DETERMINANT
KEY CONCEPT
1. The symbol a₁ b₁
a₂ b₂ is called the determinant of order two .
Its value is given by: a₁b₂ - a₂b₁
2. The symbol a₁ b₁ c₁
a₂ b₂ c₂
a₃ b₃ c₃ is called the determinant of order three .
Its value can be found as:
D= a₁| b₂ c₂| - a₂| b₁ c₁| + a₃| b₁ c₁|
b₃ c₃ b₃ c₃ b₂ c₂ OR
D= a₁| b₂ c₂| - b₁|a₂ c₂| + c₁| a₂ b₂
b₃ c₃ a₃ c₃ a₃ b₃ ....and so on.
In this manner we can expend diterminant in 6 ways using elements of ;
R₁ , R₂, R₃ or C₁, C₂, C₃.
3. Following examples of short hand writing large expression are:
i) The lines: a₁x + b₁y + c₁=0.....(1)
a₂x + b₂y + c₂ = 0...(2)
a₂x + b₃y + c₃ =0....(3)
are concurrent if a₁ b₁ c₁
a₂ b₂ c₂ = 0
a₃ b₃ c₃
Condition for consistency of three simultaneous linear equation in two variables .
ii) ax²+ 2hxy + by²+ 2gx + 2fy + c=0 represents a pair of straight lines if:
abc + 2fgh - af² - bg²- ch²= 0 =
a h g
h b f
g f c
iii) Area of a triangle whose vertices are (x₁, y₁) ; r= 1,2.3 is:
D= (1/2)| x₁ y₁ 1|
x₂ y₂ 1
x₃ y₃ 1
If D= 0, then the three points are collinear .
iv) Equation of a straight line passing through (x₁, y₁) & (x₂, y₂) is
x y 1
x₁ y₁ 1 = 0
x₂ y₂ 1₃
4) MINORS:
The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row and the column in which the given element stands. For example p, the minor of a₁ in (Key Concept 2) is
b₂ c₂ & the minor of b₂ is a₁ c₁
b₃ c₃ a₃ c₃
Hence a determinant of order two will have "4 minors" & a determinant of order three will have "9 minors".
5) COFACTOR:
if Mᵢ꜀ represents the minor of some typical elements then cofactor is defined as:
Cᵢ꜀ =(-1)ⁱ⁺ᶜ; Where I & c denotes the row & column in which the particular element lies. Note that the value of a determinants of order 3 in terms of 'Minor' & 'Cofactor' can be written as:
D= a₁₁M₁₁ - a₁₂M₁₂ + a₁₃M₁₃ OR
D= a₁₁C₁₁ + a₁₂C₁₂ + a₁₃ +C₁₃ & so on ..... !
6. PROPERTIES OF DETERMINANTS:
P-1: The value of a determinant remains unaltered, if the rows & columns are interchanged. e.g., if
D= a₁ b₁ c₁ = a₁ a₂ a₃
a₂ b₂ c₂ b₁ b₂ b₃ = D'
a₃ b₃ c₃ c₁ c₂ c₃
D & D' are transpose of each other. If D' = - D then it is skew symmetry determinant but
D'= D => 2D = 0 => D = 0 => Skew symmetric determinant of third order has the value zero.
P-2: if any two rows (or columns) of a determinant be unchanged, the value of determinant is changed in sign only. eg.,
Let D= a₁ b₁ c₁ & D'= a₂ b₂ c₂
a₂ b₂ c₂ a₁ b₁ c₁
a₃ b₃ c₃ a₃ b₃ c₃ then D' = - D.
P-3: If a determinant has any two rows (or columns) identical, then its value is zero.
eg.,
Let D= a₁ b₁ c₁
a₁ b₁ c₁
a₃ b₃ c₃ then it can be verified that D= 0.
P- 4: If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiply by that number.
As if D= a₁ b₁ c₁ and D'= Ka₁ Kb₁ Kc₁
a₂ b₂ c₂ a₂ b₂ c₂
a₃ b₃ c₃ a₃ b₃ c₃ then D'= KD
P-5: If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants. eg.,
a₁+ x b₁+ y c₁ + z =|a₁ b₁ c₁| + |x y z|
a₂ b₂ c₂ a₂ b₂ c₂ a₂ b₂ c₂
a₃ b₃ c₃ a₃ b₃ c₃ a₃ b₃ c₃
P-6: The value of a determinant is not altered by adding to the elements of any row(or column) the same multiplies of the corresponding elements of any other row(or column ). e.g.,
Let D= a₁ b₁ c₁
a₂ b₂ c₂
a₃ b₃ c₃
D'= a₁ + ma₂ b₁+ mb₂ c₁+ mc₂
a₂ b₂ c
a₃+ na₁ b₃+ nb₁ c₃+ nc₁ then D'= D.
Note: that while applying this property atleast one row (or colum) must remain unchanged.
P-7: If by putting x= a the value of a determinant vanishes then(x - a) is a factor of the determinant.
7. MULTIPLICATION OF TWO DETERMINANTS:
A) |a₁ b₁| x |l₁ m₁| = |a₁l₁ + b₁l₂ a₁m₁ + b₁m₂
a₂ b₂ l₂ m₂ a₂l₁ + b₂l₂ a₂m₁+ b₂m₂
similarly two determinants of order three are multiplied.
B) If D= a₁ b₁ c₁ D²= A₁ B₁ C
a₂ b₂ c₂ ≠0 then A₂ B₂ C₂
a₃ b₃ c₃ A₃ B₃ C₃ where A₁, B₁, C₁ are cofas
8) SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES)
a) Consistent Equation : Definite and unique solution.( intersecting lines)
b) Inconsisting Equation : No solution. (Parallel line)
c) Dependent equation : infinite solutions( identical lines)
Let a₁x + b₁y + c₁= 0 & a₂x + b₂y + c₂ = 0 then :
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ => Given equationu are inconsistent
& a₁/a₂ = b₁/b₂ = c₁/c₂ => Given equations are independent
9) Cramer's rule: (simultaneous equations involing 3 unknowns )
Let, a₁x + b₁y + c₁z = d₁ .....(1)
a₂x + b₂y + c₂z = d₂ ....(2)
a₃x + b₃y + c₃z= d₃ .....(3)
Then x= D₁/D, y= D₂/D, z= D₃/D.
where D= a₁ b₁ c₁ , D₁= d₁ b₁ c₁
a₂ b₂ c₂ d₂ b₂ c₂
a₃ b₃ c₃ d₃ b₃ c₃
D₂= a₁ d₁ c₁ & D₃= a₁ b₁ d₁
a₂ d₂ c₂ a₂ b₂ d₂
a₃ d₃ c₃ c₃ b₃ d₃
Note:
a) If D≠ 0 and at least one D₁, D₂, D₃ ≠ 0, then the given system of equations are consistent and have unique nontrivial solution.
b) If D≠ 0 & D₁= D₂= D₃= 0, then the given system of equations are consistent and have trivial solution only.
c) If D= D₁= D₂= D₃ = 0, then the given system of equations are consistent and have infinite solutions
d) If D= 0 but atleast one of D₁, D₂, D₃ is not zero then the equation are inconsistent and have no solution.
10) If x,y,z are not all zero, the condition for a₁x + b₁y + c₁z =0; a₂x + b₂y + c₂z = 0 & a₃x + b₃y + c₃z =0 to be consistent in x,y,z is that
a₁ b₁ c₁
a₂ b₂ c₂ = 0
a₃ b₃ c₃
Remember that if a given system of linear equation have only zero solution for all its variable then the given equation are said to be TRIVIAL SOLUTION.
DETERMINANT
EXERCISE - 1
1) -7 5+ 3i (2/3) - 4i
5-3i 8 4+ 5i = real
(2/3)+ 4i 4 - 5i 9
2) On which one of the parameter out of a, p, d or x the value of the determinant
1 a a²
Cos(p-d)x cospx cos(p+d)x
Sin(p- d)x sinpx sin(p+ d)x does not depend. p
3) x³+1 x² x
If y³+ 1 y² y= 0
z³+1 z² z and x, y, z are all different then, show that xyz = -1
4) a²+2a 2a +1 1
2a +1 a+2 1= (a- 1)³
3 3 1
5) 1 1 1
x y z
x³ y³ z³
= (x - y)(y - z)(z - x)(x + y + z)
6) x 1 -3/2
Let f(x)= 2 2 1
1/(x-1) 0 1/2 Find the minimum value of f(x) (given x > 1). 4
7) If a² + b² + c²+ ab + BC + ca ≤ 0 and a, b, c belongs to R, then find the value of the determinant
(a+ b+2)² a²+ b² 1
1 (b+c+2)² b²+c²
c²+ a². 1 (c+a+2)² 65
8) a b c b+ c c+a a+ d
If D=c a b & D'= a+ b b+ c c+a
b c a c+a a+b b+c then show that D'= 2D
9) 1+a²- b² 2ab -2b
2ab 1-a²+b² 2a= (1+a²+b²)³
2b -2a 1-a²-b²
10) sinx sin(x +h) sin(x+2h)
Let f(x)= sin(x+2h) sinx sin(x +h)
Sin(x +h) sin(x+2h) sinx
If lim ₕ→₀ f(x)/h² has the value equal to k(sin3x + sin³x) find k ∈ N. 3
11) (β+γ-α-δ)⁴ (β+γ-α-δ)² 1
(γ+α-β-δ)⁴ (γ+α-β-δ)² 1
(α+β-γ-δ)⁴ (α+β-γ-δ)² 1
= 64(α-β)(α-γ)(α-δ)(β-γ)(β-δ)(γ-δ)
12) If a, b, c, are the roots of the cubic x³- 3x²+ 2 = 0 then find the value of the determinant
(b+ c)² a² a²
b² (c+a)² b²
c² c² (a+ b)². 108
13) Solve:
a) x +2 2x +3 3x+4
2x+3 3x +4 4x +5 = 0
3x+5 5x +8 10x +17. -1 or -2
b) x-2 2x -3 3x - 4
x-4 2x-9 3x -16= 0
x-8 2x-27 3x -64 4
14) If a+ b +c =0, solve for x
a- x c b
c b-x a=0
b a c- x
0 or ±√{(3/2)(a²+b²+c²)
15) Let a,b,c are the solutions of the cubic x³- 5x²+ 3x -1=0, then find the value of the determinant
a b c
a- b b- c c- a
b+c c+a a+ b. 80
16) a²+ K ab ac
ab b²+ K bc
ac bc c²+ K is divisible by K² and find the other factors.
K²(a²+b²+c²+ K)
17) a² b² c² a² b² c²
(a+1)² (b+1)² (c+1)²= 4 . a b c
(a -1)² (b-1)² (c -1)² 1 1 1
18) In a ∆ ABC, determine condition under which
Cot(a/2) cot(b/2) cot(c/2)
Tan(b/2)+ tan(c/2) tan(c/2)+tan(a/2) tan(a/2)+ tan(b/2)
1 1 1
= 0 triangle ABC is isosceles
19) 0 2x -2 2x+8
If ∆(x)= x-1 4 x²+7
0 0 x+4 and f(x)= ³꜀₌₁∑ ³ᵢ₌₁∑ aᵢ꜀cᵢ꜀ , where aᵢ꜀ is the element of iᵗʰ and cᵗʰ column in ∆(x) and cᵢ꜀ is the factor and iᵢ꜀ and c, then find the greatest value of f(x), where x ∈ [-3,18]. 0
20) (a-p)² (a-q)² (a-r)²
(b-p)² (b-q)² (b -r)² =
(c-p)² (c -q)² (c -r)²
(1+ap)² (1+aq)² (1+ ar)²
(1+ bp)² (1+ bq)² (1+ br)²
(1+cp)² (1+ cq)² (1+ cr)²
21) If a,b,c are the roots of the equation x³- 12x²+ 39x -21= 0, then find the value of the determinant
a b - c c+ b
a+ c b c - a
a- c b+ a c
22) If ax₁²+ by₁²+ cz₁²= ax₂²+ by₂²+ cz₂²= ax₃²+ by₃²+ cz₃²= d and ax₂x₃ + by₂y₃ + cz₂z₃ = ax₃x₁+ by₃y₁ + cz₃z₁ = ax₁x₂ + by₁y₂ + cz₁z₂ = f, then show that
x₁ y₁ z₁
x₂ y₂ z₂ = (d-f)√[(d+2f)/abc]
x₃ y₃ z₃ where a,b,c≠ 0
23) If Sᵣ = αʳ + βʳ + γʳ then show that
S₀ S₁ S₂
S₁ S₂ S₃
S₂ S₃ S₄ = (α-β)²(β-α)²(γ-α)²
24) If u= ax²+ 2bxy + cy², u'= a'x² + 2b'xy + c'y², then show that
y² -xy x²
a b c= ax+by by+ cy
a' b' c' a'x+ b'y b'x +c'y =
(-1/y). u u'
ax+by a'x + b'y
EXERCISE - 2
1) solve the following sets of equation using Cramer's rule and remark about their consistency.
a) x+ y + z -6=0; 2x+ y - z -1 =0; x+ y -2z +3 =0. 1,2,3, consistent
b) x+ 2y + z - 1=0; 3x+ y + z -6 =0; x+ 2y =0. 2,-1, 1 consistent
c) 7x - 7y + 5z -3 =0; 3x+ y +5z -7 =0; 2x+3 y +5z -5 =0. Inconsistent
2) For what value of K do the following system of equations a non trivial (i.e, not all zero) solution over the set of relations Q ?
x+ Ky + 3z =0; 3x+ Ky - 2z =0; 2x+ 3y -4z =0. For that value of K, find all the solutions of the system. 33/2, x: y: z = -15/2: 1: -3
3) The system of equations αx+ y + z = α -1; x+ αy +z =α -1 ; x+ y + αz =α -1 has no solution. Find α. -2
4) If the equation a(y + z)= x, b(z + x)= y, c(x + y)= z have a nontrivial solutions , then find the value of 1/(1+ a) + 1/(1+ b) + 1/(1+ c). 2
5) Given x= cy + bz, y= az + cx , z= bx + ay, Where x,y,z are not all zero, then prove that a²+ b²+ c²+ 2abc = 1.
6) Given a= x/(y -z); b= y/(z - x); c= z/(x - y), where x,y,z are not all zero, prove that : 1+ ab+ bc+ ca= 0.
7) if sinq≠ cosq and x,y,z satisfy the equations
x cosp - y sinp + z = cosq +1
x sinp + y cosp + z = 1- sinq
x cos(p+ q) - y sin(p + q) + z= 2
Then find the value of x²+ y²+ z². 2
8) Investigate for what values of λ, μ the simultaneous equations x + y+ z= 6; x + 2y + 3z =10 & x + 2y + λz = μ have
a) A unique solution. λ≠ 3
b) An infinite number of solutions. λ= 3, μ =10
c) no solution. λ= 3, μ ≠ 10
9) For what values of p, the equations : x + y+ z= 1; x + 2y + 4z =p & x + 4y + 10z = p² have a solution ? Solve them completely in each case. x= 1+ 2k, y= -3K, z= K, when p=1; x= 2K, y= 1 - 3K, z= K when p= 2; Where K ∈ R
10) Solve the equations : Kx + 2y - 2z= 1; 4x + 2Ky - z =2, & 6x + 6y + Kz = 3 considering specially the case when K= 2. if K≠ 2, x/2(K+6) = y/(2K+3)= z/6(K -2) = 1/2(K²+ 2K +15), if K=2, then x=λ, y= (1- 2 λ)/2 and z= 0 where λ ∈ R
11) a) Let a, b, c d ar distinct numbers to be chosen from the set {1, 2, 3, 4, 5}. If the least possible positive solution for x to the system of equation ax+ by =1; CX + dy =2 can be expressed in the form p/q where p and q are relatively prime, then find the value of (p + q). 19
12) Find the sum of all positive integral values of a for which every solution to the sytem of equation x+ ay =3 ayax + 4y = 6 satisfy the inequality x> 1, y > 0. 4
13) If bc+ qr= ca+ rp= ab+ pq= -1 show that
ap a p
bq. b q = 0
cr c r
14) If the following system of equations (a - t)x + by+ cz=0, bx+ (c - t)y + az =0 and cx+ ay+ (b - t)z=0 has non-trivial solutions for different values of t, then show that we can express product of these values of t in the form of determinant.
a b. c
b c a
c a b
15) Solve z+ ax + a²x + a³=0 ; z+ by + b²x + b³=0; z+ cy + c²x + c³=0 where a≠ b ≠ c . -(a+ b+ c), ab+ bc+ ca, - abc
EXERCISE - 3
1) Solve: a² a 1
sin(n+1)x sinnx sin(n-1)x =0
Cos(n+1)x cosnx cos(n-1)x nπ, n ∈ I
2) Test the consistency and solve them when consistent, the following system of equations for all values of λ :
x + y + z =1; x + 3y - 2z = λ ; 3x + (λ+ 2)y - 3z = 2λ +1. If λ= 5, system is consistent with infinite solution given by z= K, y= (1/2) (3K +4) and x= (-1/2) (5K+2) where K ∈ R, if λ≠ 5, system is consistent with unique solution given by z= (1/3) (1- λ); x= (1/3) (λ+2) and y= 0
3) 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 represents a straight line.
4) The number of values of k for which the system of equations : (k +1)x + 8y= 4k; Kx + (k +3)y = 3k -1; kx + (k +3)y = 3k -1 has infinity many solutions is
a) 0 b) 1 c) 2 d) infinite b
5) The value of λ for which the system of the equations 2x - y - z =12; x - 2y + z =-4, x + y + λz = 4 has no solution is
a) 3 b) - 3 c) 2 d) - 2 d
6) a) Consider 3 point P= (-sinβ - α), - cosβ), Q= (cos(β- α), sinβ) and
R= (cos(β - α+ θ), sin(β-θ)), where 0 < α, β, θ < π/4
i) P lies on the line segment RQ
ii) Q lies on the line segment PR
iii) R lies on the line segment QP
iv) P,Q,R are non collinear d
7) Consider the system of equations x - 2y + 3z = -1; -x + y - 2z = k; x - 3y + 4z = 1.
Statement - II: The determinant
1 3 -1
-1 -2 k ≠ 0, for k≠ 3
1 4 1
i) Statement- I is true, Statement- II is true; Statement- II is correct explanation for Statement- I
ii) Statement- I is true, Statement- II is true; Statement- II is NOT a correct explanation for Statement- I.
iii) Statement- I is true, Statement- II is false .
iv) statement- I is false , Statement- II is true. i
8) The number of all possible values of θ, where 0 ≤ θ <π, for which the system of equations
(y+ z) cos3θ = (xyz) sin3θ
x sin3θ = (2 cos3θ)/y + (2 sin3θ)/z
(xyz) Sin3θ = (y + 2z) cos3θ + y sin3θ
have a solution (x₀, y₀, z₀) with y₀z₀ ≠ 0, is 3
8) 1 a a²- bc
1 b b²- ca= 0
1 c c²- ab
9) Evaluate: 18 40 89
40 89 198
89 198 440 -1
10) a- b- c 2a 2a
2b b-c -a 2b =(a+ b+ c)³
2c 2c c-a-b
11) 1+a²-b² 2ab -2b
2ab 1- a²+b² 2a =(1+a²+b²)³
2b -2a 1-a²-b²
12) tan(A+P) tan(B+ P) tan(C+ P)
Tan(A+Q) tan(B+Q) tan(C+Q)
Tan(A+R) tan(B+R) tan(C+R) Show that the determinant vanishes for all values of A, B, C, P, Q, R where A+ B+ C+ P+ Q+ R=0
13) bc bc'+ b'c b'c'
ca ca'+ c'a c'a'
ab ab'+a'b a'b' Factorise. (ab'- a'c)(bc'- b'c)(ca'- c'a)
14) For a fixed positive integer n, if
n! (n+1)! (n+2)!
D= (n+1)! (n+2)! (n+3)!
(n+2)! (n+3)! (n+4)!
Then show that [D/(n!)³ -4] is divisible by n
15) 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
16)
DETERMINANTS & MATRICES
1) A is an involuntary Matrix given by
A= 0 1 -1
4 -3 4
3 -3 4 then the inverse of A/2 will be
a) 2A b) A⁻¹/2 c) A/2 d) A²
2) if A and B are symmetric matrices, than ABA is
a) symmetric matrix
b) skewsymmetric matrix
c) diagonal matrix
d) scalar matrix
3) if a,b,c are all different from zero and
1+ a 1 1
1 1+ b 1= 0
1 1 1+c , then the value of a⁻¹ + b⁻¹ + c⁻¹ is
a) abc b) a⁻¹b⁻¹c⁻¹ c) - a- b- c d) -1
4) if the product of n matrices
[1 1][1 2][1 3].....[1 n]
0 1 0 1 0 1 0 1 is equal to the matrix
1 378
0 1 then the value of n is equal to
a) 26 b) 27 c) 377 d) 378
5) if the system of equations ax+ y + z=0, x+ by + z=0, x+ y + cz=0 (a,b,c ≠ 1) has non trivial solution, then the value of 1/(1- a) + 1/(1- b) + 1/(1- c) is
a) -1 b) 1 c) 0 d) none
6) Let A and B are two square idempotent matrices such that AB ± BA in a null Matrix, then the value of the det.(A - B), is
a) -1 b) 1 c) 0 d) 0 or 1
7) 1+ sin²x cos²x 4sin2x
Let f(x)= sin²x 1+cos²x 4sin2x
sin²x cos²x 1+4sin2x then the maximum value of f(x), is
a) 2 b) 4 c) 6 d) 8
8) if A and B are invertible mattresses, which one of the following statements is/are correct-
a) Adj(A)= |A| A⁻¹
b) det(A⁻¹)= |det(A⁻¹)|
c) (A+ B)⁻¹ = B⁻¹+ A⁻¹
d) (AB)⁻¹ = B⁻¹ A⁻¹
9) If px⁴+ qx³+ rx²+ sx + t =
|x²+3x x-1 x+3|
x+1 2 -x x-3
x-3. x+4 3x then t is
a) 33 b) 0 c) 21 d) none
10) If A= [a b]
c d satisfies the equation x²- (a+ d)x + k =0, then
a) k= bc b) k= ad c) k= a²+ b²+ c²+ d² d) ad - bc= k
11) If a,b,c > 0 and x,y,z ∈R, then the determinant
(aˣ+ a⁻ˣ)² (aˣ - a⁻ˣ)² 1
(bʸ+ b⁻ʸ)² (bʸ -b⁻ʸ)² 1
(cᶻ +c⁻ᶻ)² (cᶻ - c⁻ᶻ)² 1 is equal to
a) aˣbʸcᶻ b) a⁻ˣb⁻ʸc⁻ᶻ c) a²ˣb²ʸc²ᶻ d) zero
12) Identify the incorrect statement in respect of two square matrices A and B conformable for sum and product-
a) tᵣ(A+ B)= tᵣ(A)+ tᵣ(B)
b) tᵣ(αA)= αtᵣ(A), α ∈ R
c) tᵣ(Aᵀ)= tᵣ(A)
d) tᵣ(AB)≠ tᵣ(BA)
13) The determinant
Cos(θ+φ) -sin(θ+φ) cos2φ
sinθ cosθ sinφ
- cosθ sinθ cosφ is
a) 0 b) independent of θ c) independent of φ d) independent of θ & φ both
14) 0 sinα sinα sinβ
Let A= -sinα 0 cosα cosβ
-sinαsinβ -cosαcosβ 0
Then
a) |A| is independent of α and β
b) A⁻¹ depends only on α
c) A⁻¹ depends only on β d) none
15) For positive numbers x,y,z, the numerical value of the determinants
1 logₓy logₓc
Log ᵧx 1 logᵧc
Log꜀x log꜀y 1 is
a) zero b) log xyz c) log(x+ y+ z) d) logx logy logz
16) Which of the following is an orthogonal matrix
a) 6/7 2/7 -3/7 b) 6/7 2/7 3/7
2/7 3/7 6/7 2/7 -3/7 6/7
3/7 -6/7 2/7 3/7 6/7 -2/7
c) -6/7 -2/7 -3/7 d) 6/7. -2/7 3/7
2/7. 3/7 6/7 2/7 2/7 -3/7
-3/7 6/7 2/7 -6/7 2/7 3/7
17) The determinant
ˣC₁ ˣC₂ ˣC₃
ʸC₁ ʸC₂ ʸC₃
ᶻC₁ ᶻC₂ ᶻC₃ is equal to
a) xyz(x+y)(y+z)(z+x)/3
b) xyz(x+y-z)(y+z-x)/4
c) xyz(x-y)(y-z)(z-x)/12 d) none
18) Given the correct order of initial T or F for following statements . Use T if statement is true and F if it is false.
STATEMENT- 1: if A is an invertible 3x3 matrix and B is a 3x4 matrix, then A⁻¹B is defined .
STATEMENT- 2: It is never true that A+ B, A'- B and AB are all defined.
STATEMENT-3: Every matrix none of which entries are zero is invertible.
STATEMENT- 4: Every invertible matrix is square and has no two rows the same.
a) TFFF b) TTFF c) TFFT d) TTTF
19) If a,b,c are all different and
a a³ a⁴-1
b b³ b⁴-1= 0, the
c c³ c⁴-1
a) abc(ab+ bc+ca)= a+b+c
b) (a+b+c)(ab+bc+ca)= abc
c) abc (a+b+c)= ab+ bc+ ca d) none
20) A and B are two given matrices such that the order of A is 3x4, if A'B and B A' are both defined then
a) order B' is 3x4
b) order of B'A is 4x4
c) order of B'A is 3x3
d) B'A is undefined
21) If the determinant
a+ p 1+ x u+ f
b+ p m+ y v+ g
c+ r n +z w+ h splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K, is
a) 6 b) 8 c) 9 d) 12
22) For a given matrix A= cosθ -sinθ
sinθ cosθ which of the following statements holds good ?
a) A= A⁻¹ and θ ∈ R
b) A is symmetric, for θ = (2n+1) π/2, n ∈ I
c) A is an orthogonal matrix for θ ∈ R
d) A is a skew-symmetric. for θ = nπ; n ∈ I
23) Let a determinant is given by
A= a b c
p q r
x y z and suppose det. A= 6.
If B= p+ x q+ y r+z
a+x b+ y c+z
a+ p b+q c+ r then
a) det. B= 6
b) det. B= -6
c) det. B= 12
d) det. B= - 12
24) matrix A = x 3 2
1 y 4
2 2 z if xyz= 60 and 8x+ 4z+ 3z =20, then A(adj A) is equals to-
a) 64 0 0 b) 88 0 0 c) 68 0 0 d) 34 0 0
0 64 0 0 88 0 0 68 0 0 34 0
0 0 64 0 0 88 0 0 68 0 0 34
25) a b a+b a c a+c
D₁ =c d c+d and D₂= b d b+d
a b a-b a c a+b+c then the value of D₁/D₂ where b ≠ 0 and ad ≠ bc, is
a) -2 b) 0 c) -2b d) 2b
26) 1 2 0 2 -1 5
Let A+2B= 6 -3 3 & 2A-B= 2 -1 6
-5 3 1 0 1 2 then Tr(A) - Tr(B) has the value equals to
a) 0 b) 1 c) 2 d) none
27) If a²+ b²+ c²= -2 and
1+ a²x (1+b²)x (1+ c²)x
f(x)= (1+a²)x 1+b²x (1+ c²)x
(1+a²)x (1+b²)x 1+ c²x then f(x) is a polynomial of degree
a) 0 b) 1 c) 2 d) 3
28) The number of solutions of a matrix equation X²= I other than I, is-
a) 0 b) 1 c) 2 d) more than 2. Where I is the 2x2 unit matrix
29) The value of θ, λ for which the following equations
Sinθx - cosθy + (λ+1)z =0; cosθx + sinθy - λz =0; λx + (λ+1)y + cosθz= 0 have nontrivial solutions is
a) θ = nπ, λ ∈ R - {0}
b) θ= 2nπ, λ is any rational number
c) θ= (2n +1)π, λ ∈R⁺, n ∈ I
d) θ = (2n +1)π/2, λ ∈ R, n ∈ I
30) 0 1 2 1/2 -1/2. 1/2
If A= 1 2. 3 & A⁻¹ =-4 3 c
3 a 1 5/2 -3/2 1/2 then
a) a=1, c=-1 b) a=2, c=-1/2 c) a=-1, c=1 d) a=1/2, c=1/2
31) The system of equations :
2x cos²θ+ y sin2θ - 2sinθ = 0
x sin2θ + 2y sin²θ = - 2 cosθ
x sinθ - y cosθ = 0, for all values of θ, can
a) have a unique non trivial solution
b) not have a solution
c) have infinite solution
d) have a trivial solution
32) 3 -3 4
If A=. 2 -3 4
0. -1 1 , then A⁻¹ is equal to
a) A B) A² c) A³ d) A⁴
33) For a non zero real a,b,c
(a²+b²)/c c c
a (b²+c²)/a a= α abc
b b (c²+a²)/b
then the value of α is
a) -4 b) 0 c) 2 d) 4
34) If A= 1 2
2 3 and A²- kA - I₂=0, then value of k is
a) 4 b) 2 c) 1 d) -4
35) If the system of equation, a²x - ay = 1- a & bx +(3- 2b)y= 3+ a pocesses a unique solution x= 1, y= 1 than:
a) a=1, b= -1 b) a= -1, b= 1 c) a=0, b= 0 d) none
36) 2 2 1 -x -y z
Let A= 2 5 2 & B= 0 y 2z
1 2 2 x - y z where x,y,z ∈ R.
if BᵀAB= 8 0 0
0 27 0
0 0 42 then the number of ordered triplet (x,y,z) is
a) 2 b) 6 c) 8 d) 9
37) Number of triplets a,b,c for which system of equations , ax - by = 2a - b and (c+1)x + cy = 10- a + 3b has infinity many solutions and x=1, y=3 is one of the solution, is:
a) exactly one b) exactly two c) exactly three d) infinitely many
38) There are two possible values of A in the solution of the matrix equation
[2A+1 -5] [A-5 B]= [14 D]
-4 A 2A-2 C E F where A,B,C,D,E,F are real numbers . The absolute value of the difference of these two solutions is:
a) 8/3 b) 11/3 c) 1/3 d) 19/3
39) The number of values of k for which the system of equation (k -1)x + (3k+1)y + 2kz =0, (k-1)x+ (4k -2)y+ (k +3)z =0 and 2x + (3k+1)y +3(k -1)z=0 has a common non zero solution is
a) 0 b) 1 c) 2 d) 3
40) Number of real values of λ for which
A= λ-1 λ λ+1
2 -1 3 =0
λ+3 λ-2 λ+7 has no inverse
a) 0 b) 1 c) 2 d) infinite
41) The number of real values of X satisfying
x 3x+2 2x-1
2x-1 4x 3x+1 =0 is
7x-2 17x+6 12x -1
a) 3 b) 0 c) 1 d) infinite
42) 1 -1 1 4 2 2
Let A=2 1 -3 & 10B= -5 0 α
1 1 1 1 -2 3 If B is the inverse of Matrix A, then α is
a) -2 b) -1 c) 2 d) 5
43) 1/z 1/z -(x+y)/z²
If D= -(y+z)/x² 1/x 1/x
-y(y+z)/x²z (x+ 2y+z)/xz -y(x+ y)/xz² then, the incorrect statement is-
a) D is independent of x.
b) D is independent of y
c) D is independent of Z
d) D is independent of x,y,z
44) mx mx -p mx+ p
If f'x)= n n+ p n - p
mx+2n mx+2n+p mx+2n-p
Then y= f(x) represents
a) a straight line parallel to x-axis
b) a straight line parallel to y-axis
c) parabola
d) a straight line with negative slope
45) Let the three matrices are
A= 2 1 B= 3 4 & C= 3 -4
4 1 2 3 -2 3 then
tᵣ(A)+ tᵣ(ABC/2) + tᵣ{A(BC)²/4} + tᵣ{A(BC)³/8}+....+ ∞ is equals to
a) 6 b) 9 c) 12 d) none
46) x -1 (x -1)² x³
x -1 x² (x+1)³
x (x+1)² (x +1)³ then the coefficient of x in D(x) is
a) 5 b) -2 c) 6 d) 0
47) In a square Matrix A of order 3 the elements, ᵢᵢ's are the sum of the roots of the equation x²- (a + b)x + ab=0; aᵢ , ᵢ₊₁'s are the product of the roots, aᵢ , ᵢ₋₁'s are all unity and the rest of the elements are all zero. The value of the det.(A) is equal to
a) 0 b) (a+ b)³ c) a³- b³ d) (a²+ b²)(a+ b)
48) The set of equations
λx - y + (cosθ) z =0
3x + y + 2z =0
(Cosθ)x + y + 2z =0
0 ≤ θ < 2π, has non trivial solution(s)
a) for no value of λ and θ
b) for all values of λ and θ
c) for all values of λ and only two values of θ
d) for only one value of λ and all values of θ
49) Let a= lim ₓ→₁ x/(ln x) - 1/(x ln x) ; b= lim ₓ→₀ (x²- 16x)/(4x + x²); c= lim ₓ→₀ ln(1+ sinx)/x and d= lim ₓ→₃ (x +1)³/{3(sin(x+1) - (x+1))} then the matrix
a b is 2 0
c d 1 -2
a) Idempotent b) Involuntary C) Non singular d) Nilpotent
50) If a, b, c are real then the value of determinant
a²+1 ab ac
ab b²+1 bc = 1 if
ac bc c²+1
a) a+ b+ c b) a+ b+ c =1 c) a+ b+ c =-1 d) a = b = c = 0
51) If A=1 1
1 1 and det.(aⁿ - I) = I - λⁿ, n ∈ N then the value of λ, is
a) 1 b) 2 c) 3 d) 4
52) If the determinants
A= (1+ x)² (1- x)² -2(2+x²) &B=(1+x)² 2x+1 x+1
2x+1 3x 1-5x (1-x)² 3x 2x
x+1 2x 2-3x 1-2x 3x-2 2x-3 with relation A+ B= 0
a) has no solution
b) has 4 real solutions
c) has two real solution and two non real solutions
d) has infinite number of solutions, real or non real
53) If an idopotent matrix is also skew-symmetric then it must be
a) an important matrix
b) an identity matrix
c) an orthogonal matrix
d) a null matrix
54) The value of the determinants
a a+ b a+ 2b
a+2b a a+ b is
a+ b a+2b a
a) 9a²(a+ b)
b) 9b²(a+ b)
c) 3a²(a+ b)
d) 7a²(a+ b)
55) Let A, B , C, D be (not necessarily square) real matrix such that
Aᵀ = BCD; Bᵀ = CDA; Cᵀ = DAB and Dᵀ = ABC.
For the matrix S= ABCD, consider the two statements.
I S³ = S
II. S²= S⁴
a) II is true but not I
b) I is true but not II
c) both I and II are true
d) both I and II are false .
56) If x= a+ 2b satisfies the cubic (a,b ∈ R)
f(x)= a - x b b
b a- x b = 0
b b A'- x
then its other two roots are
a) real and different
b) real and coincident
c) imaginary
d) such that one is real other and imaginary
57) If A and B are different Matrices satisfying A³= B³ and A²B = A, then which of the following are incorrect-
a) det(A²+ B²) must be zero
b) det(A - B) must be zero
c) det (A²+ B²) as well as det(A - B) must be zero
d) at least one of det(A²+ B²) or det(A - B) must be zero
58) If the system of linear equations x+ 2ay + az= 0 ; x+ 3by + bz= 0; x+ 4cy + cz= 0 has s a non zero solution, then a, b, c-
a) are in GP b) are in HP c) satisfy a+ 2b + 3c = 0 d) are in AP
59) Give the correct order of initial T or F for following the statements . Use T if statement is true and F if it is false.
Statement- I: if the graph of two linear equations in two variables are neither parallel nor identical then there is a unique solution to the system.
Statement- II: If the system of equation ax+ by= 0, CX+ dy =0 has a non zero solution, then has infinity many solutions .
Statement- III: The system of equation x+ y + z= 1, x= y, y= 1+ z is inconsistent.
Statement- IV: If two of the equations in a system of three linear equation are inconsistent , then the whole system is inconsistence.
a) FFTT b) TTFT c) TTFF d) TTTF
60) Matrix A= 1 tanx
- tanx 1. then let us define a function f(x)= det.(Aᵀ A⁻¹) then which of the following cannot be the value of f(f(f(f(....f(x)))) is (n ≥ 2)
ⁿ ᵗᶦᵐᵉˢ
a) fⁿ(x) b) 1 c) fⁿ⁻¹(x) d) n f(x)
61). 1+x²-y²-z² 2(xy+z) 2(zx-y)
Let A= 2(xy-z) 1+y²-z²-x² 2(yz+x
2(zx+y) 2(yz-x) 1+z²-x²-y² then determinant A is equals to-
a) (1+ xy+ yz+ zx)³
b) (1+ x²+ y²+ z²)³
c) (xy + yz + zx)³
d) (1+ x³+ y³+ z³)²
62) Let the matrix A and B defined as
A= 3 2 B= 3 1
2 1 7 3 then the value of det(2A⁹B⁻¹). is
a) 2 b) 1 c) -1 d) -2
63) There are two numbers x making the value of the determinant
1 -2 5
2 x -1
0 4 2x
equals to 86. The sum of these two numbers is-
a) -4 b) 5 c) - 3 d) 9
64) a b c
Let A= p q r
x y z and suppose that det(A)= 2 then det(B) equals, where
B= 4x 2a -p
4y 2b -q
4z 2c -r
a) det(B)= -2
b) det(B)= -8
c) det(B)= -16
d) det(B)= 8
66) Let A= a 1
-1 b where a and b are real number. If A² is a null matrix then the product ab equals
a) 0 b) 1 c) -1 d)
67) Let ∆₀ = a₁₁ a₁₂ a₁₃
a₂₁ a₂₂ a₂₃
a₃₁ a₃₂ a₃₃
(Where ∆₀≠ 0) And let ∆₁ denote the determinant formed by the cofactors of elements of ∆₀ and ∆₂ denote the determinant formed by the cofactor of ∆₁ and so on ∆ₙ denotes the determinant formed by the cofactor at ∆ₙ₋₁ then the determinant value of ∆ₙ is
a) ∆²ⁿ₀ b) ∆2ⁿ₀ c) ∆n²₀ d) ∆²₀
68) Consider a matrix
A(θ)= sinθ cosθ
-cosθ sinθ then
a) A(θ) is symmetric
b) A(θ) is skew-symmetric
c) A⁻¹(θ)= A(π- θ)
d) A²(θ)= A(π/2 - 2θ)
69) If A= cos²α sinα cosα & B= cos²β sinβcosβ
Sinαcosα sin²α sinβcosβ sin²β
are such that, AB is a null matrix, then which of the following should necessarily be an odd integral multiple of π/2.
a) α b) β c) α - β d) α + β
70) If A and B are 3 x 3 Matrices and|A|≠ 0, then which of the following are true ?
a) |AB|= 0 => |B|= 0
b) |AB|= 0 => B=0
c) |A⁻¹|= | A|⁻¹
d) |A+ A|= 2|A|
71) The determinant
a² a²- (b - c)² bc
b² b²-(c -a)² ca is divisible by
c² c²- (a-b)² ab
a) a+ b+ c
b) (a+b)(b+ c)(c+ a)
c) a²+ b²+ c²
d) (a- b)(b - c)(c - a)
72) The value of θ lying between -π/4 & π/2 and 0 ≤ A ≤ π/2 and satisfying the equation of detergents
1+ sin²A cos²A 2 sin4θ
sin²A 1+ cos²A 2sin4θ. = 0 are
sin²A cos²A 1+ 2sin4θ
a) A= π/4, θ = -π/8
b) A= 3π/8 = θ
c) A= π/5, θ = -π/8
d) A= π/6, θ = 3π/8
73) If D₁ and D₂ are two 3 x 3 diagonal matrices where none of the diagonal element is zero, then -
a) D₁D₂ diagonal matrix
b) D₁D₂ = D₂D₁
c) D₁²+ D₂² is a diagonal Matrix d) none
74) Which of the following determinant(s) vanish(es)?
a) 1 bc bc(b+c) 1 ab 1/a+ 1/b
1 ca ca(c+a) 1 bc 1/b + 1/c
1 ab ab(a+b) 1 ca 1/c + 1/a
c) 0 a-b a-c d) logₓxyc logₓy logₓc
b-a 0 b-c logᵧxyc 1 logᵧc
c-a c-b 0 log꜀xyc log꜀y 1
75) The determinant
a b aα + b
b c bα + c =0 if
aα+b bα+c 0
a) a,b,c are in AP
b) a,b,c are in GP
c) α is a root of the equation ax²+ bx + c=0
d) (x - α) is a factor of ax²+ 2bx + c
76) If A and B are two 3x3 Matrices such that their product AB is a null Matrix then
a) det. A≠ 0 => B must be a null matrix.
b) det. B≠ 0 => must be e null matrix
c) if none of A and B are null matrices then atleast one of the two matrices must be singular.
d) If neither det. A nor det. B is zero then the given statement is not possible.
77) The value of θ lying between θ = 0 & θ =π/2 & satisfying the equation:
1+ sin²θ cos²θ 4 sin4θ
sin²θ 1+ cos²θ 4 sin4θ. =0 are:
sin²θ cos²θ 1+ 4 sin4θ
a) 7π/24 b) 5π/24 c) 11π/24 d) π/24
78) If p,q,r,s are in AP and
f(x)= p+ sinx q+ sinx p- r+ sinx
q+sinx r+ sinx -1+sinx
r+ sinx s+ sinx s- q + sinx
such that ²₀∫ f(x) dx = -4 then the common difference of the AP can be:
a) -1 b) 1/2 c) 1 d) 2
79) if there are three square matrices A, B, C of same order satisfying the equation A²= A⁻¹ and let B= A²ᵗʰ and C= A²(ⁿ⁻²) then which of the following statements are true ?
a) det(B - C)
b) (B + C)(B - C)=0
c) B must be equal to C d) none
80) The determinant
-2a a+ b c+ a
a+ b -2b b+ c
c+ a b+ c -2c
is divisible by
a) a+ b b) b+ c c) c+ a d) a+ b+ c
1a 2a 3d 4b 5c 6d 7c 8c 9c 10d 11d 12d 13b 14a 15a 16a 17c 18c 19a 20b 21b 22c 23c 24c 25a 26c 27c 28d 29d 30a 31b 32c 33d 34a 35a 36c 37b 38d 39c 40d 41d 42d 43d 44a 45a 46a 47d 48a 49d 50d 51b 52d 53d 54b 55c 56b 57d 58b 59b 60d 61b 62d 63a 64d 65c 66c 67b 68c 69c 70a,c 71a,c,d 72a,b,c,d 73a,b,c 74a,b,c,d 75b,d 76a,b,c,d 77a,c 78a,c 79a,b,c 80a,b,c
Miscellaneous - A
1) If a,b,c are all different & determinant
a a³ a⁴-1
b b³ b⁴-1= 0
c c³ c⁴-1
Then show that abc (ab+ bc + ca)= a+ b+ c.
2) a²+k ab ac
Show ab b²+ k bc
ac bc c²+ k is divisible k² and find the other factor. k²(a²+ b²+ c²+ k)
3) a) Without expanding show that
bc a a² 1 a² a³
ca b b² = 1 b² b³
ab c c² 1 c² c³
b) a² b² c² a² b² c²
(a+1)² (b+1)² (c+1)²= 4. a b c
(a-1)² (b-1)² (c -1)² 1 1 1
4) x-2 2x -3 3x-4
x-4 2x-9 3x-16= 0
x-8 2x-27 3x-64. 4
5) 2ʳ⁻¹ 2(3ʳ⁻¹) 4(5ʳ⁻¹)
If Dᵣ= x y z
2ⁿ -1 3ⁿ -1 5ⁿ- 1 then show ⁿᵣ₌₁∑ Dᵣ = 0
6) In a ∆ ABC, determinant condition under which
cot(A/2) cot(B/2) cot(C/2)
tan(B/2)+ tan(C/2) tan(C/2)+ tan(A/2) tan(A/2)+ tan(B/2) =0
1 1 1 ∆ ABC is isosceles
7) (a₁- b₁)² (a₁-b₂)² (a₁ - b₃)²
(a₂-b₁)² (a₂-b₃)² (a₂-b₃)²
(a₃ -b₁)² (a₃ -b₂)² (a₃-b₃)²
= 2(a₁- a₂)(a₂- a₃)(a₃- a₁)(b₁- b₂)(b₂- b₃)(b₃- b₁)
8) If ax₁²+ by₁²+ cz₁²= ax₂²+ by₂²+ cz₂²= ax₃²+ by₃²+ cz₃²= d and ax₂x₃ + by₂y₃ + cz₂z₃= ax₃x₁ + by₃y₁ + cz₃z₁ = ax₁x₂ + by₁y₂ + cz₁z₂ = f. Then show that
x₁ y₁ z₁
x₂ y₂ z₂
x₃ y₃ z₃ = (d -f)[(d+ 2f)/abc]¹⁾² a,b,c ≠ 0)
9) If Sᵣ= αʳ + βʳ + γʳ then show that
S₀ S₁ S₂
S₁ S₂ S₃
S₂ S₃ S₄
= (α -β 0)²(β-γ)²(γ - α)².
10) If u= ax²+ 2bxy+ cy², u've= a'x²+ 2b'xy + c'y², prove that
y² -xy x²
a b c = ax+by bx+ cy =-(1/y) u u'
a' b' c' a'x+b'y b'x+c'y ax+by a'x+b'y
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