Question 1 :
Find the value of the following determinant:<br/>$\begin{vmatrix}\displaystyle \frac{-4}{7} & \displaystyle \frac{-6}{35}\\ 5 & \displaystyle \frac{-2}{5}\end{vmatrix}$
Question 2 :
Find the value of the following determinant:<br/>$\begin{vmatrix}3\sqrt{6} & -4\sqrt{2}\\ 5\sqrt{3} & 2\end{vmatrix}$
Question 3 :
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 4 :
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 5 :
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 6 :
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 7 :
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 8 :
$\mathrm{If}$ $\left|\begin{array}{lll}<br>1 & 0 & 0\\<br>2 & 3 & 4\\<br>5 & -6 & x<br>\end{array}\right|$ $= 45$ $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$ $\mathrm{x}=$<br><br>
Question 9 :
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 10 :
If A $=\begin{bmatrix}<br>0 & c &-b \\ <br> -c& 0& a\\ <br>b & -a & 0<br>\end{bmatrix}$then$\left ( a^{2}+b^{2}-c^{2} \right )\left | A \right |=$
Question 11 :
If $A$ is any skew-symmetric matrix of odd order then $\left| A \right| $ equals
Question 12 :
If $A = \begin{bmatrix}1& \log_{b}a\\ \log_{a}b& 1\end{bmatrix}$ then $|A|$ is equal to<br>
Question 13 :
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 14 :
$A=\begin{bmatrix} 5 & 5a & a \\ 0 & a & 5a \\ 0 & 0 & 5 \end{bmatrix}$ If $\left| A^{ 2 } \right| =25$ then $|a|=$
Question 15 :
If $A$ is a $3\times 3$ matrix and $\text{det}  (3A)=k(\text{det}  A)$, then $k=$
Question 16 :
$D=\begin{vmatrix} 18 & 40 & 89 \\ 40 & 89 & 198 \\ 89 & 198 & 440 \end{vmatrix}=$
Question 17 :
If $\begin{vmatrix}a & -b & -c\\-a & b & -c \\ -a & -b & -c\end{vmatrix}+\lambda abc=0$, then $\lambda$ is equal to<br>
Question 18 :
If $\displaystyle{\left| {_2^{4\,}\,\,_1^1} \right|^2} = \left| {_1^3\,\,_x^2} \right| - \left| {_{ - 2}^x\,\,_1^3} \right|,$ then $x$=
Question 19 :
$\begin{vmatrix} 2^3 & 3^3 & 3.2^2+3.2+1\\ 3^3 & 4^3 & 3.3^2+3.3+1\\ 4^3 & 5^3 & 3.4^2+3.4+1\end{vmatrix}$ is equal to?
Question 20 :
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>