If \(\cos x-\sin x=\sqrt{a} \sin x\), then \(a \sin x+\cos x-\sin x=\)

Assertion (A): In \(\triangle A B C\), if \(r=6, r_2=36, R=15\) then \(c^2+a^2=b^2\) Reason (R): In \(\triangle A B C\), if \(r: R: r_2=1: 2.5: 6\) then \(\mathrm{B}=90^{\circ}\) The correct option among the following is

If a poisson variate \(X\) satisfies \(P(X=2)\) \(=P(X=3)\), then \(P(X=5)=\)

\(A, B, C, D\) are square matrices such that \(A+B\) is symmetric, \(A-B\) is skew-symmetric and \(D\) is the transpose of \(C\).
If \(A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5\end{array}\right]\) and \(C=\left[\begin{array}{ccc}0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1\end{array}\right]\), then the matrix \(B+D=\)

\(\int e^{-x}\left(x^3-2 x^2+3 x-4\right) d x=\)

A right solid circular cylinder of given volume will have the least total surface area when

If \(\sqrt{5}-i \sqrt{15}=r(\cos \theta+i \sin \theta),-\pi \lt \theta \lt \pi\), then \(r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=\)

\[ \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \cos \frac{\pi}{14} \cos \frac{3 \pi}{14} \cos \frac{5 \pi}{14}= \]

A particle of mass \(M\) is moving in a horizontal circle of radius \(R\) with wiform speed \(v\). When the particle moves from one point to a diametrically opposite point, its

The temperature difference between the ends of two cylindrical rods \(A\) and \(B\) of the same material is \(2: 3\). In steady state the ratio of the rates of flow of heat through the rods \(A\) and \(B\) is \(5: 9\). If the radii of the rods \(A\) and \(B\) are in the ratio \(1: 2\), then the ratio of lengths of the rods \(A\) and \(B\) is

Identify the correct statements from the folluwing (A) At 298 K, the potential of hydrogen electrode placed in a solution of \(\mathrm{pH}=10\), is -0.59 V (B) The limiting molar conductivity of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Cl}^{-}\)is 119 and \(76 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\) respectively. The limiting molar conductivity of \(\mathrm{CaCl}_2\) is \(195 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-\frac{1}{2}}\) (C) The correct relationship between \(\mathrm{K}_{\mathrm{c}}\) and \(\mathrm{E}_{\text {cell }}^0\) is \(\mathrm{E}_{\mathrm{ccll}}^{-}=\frac{2.303 \mathrm{RT}}{\mathrm{nF}} \log \mathrm{~K}_{\mathrm{c}}\)

If \(f(x)=\frac{|\log x|}{x^2}\), then \(\int_{1 / e}^e f(x) d x=\)

A committee of 12 members is to be formed from 9 women and 8 men. The number of committees in which the women are in majority is