Let \(S=\left\{(x, y) \in N \times N : 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}\) and \( T=\left\{(x, y) \in R \times R :(x-7)^{2}+(y-4)^{2} \leq 36\right\}\) Then \(n ( S \cap T )\) is equal to \(......\)

Let [ ] denote the greatest integer function and \( f(x)=lim_{n\rightarrow\infty}\frac{1}{n^{3}}\sum_{k=1}^{n}[\frac{k^{2}}{3^{x}}] \) Then \( 12\sum_{j=1}^{x}f(j) \) is equal to ........... .

Let O be the vertex of the parabola \(y^2=4x\) and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

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 the area of the region enclosed by the curve \(\mathrm{y}=\min \{\sin \mathrm{x}, \cos \mathrm{x}\}\) and the \(\mathrm{x}\)-axis between \(\mathrm{x}=-\pi\) to \(\mathrm{x}=\pi\) be \(\mathrm{A}\). Then \(\mathrm{A}^2\) is equal to ...........

The length of the perpendicular from the point \((1,-2,5)\) on the line passing through \((1,2,4)\) and parallel to the line \(x + y - z =0= x -2 y +3 z -5\) is.

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

Let \(y=y(x)\) be the solution of the differential equation \((x+1) y^{\prime}-y=e^{3 x}(x+1)^{2}\), with \(y(0)=\frac{1}{3}\). Then, the point \(x=-\frac{4}{3}\) for the curve \(y = y ( x )\) is

The sum to \(20\) terms of the series \(2.2^2-3^2+2.4^2-5^2+2.6^2-\ldots \ldots\) is equal to \(........\).

The area of the region given by \(A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}\) is.

The mean and the variance of five observations are \(4\) and \(5.20,\) respectively. If three of the observations are \(3, 4\) and \(4;\) then the absolute value of the difference of the other two observations, is

Let \(\mathrm{A}(\alpha, 0)\) and \(\mathrm{B}(0, \beta)\) be the points on the line \(5 x+7 y=50\). Let the point \(P\) divide the line segment \(A B\) internally in the ratio \(7: 3\). Let \(3 x-\) \(25=0\) be a directrix of the ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and the corresponding focus be \(S\). If from \(S\), the perpendicular on the \(\mathrm{x}\)-axis passes through \(\mathrm{P}\), then the length of the latus rectum of \(\mathrm{E}\) is equal to