The number of solutions of \(\sin ^2 \mathrm{x}+\left(2+2 \mathrm{x}-\mathrm{x}^2\right) \sin \mathrm{x}-3(\mathrm{x}-1)^2=0\), where \(-\pi \leq \mathrm{x} \leq \pi\), is ..........

Let \(A =\{1,2,3, \ldots, 10\}\) and \(f: A \rightarrow A\) be defined as \(f( k )=\left\{\begin{array}{cl} k +1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even }\end{array}\right.\) Then the number of possible functions \(g : A \rightarrow A\) such that \(gof=f\) is ...... .

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

Let \(A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]\). If the sum of the diagonal elements of \(\mathrm{A}^{13}\) is \(3^{\mathrm{n}}\), then \(\mathrm{n}\) is equal to ..........

The set of values of \(k\) for which the circle \(C : 4 x^{2}+4 y^{2}-12 x+8 y+k=0\) lies inside the fourth quadrant and the point \(\left(1,-\frac{1}{3}\right)\) lies on or inside the circle \(C\) is

The number of distinct real roots of the equation \(|\mathrm{x}||\mathrm{x}+2|-5|\mathrm{x}+1|-1=0\) is ....................

The mean and standard deviation of the marks of \(10\) students were found to be \(50\) and \(12\) respectively. Later, it was observed that two marks \(20\) and \(25\) were wrongly read as \(45\) and \(50\) respectively. Then the correct variance is \(............\).

If the general solution of the differential equation \(y' = \frac{y}{x} + \phi \left( {\frac{x}{y}} \right)\) , for some function \(\phi \), is given by \(y \ln \,\left| {cx} \right| = x\), where \(c\) is an arbitrary constant, then \(\phi \,(2)\) is equal to:

If \(z_1, z_2\) are two distinct complex number such that \(\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2\), then

Consider the function \(f:(0, \infty) \rightarrow R\) defined by \(f(x)=e^{-\left|\log _e x\right|}\). If \(m\) and \(n\) be respectively the number of points at which \(f\) is not continuous and \(f\) is not differentiable, then \(\mathrm{m}+\mathrm{n}\) is

If the line \(\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z\) makes a right angle with the line \(\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}\), then \(4 \lambda+9 \mu\) is equal to :

The value of \(sin\,10^o\) \(sin\,30^o\) \(sin\,50^o\) \(sin\,70^o\) is

If the area of the region \(\left\{( x , y ): x ^{\frac{2}{3}}+ y ^{\frac{2}{3}} \leq 1 x + y \geq 0, y \geq 0\right\}\) is \(A\), then \(\frac{256 A }{\pi}\)