The values of \(\alpha\), for which \(\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0\), lie in the interval

Let the function \(f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6\) have a maxima for some value of \(x < 0\) and a minima for some value of \(x > 0\). Then, the set of all values of \(p\) is \(......\)

Let z be the complex number satisfying \( |z-5|\le3 \) and having maximum positive principal argument. Then \( 34|\frac{5z-12}{5iz+16}|^{2} \) is equal to :

Let \(y=y(x)\) satisfies the equation \(\frac{d y}{d x}-|A|=0\), for all \(x>0\), where \(A=\left[\begin{array}{ccc}y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{array}\right] .\) If \(y(\pi)=\pi+2\), then the value of \(y\left(\frac{\pi}{2}\right)\) is:

The area, enclosed by the curves \(y=\sin x+\cos x\) and \(\mathrm{y}=|\cos \mathrm{x}-\sin \mathrm{x}|\) and the lines \(\mathrm{x}=0, \mathrm{x}=\frac{\pi}{2}\) is:

The locus of a point which divides the line segment joining the point \((0,-1)\) and a point on the parabola, \(\mathrm{x}^{2}=4 \mathrm{y},\) internally in the ratio \(1: 2,\) is 

If \(y(x)=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), x \in\left(\frac{\pi}{2}, \pi\right)\) then \(\frac{d y}{d x}\) at \(x=\frac{5 \pi}{6}\) is:

The coefficient of \(x^{18}\) in the expansion of \(\left(x^4-\frac{1}{x^3}\right)^{15}\) is \(...........\).

Let a circle \(C\) of radius \(5\) lie below the \(x\)-axis. The line \(L_{1}=4 x+3 y-2\) passes through the centre \(P\) of the circle \(C\) and intersects the line \(L _{2}: 3 x -4 y -11=0\) at \(Q\). The line \(L _{2}\) touches \(C\) at the point \(Q\). Then the distance of \(P\) from the line \(5 x-12 y+51=0\) 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 for \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ a & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2\). If \(|2 \operatorname{adj}(2 \operatorname{adj}(2 A ))|\) \(=32^{ n }\), then \(3 n +\alpha\) is equal to

Let \(A\) be a \(3 \times 3\) real matrix such that \(A \left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right) ; A \left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)\) and \(A \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)\). If \(X =\left( x _{1}, x _{2}, x _{3}\right)^{ T }\) and \(I\) is an identity matrix of order \(3\) , then the system \(( A -2 I ) X =\left(\begin{array}{l}4 \\ 1 \\ 1\end{array}\right)\) has

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7},\) where \(a, b, c \in z\), then \(a+b+c\) is equal to ...........