The expansion of \(\left(1+x+x^2\right)^{-3 / 2}\) in powers of \(x\) is valid if

\(\int_0^{16} \frac{\sqrt{x}}{1+\sqrt{x}} d x=\)

If \(I_n=\int_0^a \frac{x^n}{\sqrt{a^2-x^2}} d x\), then \(\frac{I_8}{I_4}=\)

If the pair of lines given by \(\left(x^2+y^2\right) \cos ^2 \theta=(x \cos \theta+y \sin \theta)^2 \quad\) are perpendicular to each other, then \(\theta\) is equal to

If all the letters of the word 'HANDLE' are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word 'HELAND' is

The solution set of the inequation \(3^x+3^{1-x}-4 \lt 0\) contained in \(\mathbf{R}\) is

\(\lim _{n \rightarrow \infty}\left[\frac{n+3}{n^2+1^2}+\frac{n+6}{n^2+2^2}+\frac{n+9}{n^2+3^2}+\ldots+\frac{2}{n}\right]=\)

The angle \(A\) of \(\triangle A B C\) is found by measurement to be \(67 \frac{1^{\circ}}{2}\) and the area of \(\triangle A B C\) is calculated from the measurements of \(b, c, A\). In measuring \(A\), an error of \(9 \mathrm{~min}\) is made then the percentage error in the area of the triangle is

\(\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x\) is equal to

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

The following results have been obtained during the kinetic studies of reaction:
$2\mathrm{NO}+2{\mathrm{H}}_2⟶{\mathrm{N}}_2+2{\mathrm{H}}_2\mathrm{O}$

Expt	$\frac{-\mathrm{d}\left[\mathrm{NO}\right]}{\mathrm{dt}}$ ${\mathrm{molL}}^{-1} {\mathrm{s}}^{-1}$	$\left[\mathrm{NO}\right]$ ${\mathrm{molL}}^{-1}$	$\left[{\mathrm{H}}_2\right]$ ${\mathrm{molL}}^{-1}$
$1$.	$4.8\times {10}^{-5}$	$1\times {10}^{-2}$	$1\times {10}^{-3}$
$2$.	$43.2\times {10}^{-5}$	$3\times {10}^{-2}$	$1\times {10}^{-3}$
$3$.	$86.4\times {10}^{-5}$	$3\times {10}^{-2}$	$2\times {10}^{-3}$

 

If \(A(2,-1)\) and \(B(6,5)\) are two points the ratio in which the foot of the perpendicular from \((4,1)\) to \(A B\) divides it, is