$a,b,c$ are in G.P. and $a+b+c=xb$, $x$ can not be (a) $2$ (b) $-2$ (c) $3$ (d) $4$

${\int }_0^{\pi }{\left|\cos x\right|}^{3}dx$ is equal to (a) $\frac{4}{3}$ (b) $\frac{2}{3}$ (c) $0$ (d) $\frac{-8}{3}$

There are $5$ girls and $7$ boyes and $2$ girls is to be formed such that no two specific boys are in the same team. Number of way to do so (a) $400$ (b) $250$ (c) $200$ (d) $300$

The equation $x^2+2x+2=0$ has root $\alpha$ and $\beta$ . Then value of ${\alpha }^{15}+{\beta }^{15}$ is (a) $512$ (b) $256$ (c) $-512$ (d) $-256$

Average height & variance of $5$ students in a class is $150$ and $18$ respectively. A new student whose height is $156cm$ is added to the group. Find new variance. (a) $20$ (b) $22$ (c) $16$ (d) $14$

if $\vec{A}=i-j,\vec{B}=i+j+k$ are two vectors and $\vec{C}$ is another vector such that $\vec{A}\times \vec{C}+\vec{B}=\vec{0}$ and $\vec{A}.\vec{C}=0$ then ${\left|\vec{C}\right|}^2=$ (a) $9$ (b) $8$ (c) $\frac{19}{2}$ (d) $\frac{17}{2}$

If $x^2\ne n\pi -1,n\in N$. Then, the value of $\int x\sqrt{\frac{2\sin \left(x^2+1\right)-\sin 2\left(x^2+1\right)}{2\sin \left(x^2+1\right)+\sin 2\left(x^2+1\right)}}dx$ is equal to:

$f\left(x\right)=\{5;x\le 1,a+bx;1<x<3,b+5x;3\le x<5,30x\ge 5$ then

$\left\{\frac{2^{403}}{15}\right\}=\frac{k}{15}$ then find $k$. (a) $2$ (b) $8$ (c) $1$ (d) $4$

The value of $3{\left(\cos \theta -\sin \theta \right)}^4+6{\left(\sin \theta +\cos \theta \right)}^2+4{\sin }^6\theta$ is where $\theta \in \left(\frac{\pi }{4},\frac{\pi }{2}\right)$ (a) $13-4{\cos }^4\theta$ (b) $13-4{\cos }^6\theta$ (c) $13-4{\cos }^6\theta +2{\sin }^4\theta {\cos }^2\theta$ (d) $13-4{\cos }^4\theta +2{\sin }^4\theta {\cos }^2\theta$