Derivative of a Determinant

Q. If the determinant
$\left|\begin{array}{ccc}xp+y& x& y\\ yp+z& y& z\\ 0& xp+y& yp+z\end{array}\right|=0$ and $x,y,z,p\in R^{+}$ then

1. $\text{x, y, z are in A.P.}$
2. $\text{x, y, z are in G.P.}$
3. $\text{x, y, z are in H.P.}$
4. $\text{xy, yz, zx are in A.P.}$

Q.

$\left|\begin{array}{ccc}1+a^2-b^2& 2ab& -2b\\ 2ab& 1-a^2+b^2& 2a\\ 2b& -2a& 1-a^2-b^2\end{array}\right|=(1+a^2+b^2{)}^3$

Q. lf $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, then $(aI+bE{)}^3=$

1. $a^{3}I+3a{b}^{2}E$
2. $a^{3}I+3a^{2}bE$
3. $aI+bE$
4. $a^{3}I+b^{3}E$

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a,b$ are even for $i=1,2,3$. then find $f^{\prime }\left(2\right)$

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a_i,b_i$ are even for $i=1,2,3,$ then $f^{\prime }\left(2\right)$ is.

Q. If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

1. 1
2. 2
3. 5
4. 3

Q. If $a,b,c>0$ and $x,y,zϵR$, then the determinant $\left|\begin{array}{ccc}(a^x+a^{-x}{)}^2& (a^x-a^{-x}{)}^2& 1\\ (b^y+b^{-y}{)}^2& (b^y-b^{-y}{)}^2& 1\\ (c^z+c^{-z}{)}^2& (c^z-c^{-z}{)}^2& 1\end{array}\right|$ is equal to

1. $a^{2x}{b}^{2y}{c}^{2z}$
2. $a^x{b}^y{c}^x$
3. zero
4. $a^{-x}{b}^{-y}{c}^{-z}$

Q. Reduce the expression $\frac{1}{\frac{1}{x-4}-\frac{1}{x+6}}$ in simplified form.

1. $10$
2. $\frac{10(x-4)}{x+6}$
3. $\frac{x^2+2x-24}{10}$
4. $\frac{10}{x^2+2x-24}$

Q. $A=\left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right],A^{100}=?$

1. $\left[\begin{array}{ccc}0& 0& x^{100}\\ 0& x^{100}& 0\\ x^{100}& 0& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right]$
3. $\left[\begin{array}{ccc}0& x^{100}& 0\\ 0& 0& x^{100}\\ x^{100}& 0& 0\end{array}\right]$
4. $\left[\begin{array}{ccc}0& x^{100}& 0\\ x^{100}& 0& 0\\ 0& 0& x^{100}\end{array}\right]$

Q. If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right].E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$ then show that $(aI+bE{)}^3=a^{3}I+3a^{2}bE$.

Q. If $z_1=x_1+i{y}_1,z_2=x_2+i{y}_2$, then

1. $|z_1{|}^2-|z_2{|}^2$
2. $|z_1{|}^2-|z_1-z_2{|}^2$
3. $\overline{z_1}{z}_2-z_1\overline{z_2}$
4. $z_1\overline{z_2}-z_2\overline{z_1}$

Q. If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, then which of the following statements is/are correct?

1. All three I, II and III
2. Only I and II
3. Only II and III
4. only I

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a,b$ are even for $i=1,2,3$. then find $f^{\prime }\left(2\right)$

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a_i,b_i$ are even for $i=1,2,3,$ then $f^{\prime }\left(2\right)$ is.

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a,b$ are even for $i=1,2,3$. then find $f^{\prime }\left(2\right)$

Q. If $f\left(y\right)=\left|\begin{array}{ccc}(1-y{)}^{a_1{b}_1}& (1-y{)}^{a_1{b}_2}& (1-y{)}^{a_1{b}_3}\\ (1-y{)}^{a_2{b}_1}& (1-y{)}^{a_2{b}_2}& (1-y{)}^{a_2{b}_3}\\ (1-y{)}^{a_3{b}_1}& (1-y{)}^{a_3{b}_2}& (1-y{)}^{a_3{b}_3}\end{array}\right|$ and $a_i,b_i$ are even for $i=1,2,3,$ then $f^{\prime }\left(2\right)$ is.

Q. If $x\ne y\ne z$ and $\left|\begin{array}{ccc}1+x^3& x^2& x\\ 1+y^3& y^2& y\\ 1+z^3& z^2& z\end{array}\right|=0$, then the value of $xyz$ is

1. $-1$
2. $-2$
3. $2$
4. $1$