The integral \(\int_{-1}^{\frac{3}{2}}\left(\left|\pi^2 x \sin (\pi x)\right|\right) d x\) is equal to:

The term independent of \(x\) in expansion of \({\left( {\frac{{x + 1}}{{{x^{2/3}} - {x^{\frac{1}{3}}} + 1\;}}--\frac{{x - 1}}{{x - {x^{1/2}}}}} \right)^{10}}\) is

Let \(S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a, a \in \mathbb{Z}\}\). Then \(n(S)\) is equal to :

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to

Let \(\mathrm{A}(x, y, z)\) be a point in \(x y\)-plane, which is equidistant from three points \((0,3,2),(2,0,3)\) and ( \(0,0,1\) ).
Let \(\mathrm{B}=(1,4,-1)\) and \(\mathrm{C}=(2,0,-2)\). Then among the statements
(S1) : \(\triangle \mathrm{ABC}\) is an isosceles right angled triangle, and
(S2) : the area of \(\triangle \mathrm{ABC}\) is \(\frac{9 \sqrt{2}}{2}\),

If the domain of the function \(f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)\) is \([\alpha, \beta) \cup(\gamma, \delta]\), then \(|3 \alpha+10(\beta+\gamma)+21 \delta|\) is equal to \(.......\).

\(\left( {\left( {\begin{array}{*{20}{c}}
{21}\\
1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
1
\end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
2
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
2
\end{array}} \right)} \right)\)\( + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
3
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
3
\end{array}} \right)} \right) + \;.\;.\;.\)\( + \left( {\left( {\begin{array}{*{20}{c}}
{21}\\
{10}
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
{10}\\
{10}
\end{array}} \right)} \right) = \)

The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) is

The numbers of pairs \((a, b)\) of real numbers, such that whenever \(\alpha\) is a root of the equation \(x^{2}+a x+b=0, \alpha^{2}-2\) is also a root of this equation, is :

If \((a+\sqrt{2} b \cos x)(a-\sqrt{2} b \cos y)=a^{2}-b^{2}\) where \(a>b>0,\) then \(\frac{d x}{d y}\) at \(\left(\frac{\pi}{4}, \frac{\pi}{4}\right)\) is

Let \([x]\) denote the greatest integer \(\leq x\), where \(x \in R\). If the domain of the real valued function \(\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}\) is \((-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}\), then the value of \(\mathrm{a}+\mathrm{b}+\mathrm{c}\) is:

Let \(A=\{-3,-2,-1,0,1,2,3\),\(\} . Let R\) be a relation on A defined by \(x R y\) if and only if \(0 \leq x^2+2 y \leq 4\).
Let \(l\) be the number of elements in R and \(m\) be the minimum number of elements required to be added in R to make it a reflexive relation. then \(l+m\) is equal to

Let \(\mathrm{ABC}\) be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle \(\mathrm{ABC}\) and the same process is repeated infinitely many times. If \(\mathrm{P}\) is the sum of perimeters and \(Q\) is be the sum of areas of all the triangles formed in this process, then :