\(\mathop \smallint \limits_{ - \pi /2}^{\pi /2} \frac{{{{\sin }^2}x}}{{1 + {2^x}}}dx =\) . ..  . 

Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{ r } \cdot(\hat{ i }+3 \hat{ j }-\hat{ k })=6\) and \(\overrightarrow{ r } \cdot(-6 \hat{ i }+5 \hat{ j }-\hat{ k })=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^{2}}{ d ^{2}}\) is equal to

For \(x \in \left( {0,\frac{3}{2}} \right)\), let \(f\left( x \right) = \sqrt x \), \(g\left( x \right) = \tan \,x\) and \(h\left( x \right) = \frac{{1 - {x^2}}}{{1 + {x^2}}}\). If \(\phi \left( x \right) = \left( {\left( {hof} \right)og} \right)\left( x \right)\), then \(\phi \left( {\frac{\pi }{3}} \right)\) is equal to

The area (in sq. units) of the smaller portion enclosed between the curves, \(x^2 + y^2 = 4\) and \(y^2 =3x\), is

Let \(\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=((\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}+\hat{\mathrm{j}})) \times \hat{\mathrm{i}}) \times \hat{\mathrm{i}}\). Then the square of the projection of \(\vec{a}\) on \(\vec{b}\) is :

If the image of the point \(P\left( {1, - 2,3} \right)\) in the plane , \(2x + 3y - 4z + 22 = 0\) measured parallel to line , \(\frac{x}{1} = \frac{y}{4} = \frac{z}{5}\) is \(Q\) , then \(PQ \) is equal to :

If the shortest distance between the lines \(\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}\) and \(\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}\) is \(\frac{6}{\sqrt{5}}\), then the sum of all possible values of \(\lambda\) is :

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

Let \(y=p(x)\) be the parabola passing through the points \((-1,0),(0,1)\) and \((1,0)\). If the area of the region \(\left\{(x, y):(x+1)^2+(y-1)^2 \leq 1, y \leq p(x)\right\}\) is \(A\), then \(12(\pi-4 A )\) is equal to \(.........\).

Let a triangle \(PQR\) be such that \(P\) and \(Q\) lie on the line \(\dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2}\) and are at a distance of \(6\) units from \(R(1, 2, 3)\). If \((\alpha, \beta, \gamma)\) is the centroid of \(\triangle PQR\), then \(\alpha + \beta + \gamma\) is equal to :

The sum \(1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots\) upto \(\infty\) terms, is equal to

Consider a circle \(C_1: x^2+y^2-4 x-2 y=\alpha-5\).Let its mirror image in the line \(y=2 x+1\) be another circle \(C _2: 5 x ^2+5 y ^2-10 fx -10 gy +36=0\).Let \(r\) be the radius of \(C _2\). Then \(\alpha+ r\) is equal to \(......\).

If the image of the point \(\mathrm{P}(1,0,3)\) in the line joining the points \(\mathrm{A}(4,7,1)\) and \(\mathrm{B}(3,5,3)\) is \(\mathrm{Q}(\alpha, \beta, \gamma)\), then \(\alpha+\beta+\gamma\) is equal to