Question 1 :
Find the value of $x$ in $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$.
Question 2 :
If $A = \begin{bmatrix}1& \log_{b}a\\ \log_{a}b& 1\end{bmatrix}$ then $|A|$ is equal to<br>
Question 3 :
If $\begin{bmatrix} x & 1 & 1\\ 2 & 3 & 4\\ 1 & 1 & 1\end{bmatrix}$ has no inverse, then $x=$
Question 4 :
The determinant $\begin{vmatrix}a & b & a\alpha +b\\ b & c & b\alpha +c\\ a\alpha +b & b\alpha +c & 0\end{vmatrix}$ is equal to zero, if.
Question 5 :
Find the value of the following determinant:<br/>$\begin{vmatrix}1.2 & 0.03\\ 0.57 & -0.23\end{vmatrix}$
Question 6 :
If the value of the determinant $\begin{vmatrix}m & 2\\ -5 & 7\end{vmatrix}$ is $31$, find $m$.
Question 7 :
$\displaystyle \Delta = \begin{vmatrix}1 & \log_{x}y  & \log_{x}z \\ \log_{y}x & 1 & \log_{y}z \\ \log _{z}x  &\log _{z}y  & 1\end{vmatrix}$ is equal to <br/> <br/>
Question 8 :
The value of the determinant$\begin{vmatrix} 5 & 1 \\ 3 & 2 \end{vmatrix}$
Question 9 :
If the trivial solution is the only solution of the system of equations$x-ky+z=0,kx+3y-kz=0, 3x+y-z=0$. Then the set of all values of k is:<br>
Question 10 :
If $A$ is a skew symmetric matrix, then $\left| A \right| $ is
Question 11 :
$\mathrm{If}$ $\left|\begin{array}{lll}<br>1 & 0 & 0\\<br>2 & 3 & 4\\<br>5 & -6 & x<br>\end{array}\right|$ $= 45$ $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$ $\mathrm{x}=$<br><br>
Question 12 :
If $A$ is a $3\times 3$ matrix and $\text{det}  (3A)=k(\text{det}  A)$, then $k=$
Question 13 :
If $\begin{vmatrix} x & y\\ 4 & 2 \end{vmatrix}=7$ and $\begin{vmatrix} 2 & 3\\ y & x \end{vmatrix}=4$ then<br>
Question 14 :
Find the values of x, if <br>$\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix}$= $\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$
Question 15 :
If $\displaystyle{\left| {_2^{4\,}\,\,_1^1} \right|^2} = \left| {_1^3\,\,_x^2} \right| - \left| {_{ - 2}^x\,\,_1^3} \right|,$ then $x$=
Question 16 :
If $\omega$ is a non-real cube root of unity and n is not a multiple of 3, then $\displaystyle \Delta =\left | \begin{matrix}<br>1 & \omega^{n} &\omega^{2n} \\ <br>\omega^{2n}&1 &\omega^{n} \\ <br>\omega^{n}&\omega^{2n} &1 <br>\end{matrix} \right |$ is equal to<br>
Question 17 :
If abc $\neq $0 and if $\begin{vmatrix}<br/>a & b & c\\ <br/>b & c & a\\ <br/>c & a & b<br/>\end{vmatrix}$ = 0 then $\dfrac{a^{3}+b^{3}+c^{3}}{abc}$ 
Question 18 :
Find the value of the following determinant:<br/>$\begin{vmatrix}3\sqrt{6} & -4\sqrt{2}\\ 5\sqrt{3} & 2\end{vmatrix}$
Question 19 :
If $\begin{vmatrix}a & -b & -c\\-a & b & -c \\ -a & -b & -c\end{vmatrix}+\lambda abc=0$, then $\lambda$ is equal to<br>
Question 20 :
$x = \left| \begin{gathered}   - 1\,\,\,\,\,\, - 2\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\, - 2 \hfill \\  \,\,\,2\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,1 \hfill \\ \end{gathered}  \right|$, then $x=$
Question 21 :
Let the matrix A and B be defined as $A =\begin{vmatrix} 3 & 2 \\ 2 & 1 \end{vmatrix}$ and $B =\begin{vmatrix} 3 & 1 \\ 7 & 3 \end{vmatrix}$ then the value of Det.$(2A^9B^{-1})$, is
Question 22 :
Find x if it is given that:$\det \left[\begin{array}{lll}<br/>2 & 0 & 0\\<br/>4 & 3 & 0\\<br/>4 & 6 & x<br/>\end{array}\right]=42$<br/>
Question 23 :
If $\Delta_1=\begin{vmatrix} 1 & 0\\ a & b\end{vmatrix}$ and $\Delta_2=\begin{vmatrix} 1 & 0\\ c & d\end{vmatrix}$ then $\Delta_2 \Delta_1$ is equal to<br>
Question 24 :
$D=\begin{vmatrix} 18 & 40 & 89 \\ 40 & 89 & 198 \\ 89 & 198 & 440 \end{vmatrix}=$
Question 25 :
If $A$ is any skew-symmetric matrix of odd order then $\left| A \right| $ equals
Question 26 :
Let a, b, c be three complex numbers, and let<br>$z=\begin{vmatrix}<br>0 & -b & -c\\ <br>b & 0 & -a\\ <br>c & a & 0<br>\end{vmatrix}$<br>then z equal<br>
Question 27 :
For positive numbers $x, y$ and $z$ the numerical value of the determinant $\begin{vmatrix} 1 & \log_x y & \log_x z\\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix}$ is<br>
Question 28 :
<i></i>What is the determinant of the matrix $\left [\begin{matrix} 3& 6\\ -1 & 2\end {matrix} \right]$?<br/>
Question 29 :
Let $\omega\neq{1}$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $ \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ { \omega }^{ 2 } & \omega & 1 \end{bmatrix}$Where each of $a,\ b$ and $c$ is either $\omega$ or ${\omega}^{2}$. Then the number of distinct matrices in the set $S$ is
Question 30 :
If $\begin{vmatrix} 6i & -3i & 1\\ 4 & 3i & -1 \\ 20 & 3 & i\end{vmatrix}=x+iy$, then<br>
Question 31 :
$A=\begin{bmatrix} 5 & 5a & a \\ 0 & a & 5a \\ 0 & 0 & 5 \end{bmatrix}$ If $\left| A^{ 2 } \right| =25$ then $|a|=$
Question 32 :
$\begin{vmatrix} 2^3 & 3^3 & 3.2^2+3.2+1\\ 3^3 & 4^3 & 3.3^2+3.3+1\\ 4^3 & 5^3 & 3.4^2+3.4+1\end{vmatrix}$ is equal to?
Question 33 :
$\displaystyle \Delta = \begin{vmatrix}1 & \log_{x}y  & \log_{x}z \\ \log_{y}x & 1 & \log_{y}z \\ \log _{z}x  &\log _{z}y  & 1\end{vmatrix}$ is equal to <br/> <br/>
Question 34 :
The product of a matrix and its transpose is an identity matrix. The value of determinant of this matrix is
Question 35 :
If $\begin{vmatrix} a & a & x\\ m & m & m\\ b & x & b\end{vmatrix}=0$ then $x=?$
Question 36 :
$\begin{vmatrix} 10! & 11! & 12!\\ 11! & 12! & 13! \\ 12! & 13! & 14!\end{vmatrix}=$ __________.
Question 37 :
If bc +qr = ca + rp = ab + pq = -1 then the value of$\displaystyle \begin{vmatrix}ap& a & p\\bq & b & q\\cr & c & r\end{vmatrix}$ is
Question 38 :
$\begin{vmatrix} \cos { { 15 }^{ o } } & \sin { { 15 }^{ o } } \\ \sin { { 15 }^{ o } } & \cos { { 15 }^{ o } } \end{vmatrix}=$?
Question 39 :
lf $\alpha,\ \beta$ are the roots of the equation$x^{2}+x+1=0$ and $S_k={\alpha}^k+{\beta}^k ; k=1,2,3,4$ , then<br>$\left| \begin{matrix} 3 & 1+{ S }_{ 1 } & 1+{ S }_{ 2 } \\ 1+{ S }_{ 1 } & 1+{ S }_{ 2 } & 1+{ S }_{ 3 } \\ 1+{ S }_{ 2 } & 1+{ S }_{ 3 } & 1+{ S }_{ 4 } \end{matrix} \right| =\\$<br>
Question 40 :
If A is a square matrix of order n such that itselements are polynomials in x and its r-rows becomeidentical for $x =\alpha$ then
Question 41 :
Let $\begin{Bmatrix}\Delta_{1},\Delta_{2},\Delta_{3},.....\Delta_{k}\end{Bmatrix}$ be the set of third orderdeterminants that can be made with thedistinct nonzero real numbers $a_{1},a_{2},a_{3},.......a_{9}.$Then<br>
Question 42 :
The number of values of k for which the system of linear equations, $(2k+1)x+5ky=k+2$ and $kx+(k+2)y=2$ has no solution, is:
Question 43 :
The value of the determinant<br/>$\displaystyle \Delta =\left| \begin{matrix} \log { x }  \\ \log { 2x }  \\ \log { 3x }  \end{matrix}\,\,\,\begin{matrix} \log { y }  \\ \log { 2y }  \\ \log { 3y }  \end{matrix}\,\,\,\begin{matrix} \log { z }  \\ \log { 2z }  \\ \log { 3z }  \end{matrix} \right| $<br/>
Question 44 :
If ${ \Delta }_{ r }=\begin{vmatrix} { a }^{ r } & { b }^{ r } & { c }^{ r } \\ a & b & c \\ 1-a & 1-b & 1-c \end{vmatrix}$. If $\displaystyle \sum _{ r=0 }^{ \infty }{ { \Delta }_{ r } } =0$, then which statement is true.
Question 45 :
If $\left |\begin{array}{111}6i & -3i & 1 \\4 & 3i & -1 \\20 & 3 & i \\\end {array}\right | =x+iy$, then
Question 46 :
If $f(x)=\left| \begin{matrix} \cos { x } \\ 2\sin { x } \\ \tan { x } \end{matrix}\begin{matrix} x \\ { x }^{ 2 } \\ x \end{matrix}\begin{matrix} 1 \\ 2x \\ 1 \end{matrix} \right|$, then$\lim _{ x\rightarrow 0 }{ \frac { f^{ \prime }\left( x \right) }{ x } }$
Question 47 :
For a fixed $+ive$ integer $n$, let $D=$<br>$\left| \begin{matrix} \left( n-1 \right) ! & \left( n+2 \right) ! & { \left( n+3 \right) ! }/{ n\left( n+1 \right) ! } \\ \left( n+1 \right) ! & \left( n+3 \right) ! & { \left( n+5 \right) ! }/{ \left( n+2 \right) !\left( n+3 \right) ! } \\ \left( n+3 \right) ! & \left( n+5 \right) ! & { \left( n+7 \right) ! }/{ \left( n+4 \right) !\left( n+5 \right) ! } \end{matrix} \right| $<br>then $\dfrac { D }{ \left( n-1 \right) !\left( n+1 \right) !\left( n+3 \right) ! } $ is equal to
Question 48 :
If $\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix}=x+iy$, then<br>
Question 49 :
What is the value of a + b + c + d?
Question 50 :
One of the roots of $\begin{vmatrix}x + a & b & c\\ a & x + b & c\\ a & b & x + c\end{vmatrix} = 0$ is
Question 51 :
If $A=\begin{vmatrix} 10 & 2 \\ 30 & 6 \end{vmatrix}\\ $ then $\left|A\right|=$
Question 52 :
Assertion: If $A$ is skew symmetric matrix of order $3\times 3$, then $det (A)=0$ or $\left| A \right| =0$
Reason: If $A$ is a square matrix, then $det (A)=det ({A}){'}=det ({-A}^{'})$
Question 53 :
If $\Delta =\begin{vmatrix}<br>2 & -\sin \theta & 1\\ <br>-\sin \theta & 2 & \sin \theta \\ <br>-1 & -\sin \theta & 2<br>\end{vmatrix}$ then<br>
Question 54 :
Let $\displaystyle \omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}$,then the value of the determinant<br>$\left | \begin{matrix}<br>1 & 1 &1 \\ <br>1& -1-\omega ^{2} &\omega ^{2} \\ <br>1& \omega ^{2} & \omega ^{4}<br>\end{matrix} \right |,$is<br>
Question 55 :
$A=\left[ \begin{matrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{matrix} \right] $; If $\left| { A }^{ 2 } \right| =25$, then $\left| \alpha \right| =$
Question 56 :
If $\displaystyle p+q+r= a+b+c=0$ then the determinant<br>$\displaystyle \Delta =\begin{vmatrix}<br>pa & qb & rc \\ <br>qc & ra & pb \\ <br>rb & pc & qa<br>\end{vmatrix}$ equals<br>
Question 57 :
If $f\left( x \right) =\begin{vmatrix} \cos ^{ 2 }{ x }  & \cos { x } .\sin { x }  & -\sin { x }  \\ \cos { x } .\sin { x }  & \sin ^{ 2 }{ x }  & \cos { x }  \\ \sin { x }  & -\cos { x }  & 0 \end{vmatrix}$, then for all $x \epsilon R $, the value of $f(x)=$
Question 58 :
The value of $\triangle$ = $<br>\left |<br>\begin{array}{111}<br>2 & a+r+2 & a+r \\<br>a+r+2 & 2(a+1)(r+1) & a(r+1)+r(a+1) \\<br>a+r & a(r+1)+r(a+1) & 2ar \\<br>\end {array}<br>\right |<br>$
Question 59 :
The matrix $\begin{bmatrix}1 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & 1 & 1\end{bmatrix}$ is
Question 60 :
If $y=\sin { mx } $, then the value of the determinant $\begin{vmatrix} y & { y }_{ 1 } & { y }_{ 2 } \\ { y }_{ 3 } & { y }_{ 4 } & { y }_{ 5 } \\ { y }_{ 6 } & { y }_{ 7 } & { y }_{ 8 } \end{vmatrix}$, where ${ y }_{ n }=\cfrac { { d }^{ n }y }{ d{ x }^{ n } } $, is <br>
Question 61 :
If the system of equations of $3x-2y+z=0, kx-14y+15z=0,x+2y+3z=0$ has non trivial solution then $k=$
Question 62 :
If $A + B + C = \pi$, then $\begin{vmatrix}\sin (A + B + C)& \sin B& \cos C\\ -\sin B& 0 & \tan A\\ \cos (A + B)& -\tan A& 0\end{vmatrix}$ is equal to<br>
Question 63 :
If $f(x)=\begin{vmatrix} 1 & x & x+1\\ 2x & x(x-1) & (x+1)x\\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1)\end{vmatrix}$ then $f(100)$ is equal to?
Question 64 :
If a, b, c, d denote the determinants of matricesA, B, C, D where <br>A= $\begin{bmatrix}<br>0 & x-y & x-z\\ <br>y-x&0 &y-z \\ <br>z-x&z-y &0 <br>\end{bmatrix}$<br>B= $\begin{bmatrix}<br>1& 1 & 1\\ <br>2&3 &4 \\ <br>4&9 &16 <br>\end{bmatrix}$<br>C=$\begin{bmatrix}<br>0 & 0& 2\\ <br>0&5 &0 \\ <br>7&0 &0 <br>\end{bmatrix}$<br>D=$\begin{bmatrix}<br>1 &1 & 1\\ <br>1&2 &8 \\ <br>1&3 &27 <br>\end{bmatrix}$<br>Then the ascending order of a, b, c, d<br>
Question 65 :
If $A,B,C$ are the angles of a triangle, then the value of determinant<br/>$\begin{vmatrix} -1+\cos { B }  & \cos { C } +\cos { B }  & \cos { B }  \\ \cos { C } +\cos { A }  & -1+\cos { A }  & \cos { A }  \\ -1+\cos { B }  & -1+\cos { A }  & -1 \end{vmatrix}$ is
Question 66 :
If $\Delta =\begin{vmatrix}<br>a & 5-i & 7+i\\ <br>5+i & b & 3+i\\ <br>7-i & 3-i & c<br>\end{vmatrix}$, then $\Delta $ is always<br>
Question 67 :
If one of the roots of $\left |\begin{matrix} 3 & 5 & x\\ 7 & x & 7\\ x & 5 & 3\end{matrix}\right |=0$  is $- 10$, the other roots are
Question 68 :
For a positive numbers $x, y$ and $z$ the numerical value of the determinant $\begin{bmatrix}1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z\\ \log_{z} x & \log_{z} y & 1\end{bmatrix}$ is:
Question 69 :
State true or false. $\left| \begin{array} { c c c } { \sin \theta } & { \cos \theta } & { 1 } \\ { - \cos \theta } & { \sin \theta } & { 1 } \\ { 1 } & { 1 } & { 1 } \end{array} \right| = 1 + 2 \sin \theta$
Question 70 :
If $ A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} $ and $ B = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$ then what is determinant of AB ?
Question 71 :
If the value of determinant $\displaystyle \begin{vmatrix}a & 1 & 1 \\1 & b & 1 \\1 & 1 & c\end{vmatrix}$ is positive then<br>
Question 72 :
If C = $2 \cos \theta$, then the value of the determinant to $\Delta = \left |\begin{matrix} c & 1 & 0 \\ 1 & c & 1 \\ 6 & 1 & c \end{matrix}\right |$ is
Question 73 :
<br/>Let $\displaystyle f(\theta)=\begin{vmatrix}\cos\dfrac{\theta}{2}<br/> &1  &1 \\1 &\cos \dfrac{\theta}{2}  &-\cos\dfrac{\theta}{2} \\-\cos\dfrac{\theta}{2}  &1  &-1 \end{vmatrix}$ <br> $\displaystyle f(\pi) +f(-\pi) $ is equal to
Question 74 :
The value of $\begin{vmatrix}<br/>-a^{2} &ab &ac \\ <br/> ab& -b^{2} &bc \\ <br/>ac & bc& -c^{2}<br/>\end{vmatrix}$ is <br/>
Question 75 :
The value of the determinant $\begin{vmatrix} \sin ^{ 2 }{ { 36 }^{ o } } & \cos ^{ 2 }{ { 36 }^{ o } } & \cot { { 135 }^{ o } } \\ \sin ^{ 2 }{ { 53 }^{ o } } & \cot { { 135 }^{ o } } & \cos ^{ 2 }{ { 53 }^{ o } } \\ \cot { { 135 }^{ o } } & \cos ^{ 2 }{ { 25 }^{ o } } & \cos ^{ 2 }{ { 65 }^{ o } } \end{vmatrix}$ is
Question 76 :
If the coordinates of the vertices of a triangle are (0,0) , (0,2) and(3,1) , then area of the triangle is<br>
Question 77 :
The sum of the real roots of the equation<br>$\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0$ is equal to
Question 78 :
If $\theta = \dfrac{\pi }{12}$ and <br/>A=$\begin{bmatrix}<br/>\cos\theta  & \sin\theta \\ <br/> \sin\theta & \cos\theta <br/>\end{bmatrix}$ then det $(A^{6})$ =<br/>
Question 79 :
If a. b, c are negative and different real numbers then $\displaystyle \Delta=\begin{vmatrix}a &b &c \\ b &c &a \\c &a &b \end{vmatrix}$ is<br>
Question 80 :
A determinant of second order is made with the elements $0$ and $1.$ The number of determinants with non-negative values is:
Question 81 :
If $A+B+C= \pi$, then $ \displaystyle \left| \begin{matrix} \tan { \left( A+B+C \right)  }  & \tan { B }  & \tan { C }  \\ \tan { (A+C) }  & 0 & \tan { A }  \\ \tan { (A+B) }  & -\tan { A }  & 0 \end{matrix} \right| $ is equal to<br/>
Question 82 :
If the determinant of the adjoint of a (real) matrix of order 3 is 25, then the determinant of the inverse of the matrix is.<br>
Question 83 :
If A = $\begin{bmatrix}-8 & 5\\ 2 & 4\end{bmatrix}$ satisfies the equation $x^2\, +\, 4x\, -\, p\, =\, 0$, then p =
Question 84 :
$f(x)=\begin{vmatrix} \cos { x }  & x & 1 \\ 2\sin { x }  & { x }^{ 2 } & 2x \\ \tan { x }  & x & 1 \end{vmatrix}$. The value of $\displaystyle \lim _{ x\rightarrow 0 }{ \cfrac { f(x) }{ x }  } $ is equal to<br/>
Question 85 :
The value of the determinant $\displaystyle \left | \begin{matrix}<br/>1 &\omega ^{3}  &\omega ^{5} \\ <br/> \omega ^{3}&1  &\omega ^{4} \\ <br/> \omega ^{5}&\omega ^{4}  &1 <br/>\end{matrix} \right |$ , where $\omega$ is an imaginary cube root of unity,is<br/>
