If $\sin x+{\sin }^{2}x=1$, $x\in \left(0,\frac{\pi }{2}\right)$, then $({\cos }^{12}x+{\tan }^{12}x)+3({\cos }^{10}x+{\tan }^{10}x+{\cos }^{8}x+{\tan }^{8}x)+({\cos }^{6}x+{\tan }^{6}x)$ is equal to:

If $\sum _{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\right\}=a\sqrt{3}+b,a,b\in Z$, then $a^2+b^2$ is equal to :

The value of $\left(\sin 70^{\circ }\right)\left(\cot 10^{\circ }\cot 70^{\circ }-1\right)$ is

Let the range of the function $f(x)=6+16\cos x\cdot \cos \left(\frac{\pi }{3}-x\right)\cdot \cos \left(\frac{\pi }{3}+x\right)\cdot \sin 3x\cdot \cos 6x,x\in \mathbf{R}$ be $[\alpha ,\beta ]$. Then the distance of the point $(\alpha ,\beta )$ from the line $3 x+4 y+12=0$ is :

If for $\theta \in \left[-\frac{\pi }{3},0\right]$, the points $(x,y)=\left(3\tan \left(\theta +\frac{\pi }{3}\right),2\tan \left(\theta +\frac{\pi }{6}\right)\right)$ lie on $xy+\alpha x+\beta y+\gamma =0$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to :

If $10{\sin }^4\theta +15{\cos }^4\theta =6$, then the value of $\frac{27{\mathrm{cosec}}^6\theta +8{\sec }^6\theta }{16{\sec }^8\theta }$ is

Let $A=\left[\begin{array}{c}{a}_{ij}\end{array}\right]=\left[\begin{array}{cc}{\log }_{5}128& {\log }_{4}5\\ {\log }_{5}8& {\log }_{4}25\end{array}\right]$. If $A_{ij}$ is the cofactor of $a_{ij}$, $C_{ij}=\sum _{k=1}^2{a}_{ik}{A}_{jk},1\le i,j\le 2$, and $C=[C_{ij}]$, then $ 8|C| $ is equal to :

Let $A=[a_{ij}]$ be a $2\times 2$ matrix such that $a_{ij}\in \{0,1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

Let $\alpha ,\beta (\alpha \ne \beta )$ be the values of $ m $, for which the equations $ x+y+z=1 $, $ x+2y+4z=m $ and $x+4y+10z=m^2$ have infinitely many solutions. Then the value of $\sum _{n=1}^{10}(n^{\alpha }+n^{\beta })$ is equal to :

Let $\mathrm{A}=\left[a_{ij}\right]$ be a matrix of order $3\times 3$, with $a_{ij}=(\sqrt{2}{)}^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha +\beta \sqrt{2},\alpha ,\beta \in \mathbf{Z}$, then $\alpha +\beta$ is equal to :

Let M and m respectively be the maximum and the minimum values of $f(x)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 4x\end{array}\right|,x\in R$ Then $M^4-m^4$ is equal to :

If the system of linear equations : $$\begin{array}{rl}& x+y+2z=6\\ & 2x+3y+\mathrm{az}=\mathrm{a}+1\\ & -x-3y+\mathrm{b}z=2\mathrm{b}\end{array}$$ where $a,b\in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

For a $3\times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3\times 3$ matrix such that $|A|=\frac{1}{2}$ and trace $(A)=3$. If $B=\mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B|+$ trace $(B)$ equals :

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -2\\ 0& 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right],\theta >0$. If $\mathrm{B}=\mathrm{PAP}{}^{\top },\mathrm{C}={\mathrm{P}}^{\top }{\mathrm{B}}^{10}\mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $m+n$ is :

If the system of equations $$\begin{array}{rl}& (\lambda -1)x+(\lambda -4)y+\lambda z=5\\ & \lambda x+(\lambda -1)y+(\lambda -4)z=7\\ & (\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9\end{array}$$ has infinitely many solutions, then ${\lambda }^2+\lambda$ is equal to

If $\mathrm{A},\mathrm{B},\mathrm{and}\left(\mathrm{adj}\left({\mathrm{A}}^{-1}\right)+\mathrm{adj}\left({\mathrm{B}}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $A{\left(\mathrm{adj}\left(A^{-1}\right)+\mathrm{adj}\left(B^{-1}\right)\right)}^{-1}B$, is equal to

The system of equations $$\begin{array}{rl}& x+y+z=6,\\ & x+2y+5z=9,\\ & x+5y+\lambda z=\mu ,\end{array}$$ has no solution if

Let $A=\left[a_{ij}\right]$ be a $3\times 3$ matrix such that $A\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right],A\left[\begin{array}{l}4\\ 1\\ 3\end{array}\right]=\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]$ and $A\left[\begin{array}{l}2\\ 1\\ 2\end{array}\right]=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right]$, then $a_{23}$ equals :

If the system of equations $$\begin{array}{rl}& 2x-y+z=4\\ & 5x+\lambda y+3z=12\\ & 100x-47y+\mu z=212\end{array}$$ has infinitely many solutions, then $\mu -2\lambda$ is equal to

For some $a, b,$ let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x}& 1& \mathrm{b}\\ \mathrm{a}& 1+\frac{\sin x}{x}& \mathrm{b}\\ \mathrm{a}& 1& \mathrm{b}+\frac{\sin x}{x}\end{array}\right|,x\ne 0,\underset{x\to 0}{\lim }f(x)=\lambda +\mu \mathrm{a}+\nu \mathrm{b}.$ Then $(\lambda +\mu +v{)}^2$ is equal to :