Let \(A\) be the point \((1,2)\) and \(B\) be any point on the curve \(x^2+y^2=16\). If the centre of the locus of the point \(P\), which divides the line segment \(A B\) in the ratio \(3: 2\) is the point \(C(\alpha, \beta)\), then the length of the line segment \(AC\) is

Let the line \(x=-1\) divide the area of the region \(\{(x,y):1+x^{2}\le y\le3-x\}\) in the ratio \(m:n\), \(\gcd(m,n)=1\). Then \(m+n\) is equal to

The frequency distribution of the age of students in a class of \(40\) students is given below.

Age	\(15\)	\(16\)	\(17\)	\(18\)	\(19\)	\(20\)
No. of students	\(5\)	\(8\)	\(5\)	\(12\)	\(X\)	\(Y\)

If the mean deviation about the median is \(1.25\) , then \(4 x+5 y\) is equal to :

Let the foot of perpendicular from the point \(A (4,3\), 1) on the plane \(P: x-y+2 z+3=0\) be N. If \(B(5\), \(\alpha, \beta), \alpha, \beta \in Z\) is a point on plane \(P\) such that the area of the triangle \(A B N\) in \(3 \sqrt{2}\), then \(\alpha^2+\beta^2+\alpha \beta\) is equal to \(...........\).

A  man \(X\)  has \(7\)  friends, \(4\)  of them are ladies and  \(3\) are men. His wife \(Y\) also has \(7\) friends, \(3\) of  them are  ladies and \(4\) are men. Assume \(X\) and \(Y\) have no comman friends. Then the total number of ways in which \(X\) and \(Y\) together  can throw a party inviting \(3\) ladies and \(3\) men, so that \(3\) friends of each of \(X\) and \(Y\) are in this party is :

If the coefficients of \(x^7\) in \(\left( ax ^2+\frac{1}{2 bx }\right)^{11}\) and \(x ^{-7}\) in \(\left(a x-\frac{1}{3 b x^2}\right)^{11}\) are equal, then

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

If the area of the bounded region \(R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) is, \(\alpha\left(\log _{e} 2\right)^{-1}+\beta\left(\log _{e} 2\right)+\gamma\), then the value of \((\alpha+\beta-2 \gamma)^{2}\) is equal to:

Let \(x^{2}+y^{2}+A x+B y+C=0\) be a circle passing through \((0,6)\) and touching the parabola \(y = x ^{2}\) at \((2,4)\). Then \(A + C\) is equal to

If all the words (with or without meaning) having five letters, formed using the letters of the word \(SMALL\) and arranged as in a dictionary; then the position of the word \(SMALL\) is :

Let the domains of the functions
\(\mathrm{f}(\mathrm{x})=\log _4 \log _3 \log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)\) and \(g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)\) be \((\alpha, \beta)\) and \([\gamma, \delta]\), respectively. Then \(\alpha^2+\beta^2+\gamma^2+\delta^2\) is equal to :-

The equation of the line through the point \((0,1,2)\) and perpendicular to the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}\) is

If the shortest between the lines \(\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}\) and \(\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{z+2 \sqrt{6}}{5}\) is \(6\), then the square of sum of all possible values of \(\lambda\) is