If \(m, l, r, s, n\) are integers such that, \(9 \gt m \gt l \gt s \gt n \gt r \gt 2\) and \(\begin{aligned} & \int_{-2 \pi}^{2 \pi} \sin ^m x \cos ^n x d x=4 \int_0^\pi \sin ^m x \cos ^n x d x, \ & \int_{-\pi}^\pi \sin ^r x \cos ^s x d x=4 \int_0^{\pi / 2} \sin ^r x \cos ^s x d x \ & \text { and } \int_{-\pi / 2}^{\pi / 2} \sin ^l x \cos ^m x d x=0 \text {, the }\end{aligned}\)

If the term independent of \(x\) in the expansion of \(\left(\sqrt{\mathrm{x}}-\frac{\mathrm{k}}{\mathrm{x}^2}\right)^{10}\) is 405 , then \(k=\)

If \(y=\tan ^{-1}\left[\frac{\sin ^3(2 x)-3 x^2 \sin (2 x)}{3 x \sin ^2(2 x)-x^3}\right]\), then \(\frac{d y}{d x}=\)

If \(m\) and \(\sigma^2\) are the mean and variance of the random variable \(X\), whose distribution is given by
Then

Let \([A]_{3 \times 3}\) be a non-singular matrix such that \(A^{-1}=\frac{1}{3}\left(A^2-5 A+7 I\right)\).
Then \(17 A^8-85 A^7+119 A^6-51 A^5-19 A^4\) \(+95 A^3-133 A^2+58 A+I=\)

If \(X\) is a poisson variate with \(P(X=0)=0.8\), then the variance of \(X\) is

If $0<A<B<\frac{\pi }{4}, \cos (A+B)=\frac{11}{61}$ and $\sin (A-B)=\frac{24}{25},$ then $\sin 2A+\sin 2B=$

\[ \begin{aligned} & \text { Given } \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{H}_2 \mathrm{O}(l) ; \Delta \mathrm{H}=-285 \mathrm{~kJ} \ & \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(l) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-76.6 \mathrm{~kJ} \ & \mathrm{~N}_2(\mathrm{~g})+3 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \rightarrow 2 \mathrm{HNO}_3(l) ; \Delta \mathrm{H}=-348.2 \mathrm{~kJ} \ & \text { Calculate the } \Delta \mathrm{H} \text { of } 2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g}) \end{aligned} \] Calculate the \(\Delta \mathrm{H}\) of \(2 \mathrm{~N}_2(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{~N}_2 \mathrm{O}_5(\mathrm{~g})\).

If \(Q\) denotes the set of all rational numbers and \(f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}\) for any \(\frac{p}{q} \in Q\), then observe the following statements. I. \(f\left(\frac{p}{q}\right)\) is real for each \(\frac{p}{q} \in Q\) II. \(f\left(\frac{p}{q}\right)\) is a complex number for each \(\frac{p}{q} \in Q\). Which of the following is correct?

A number \(n\) is chosen at random from the natural numbers 2 to 1001 . The probability that \(n\) is a number that leaves remainder 1 when divided by 7 , is

If the angles of a \(\triangle A B C\) are in \(\mathrm{AP}\), then

If $\cos x+\cos y=p,\sin x+\sin y=q$, then $\cos \left(\frac{x-y}{2}\right)=$

Statement (I) : The temperature coefficient of resistance for most of meals in pure form is positive. Statement (II) : A metal wire \(2 \mathrm{~mm}\) in diameter carries a charge of \(360 \pi \mathrm{C}\) in two hours. If the metal contains \(5 \times 10^{22}\) free electrons \(/ \mathrm{cm}^3\), then drift velocity of the electrons in the wire is \(6.25 \times 10^{-6} \mathrm{~m} / \mathrm{s}\). Statement (III) : Semiconductors like pure germanium does not obey Ohm's law for all range of electric field values. Which of the following is correct?