ 3x² 3x 1
x² x²+2x 2x+1 1 = (x -1)⁶
x 2x+1 x+2 1
1 3 3 1
30) x²+ 2x 2x+1 1
2x+1 x+2 1= (x -1)²
3 3 1
31) (x - 2)² (x -1)² x²
(x -1)² x² (x+1)²= -8
x² (x+1)² (x+2)²
32) 0 x y z
-x 0 c b = (ax - by + cz)²
-y -c 0 a
-z -b -a 0
33) 5 6 7 a
6 7 8 b
7 8 9 c = (a - 2b + c)³
a b c 0
Is this statement true or false?
34) a b - c c+ b
a+ c b c- a
a- b b+ a c
= (a+ b+ c)(a²+ b²+ c²).
35) (b+ c)² a² a²
b² (c+ a)² b²
c² c² (a+ b)²
= 2abc(a+ b + c)³
36) (a+ b)² ca cb
ca (b+ c)² ab
bc ab (c+ a)¹
= 2abc(a+ b+c)³
37) If 2s= a+ b+ c, show that
a² (s- a)² (s - a)²
(s- b)² b² (s- b)²
(s- c)² (s- c)² c²
= 2s³(s- a)(s- b)(s- c).
38) x²+ x x+1 x-2
2x²+3x -1 3x 3x-3= xA+ B
x²+2x+3 2x -1 2x -1
Where A and B are determinants of order 3 not involving x.
39) sin²A sinA cosA cos²A
sin²B sinB cosB cos²B
sin²C sinC cosC cos²C
Given that A+ B+ C=π
40) If A, B, C be the angles of a triangle and
Cos(A- B) cos(B -C) cos(C - A)
cos(A+B) cos(B+ C) cos(C+A)= 0
sin(A+B) sin(B+C) sin(C+A)
Then show that the triangle is an isosceles triangle.
41) If A, B, C be the angles of a triangle and
1 1 1
1+sinA 1+ sinB 1+ sinC=0
SinA+sin²A sinB+sin²B sinC+sin²C
Then show that ∆ must be isosceless.
42) 1 cosα cos²α
1 cosβ cos²β
1 cosγ cos²γ
= 2 sin{(α+β)/2} sin{(β+γ)/2} sin{γ+α)/2 [sin(α-β)+ sin(β- γ)+ sin(γ- α)
43) sinα sin2α sin3α
sinβ sin2β sin3β
Sinγ sin2γ sin3γ
= 16(sinα sinβ sinγ) sin{(α+β)/2} sin{(β+γ)/2} sin{γ+α)/2 [sin(α-β) sin(β- γ)+ sin(γ- α)]
44) If y= sin px, show that
y y₁. y₂
y₃ y₄ y₅ = 0
y₆ y₇ y₈
Where yᵣ means rth differential coefficient of y.
45) - bc b²+ bc c²+ bc
a²+ ac -ac c²+ ac
a²+ ab b²+ab -ab
= (bc + ca+ ab)³.
46) ax- by - cz ay+ bx cx+ az
ay+ bx by- cz- ax bz + cy
cx+ az bz+cy cz- ax - by
= (x²+ y²+ z²)(a²+ b²+ c²)+ax + by+ cz).
47) let a, b, c be real numbers with a²+ b²+ c²= 1. Show that the equation
ax - by - c bx + ay cx + a
bx+ ay -ax+ by- c cy+ b = 0
cx + a cy+ b -ax- by+c
Represent a straight line.
48) ˣCᵣ ˣCᵣ₊₁ ˣCᵣ₊₂
ʸCᵣ ʸCᵣ₊₁ ʸCᵣ₊₂
ᶻCᵣ ᶻCᵣ₊₁ ᶻCᵣ₊₂ =
ˣCᵣ ˣ⁺¹Cᵣ₊₁ ˣ⁺²Cᵣ₊₂
ʸCᵣ ʸ⁺¹Cᵣ₊₁ ʸ⁺²Cᵣ₊₂
ᶻCᵣ ᶻ⁺¹Cᵣ₊₁ ᶻ⁺²Cᵣ₊₂
49) If both n and r be greater than 1, find the value of x if
ˣCᵣ ⁿ⁻¹Cᵣ ⁿ⁻¹Cᵣ₋₁
ˣ⁺¹Cᵣ ⁿCᵣ ⁿCᵣ₋₁ =0
ˣ⁺²Cᵣ ⁿ⁺¹Cᵣ ⁿ⁺¹Cᵣ₋₁
50) If n and p be two+ve integers such that n≥ p+2 and.
ⁿCₚ ⁿCₚ₊₁ ⁿCₚ₊₂
D(n,p)= ⁿ⁺¹Cₚ ⁿ⁺¹Cₚ₊₁ ⁿ⁺¹Cₚ₊₂
ⁿ⁺²Cₚ ⁿ⁺²Cₚ₊₁ ⁿ⁺²Cₚ₊₂
Then show that
D(n,p)= ⁿ⁺²C₃/ᵖ⁺²C₃ D(n -1, p-1)
51) if n> 2 then sum the series
ⁿᵣ₌₂ ∑(-2)ʳ= ⁿ⁻²Cᵣ₋₂ ⁿ⁻²Cᵣ₋₁ ⁿ⁻²Cᵣ
-3 1 1
2 -1 0
52) If 3ⁿ is a factor of the determinant
1 1 1
ⁿC₁ ⁿ⁺³C₁ ⁿ⁺⁶C₁
ⁿC₂ ⁿ⁺³C₂ ⁿ⁺⁶C₂
Then the maximum value of n is 3.
53) Evaluate
ˣC₁ ˣC₂ ˣC₃
ʸC₁ ʸC₂ ʸC₃
ᶻC₁ ᶻC₂ ᶻC₃
54) Suppose three digit numbers A28 , 3B9, and 62C where A, B and C are integers between 0 and 9 are divisible by a fixed integer k. Show that the determinant
A 3 6
8 9 C
2 B 2
is also divisible by k.
55) If abc, lmn, and pqr be any three digit numbers each of which is divisible by k, then show that
a b c
l m n
p q r
is also divisible by k.
56) For a fixed positive integer n, if
n! (n+1)! (n+2)!
(n+1)! (n+2)! (n+3)!
+n+2)! (n+3)! (n+4)!
Then show that [D/(n!)³ -4] is divisible by n.
57) Let a> 0, d> 0. Find the value of the determinant
1/a 1/a(a+d) 1/(a+d)(a+2d)
1/(a+d) 1/(a+d)(a+2d) 1/(a+2d)(a+3d)
1/(a+2d) 1/(a+2d)(a+3d) 1/(a+3d)(a+4d)
58) If d(x)= (x- a)+x - b)(x -c)+x - d) then show that
a x x x
x b x x
x x c x
x x x d
= f(x) - x f'(x).
