Let \(x=2 t, y=\frac{i}{3}\) be a conic. Let \(S\) be the focus and \(B\) be the point on the axis of the conic such that \(SA \perp BA\), where \(A\) is any point on the conic. If \(k\) is the ordinate of the centroid of \(\Delta SAB\), then \(\lim _{ t \rightarrow 1} k\) is equal to

If the curves \(\frac{{{x^2}}}{\alpha } + \frac{{{y^2}}}{4} = 1\) and \({y^3} = 16x\) intersect at right angles, then a value of \(\alpha \) is

\( \text { If } S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\ldots . \) \( +60(1+x)^{60}, x \neq 0 \text {, and }(60)^2 S(60)=a(b)^b+b\) where \(a, b N\), then \((a+b)\) equal to ...............

Let \(f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10\). The number of points of local maxima of \(f\) in interval \((0,2 \pi)\) is:

Consider the set of all lines \(px + qy + r = 0\) such that \(3p + 2q + 4r = 0\) . Which one of the following statements is true?

Let \(P =\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]\) where \(\alpha \in R .\) Suppose \(Q =\left[ q _{ ij }\right]\) is a matrix satisfying \(PQ = kI _{3}\) for some non-zero \(k \in R .\) If \(q_{23}=-\frac{k}{8}\) and \(|Q|=\frac{k^{2}}{2}\), then \(\alpha^{2}+ k ^{2}\) is equal to...........

Statement \(- 1:\) The function \(x^2 (e^x + e^{-x})\) is increasing for all \(x > 0.\) Statement \(-2:\) The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x > 0\) and the sum of two increasing functions in any interval \((a, b)\) is an increasing function in \((a, b).\)

For the frequency distribution :

Variate \(( x )\)	\(x _{1}\)	\(x _{1}\)	\(x _{3} \ldots \ldots x _{15}\)
Frequency \((f)\)	\(f _{1}\)	\(f _{1}\)	\(f _{3} \ldots f _{15}\)

where \(0< x _{1}< x _{2}< x _{3}<\ldots .< x _{15}=10\) and \(\sum \limits_{i=1}^{15} f_{i}>0,\) the standard deviation cannot be 

Let \(A\) be a square matrix such that \(A A^T=I\). Then \(\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]\) is equal to

The value of \(12 \int \limits_0^3\left|x^2-3 x+2\right| d x\) is \(.............\)

Let \(\quad S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.\) is a real number \(\}\). Then which of the following is NOT correct?

\(ABC\) is a triangular park with \(AB = AC = 100\) \(metres\). A vertical tower is situated at the mid-point of \(BC\). If the angles of elevation of the top of the tower at \(A\) and \(B\) are \({\cot ^{ - 1}}\left( {3\sqrt 2 } \right)\) and \(\cos e{c^{ - 1}}\left( {2\sqrt 2 } \right)\) respectively, then the height of the tower (in metres) 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 _______