NDA MATHEMATICS QUIZ-2

In the polynomial $(\alpha -1)(\alpha-2)(\alpha-3) \ldots . .(\alpha-100)$. The coefficient of $\alpha^{99}$ is:

$\operatorname{Let} A=\left[\begin{array}{cc}\cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right]$and $X=\left[\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{2}\end{array}\right]$then$A^{3} X$ is equal to

If X = {a,b,c,d}, Y = {b,d,f} then the number of elements of  set Z satisfying X $\cap$ Y $\subseteq$ Z $\subseteq$ X $\cup$ Y is:

In a triangle $\frac{2 \cos A}{a}+\frac{\cos B}{b}+\frac{2 \cos C}{c}$ $=\frac{a}{b c}+\frac{b}{a c}$ then the value of angle $\mathrm{A}$ is

If $r^{2}=x^{2}+y^{2}+z^{2}$   then   $\tan ^{-1}\left(\frac{y z}{x r}\right)+\tan ^{-1}\left(\frac{x z}{y r}\right)+\tan ^{-1}\left(\frac{x y}{z r}\right)$ equals

The equation $\cos 2 \theta+p \sin \theta=2 p-7$ has a solution

$\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x \cdot d x}{\cos ^{\frac{3}{2}} x+\sin ^{\frac{3}{2}} x}=\ldots \ldots$

If $y=\cos ^{-1}\left(\frac{x-x^{-1}}{x+x^{-1}}\right)$ then $\frac{d y}{d x}$ will be:

If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$ then $2 x \frac{d y}{d x}+y$ is equal to

$\operatorname{Lt}_{x \rightarrow 0} \frac{x \cos x-\log (1+x)}{x^{2}}$ equals: