Mathematics - Determinants and Matrices
Exam Duration: 45 Mins Total Questions : 30
If a,b,c>0 then the value o the determinant
\(\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| \) is
- (a)
always positive
- (b)
always negative
- (c)
always zero
- (d)
none of these
The solution set of the equation
\(\left| \begin{matrix} 1 & 4 & 20 \\ 1 & -2 & 5 \\ 1 & 2x & 5{ x }^{ 2 } \end{matrix} \right| =0\)is
- (a)
0,1
- (b)
1,2
- (c)
1,5
- (d)
none of these
For all values of A,B,C and P,Q,R value of the determinant
\(\left| \begin{matrix} cos(A-P) & cos(B-P) & cos(C-P) \\ cos(A-Q) & cos(B-Q) & cos(C-Q) \\ cos(A-R) & cos(B-R) & cos(C-R) \end{matrix} \right| \), IS
- (a)
cos A cos B cos C
- (b)
cos P cos Q cos R
- (c)
sin A sin B sin C
- (d)
0
If the system of linear equations
x+2ay+az=0
x+3by+bz=0
x+4cy+cz=0
has a non -zero solution, then a,b,c
- (a)
are in A.P.
- (b)
are in G.P.
- (c)
are in H.P.
- (d)
satisfy a+2b+3c=0
If a,b,c are in A.P. then value of
\(\triangle =\left| \begin{matrix} 4 & 5 & 6\quad a \\ 5 & 6 & 7\quad b \\ 6 & 7 & 8\quad c \\ x & y & z\quad 0 \end{matrix} \right| \) is
- (a)
0
- (b)
1
- (c)
2
- (d)
5
If \(A=\left[ \begin{matrix} 1 & 2 \\ -3 & 4 \end{matrix} \right] \), then A2 equals
- (a)
\(\left[ \begin{matrix} 1 & 4 \\ 9 & 16 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 3 & 2 \\ 7 & 4 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 7 & 10 \\ 12 & 24 \end{matrix} \right] \)
- (d)
none of these
The matrix A satisfying the equation
\(\left[ \begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix} \right] A=\left[ \begin{matrix} 1 & 1 \\ 0 & -1 \end{matrix} \right] \) is
- (a)
\(\left[ \begin{matrix} 1 & 4 \\ -1 & 0 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 1 & -4 \\ 1 & 0 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 & 4 \\ 0 & -1 \end{matrix} \right] \)
- (d)
none of these
If A and B are two matrices, then A+B and AB are both defined as
- (a)
number of columns of A is equal to number of rows of B
- (b)
A and B are square matrices of same order
- (c)
A and B are two matrices not necessarily of same order
- (d)
None of the above
If A=\(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\), then An is equal to
- (a)
2n-1A-(n-1)I
- (b)
nA-(n-1)I
- (c)
2n-1A+(n-1)I
- (d)
nA+(n-1)I
Let \(\Delta =\begin{vmatrix} AX & x^{ 2 } & 1 \\ By & y^{ 2 } & 1 \\ Cz & z^{ 2 } & 1 \end{vmatrix}\) and \(\Delta _{ 1 }=\begin{vmatrix} A & B & C \\ x & y & z \\ zy & zx & xy \end{vmatrix}\) then
- (a)
\(\Delta_1=-\Delta\)
- (b)
\(\Delta\ne\Delta_1\)
- (c)
\(\Delta-\Delta_1=0\)
- (d)
None of the above
If the equation are x+y-3=0, \((1+\beta)x+(2+\beta)y-8=0\) and \(x-(1+\beta)y+(2+\beta)=0\) then the value of \(\beta\) for the consistent solution is
- (a)
2
- (b)
\(16\over3\)
- (c)
\(-5\over3\)
- (d)
\(5\over3\)
Let A and B are two matrices of same order 3 x 3, where A = \(\left( \begin{matrix} 1 & 3 & \lambda +2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{matrix} \right) \), B = \(\left( \begin{matrix} 3 & 2 & 4 \\ 3 & 2 & 5 \\ 2 & 1 & 4 \end{matrix} \right) \)
The correct statement is
- (a)
(A + B) (A - B) = A2 + B2 - 2AB
- (b)
(A+B)2 = A2 + B2 + AB + BA
- (c)
(A + B)2 = A2 + B2 + 2AB
- (d)
none of the above
If \({ \triangle }_{ 1 }=\left| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \end{matrix} \right| ,{ \triangle }_{ 2 }=\left| \begin{matrix} 1 & bc & a \\ 1 & ca & b \\ 1 & ab & c \end{matrix} \right| \) then
- (a)
\({ \triangle }_{ 1 }+{ \triangle }_{ 2 }=0\)
- (b)
\({ \triangle }_{ 1 }+{ 2\triangle }_{ 2 }=0\)
- (c)
\({ \triangle }_{ 1 }={ \triangle }_{ 2 }\)
- (d)
\({ \triangle }_{ 1 }=2{ \triangle }_{ 2 }\)
If a, b, c are sides of a triangle and \(\left| \begin{matrix} { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ { \left( a+1 \right) }^{ 2 } & { \left( b+1 \right) }^{ 2 } & { \left( c+1 \right) }^{ 2 } \\ { \left( a-1 \right) }^{ 2 } & { \left( b-1 \right) }^{ 2 } & { \left( c-1 \right) }^{ 2 } \end{matrix} \right| =0\) then
- (a)
\(\Delta ABC\) is an equilateral triangle
- (b)
\(\Delta ABC\) is a right-angled isosceles triangle
- (c)
\(\Delta ABC\) is an isosceles triangle
- (d)
none of the above
TIlt' values of ex for which the system of equations x + y + z = 1, x + 2y + 4z = \(\alpha\), x + 4y + 10z = \(\alpha\) 2 consistent, are given by
- (a)
1, 2
- (b)
-1, 2
- (c)
1, -2
- (d)
-1, -2
If f(x)=\(\left| \begin{matrix} x & \cos { x } & { e }^{ { { x }^{ 2 } } } \\ \sin { x } & { x }^{ 2 } & \sec { x } \\ \tan { x } & 1 & 2 \end{matrix} \right| \) then the value of \(\int _{ -{ \pi }/{ 2 } }^{ { \pi }/{ 2 } }{ f\left( x \right) dx } \) is equal to
- (a)
5
- (b)
3
- (c)
1
- (d)
0
If A = [aij]2\(\times\)2 where aij =i+j, then A is equal to
- (a)
\(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
- (b)
\(\begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}\)
- (c)
\(\begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}\)
- (d)
\(\begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix}\)
The order of the single matrix obtained from is \(\left[ \begin{matrix} 1 & -1 \\ 0 & 2 \\ 2 & 3 \end{matrix} \right] \left\{ \left[ \begin{matrix} -1 & 2 & 1 \\ 2 & 0 & 1 \end{matrix} \right] -\left[ \begin{matrix} 0 & 1 & 23 \\ 1 & 0 & 21 \end{matrix} \right] \right\} \)
- (a)
2\(\times\)3
- (b)
2\(\times\)2
- (c)
3\(\times\)2
- (d)
3\(\times\)3
If A is a square matrix such that A2=A then (I+A)3-7A is equal to
- (a)
A
- (b)
I-A
- (c)
I
- (d)
3A
If A =\(\left[ \begin{matrix} 1 & 2 & -1 \\ 3 & 0 & 2 \\ 4 & 5 & 0 \end{matrix} \right] \) , B =\(\left[ \begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 2 \\ 0 & 1 & 3 \end{matrix} \right] \) ,thten AB is equal to
- (a)
\(\left[ \begin{matrix} 5 & 1 & -3 \\ 3 & 2 & 6 \\ 14 & 5 & 0 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 11 & 4 & 3 \\ 1 & 2 & 3 \\ 0 & 3 & 3 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 & 8 & 4 \\ 2 & 9 & 6 \\ 0 & 2 & 0 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 0 & 1 & 2 \\ 5 & 4 & 3 \\ 1 & 8 & 2 \end{matrix} \right] \)
If A and B are square matrices of same order and A' denotes the transpose of A, then
- (a)
(AB)' =B'A'
- (b)
(AB)' =A'B'
- (c)
AB=0 ⇒ |A| = 0 and |B| = 0
- (d)
AB= O ⇒ A = O or B= O
If A =\(\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{matrix} \right] \) is a matrix satisfying AAT= 9I3, then find the values of a and b respectively
- (a)
1,2
- (b)
-2,-1
- (c)
-1,2
- (d)
-2,1
If \(\left| \begin{matrix} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{matrix} \right| =k\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| ,\) then k =
- (a)
0
- (b)
1
- (c)
2
- (d)
3
\(\left| \begin{matrix} a+1 & a+2 & a+4 \\ a+3 & a+5 & a+8 \\ a+7 & a+10 & a+14 \end{matrix} \right| =\)k(a+b+c)3, then k is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
\(\left| \begin{matrix} { b }^{ 2 }{ c }^{ 2 } & { bc } & b+c \\ { c }^{ 2 }{ a }^{ 2 } & ca & c+a \\ { a }^{ 2 }{ b }^{ 2 } & ab & a+b \end{matrix} \right| =\)
- (a)
a7+b7+c7
- (b)
(a+b+c)7
- (c)
(a2+b2+c2)(a5+b5+c5)
- (d)
0
If a, b, care cube roots of unity, then \(\left| \begin{matrix} { e }^{ a } & { e }^{ 2a } & { e }^{ 3a }-1 \\ { e }^{ b } & { e }^{ 2b } & { e }^{ 3b }-1 \\ { e }^{ c } & { e }^{ 2c } & { e }^{ 3c }-1 \end{matrix} \right| \)=
- (a)
0
- (b)
e
- (c)
e2
- (d)
e3
Find the value of the following determinants. \(\left| \begin{matrix} 1 & { a }^{ 2 }+bc & { a }^{ 3 } \\ 1 & { b }^{ 2 }+ca & { b }^{ 3 } \\ 1 & { c }^{ 2 }+ab & { c }^{ 3 } \end{matrix} \right| \)
- (a)
-(a-b)(b-c)(c-a)(a2+b2+c2)
- (b)
(a-b)(b-c)(c-a)
- (c)
(a2+b2+c2)
- (d)
(a-b)(b-c)(c-a)(a2+b2+c2)
The determinant\(\left| \begin{matrix} { b }^{ 2 }-ab & b-c & bc-ac \\ ab-{ a }^{ 2 } & a-b & { b }^{ 2 }-ab \\ bc-ac & c-a & ab-{ a }^{ 2 } \end{matrix} \right| \)equals
- (a)
abc(b-c)(c-a)(a-b)
- (b)
(b-c)(c-a)(a-b)
- (c)
(a+b+c)(b-c)(c-a)(a-b)
- (d)
None of these
let f(t)=\(\left| \begin{matrix} cost & t & 1 \\ 2sint & t & 2t \\ sint & t & t \end{matrix} \right| \),then \(\underset { t\rightarrow 0 }{ lim } \cfrac { f(t) }{ { t }^{ 2 } } \) is equal to
- (a)
0
- (b)
-1
- (c)
2
- (d)
3
There are two values of a which makes determinant,\(\Delta =\left| \begin{matrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{matrix} \right| =]86,\) then sum of these number is
- (a)
4
- (b)
5
- (c)
-4
- (d)
9