Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f^{\prime \prime}(\mathrm{x})>0$ for all $\mathrm{x} \in \mathbf{R}$ and $f^{\prime}(\mathrm{a}-1)=0$, where a is real number. Let $\mathrm{g}(\mathrm{x})=f\left(\tan ^{2} \mathrm{x}-2 \tan \mathrm{x}+\mathrm{a}\right)$, $0<\mathrm{x}<\frac{\pi}{2}.\\$ Consider the following two statements :$\\$(I) g is increasing in $\left(0, \frac{\pi}{4}\right)\\$ (II) g is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Let the line L pass through the point $(-3,5,2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point ( $-2, \mathrm{r}, 1$ ) is , then the sum of all possible values of is :

If P is a point on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=4, \mathrm{Q}$ is a point on the straight line and is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such point P is .

The positive integer n , for which the solutions of the equation\\ $\mathrm{x}(\mathrm{x}+2)+(\mathrm{x}+2)(\mathrm{x}+4)+\ldots .+(\mathrm{x}+2 \mathrm{n}-2)(\mathrm{x}+2 \mathrm{n})$\\ \\ are two consecutive even integers, is :

Let the line $L_{1}$ be parallel to the vector $-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and pass through the point and the line be parallel to the vector and pass through the point . If the line is parallel to the vector and intersects the lines and at the points C and D , respectively, then is equal to :

Let $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2} f^{\prime}(1)+2 \mathrm{x} f^{\prime \prime}(2)+f^{\prime \prime \prime}(3)$, . Then the value of is :

If\\ $\left(\frac{1}{{ }^{15} \mathrm{C}_{0}}+\frac{1}{{ }^{15} \mathrm{C}_{1}}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_{1}}+\frac{1}{{ }^{15} \mathrm{C}_{2}}\right) \ldots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right) =\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_{0}{ }^{14} \mathrm{C}_{1} \ldots{ }^{14} \mathrm{C}_{12}}$, then is equal to

Let one end of a focal chord of the parabola $y^{2}=16 x$ be (16, 16). If $\mathrm{P}(\alpha, \beta)$ divides this focal chord internally in the ratio , then the minimum value of is equal to :

Let [ • denote the greatest integer function and $f(\mathrm{x})=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^{3}} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^{2}}{3^{\mathrm{x}}}\right]$. Then is equal to

Let $\alpha$ and $\beta$ be the roots of equation $x^{2}+2 a x+(3 a+10) =0$ such that . Then the set of all possible values of is :

A random variable $X$ takes values $0,1,2,3$ with probabilities $\frac{2 \mathrm{a}+1}{30}, \frac{8 \mathrm{a}-1}{30}, \frac{4 \mathrm{a}+1}{30}$, b respectively, where . Let and respectively be the mean and standard deviation of such that Then is equal to :

For a triangle $A B C$, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and . If and , where is the angle between and , then is equal to:

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then is equal to:

In the line $\alpha x+4 y=\sqrt{7}$, where , touches the ellipse at the point in the first quadrant, then one of the focal distances of P is :

Let $\mathrm{y}^{2}=12 \mathrm{x}$ be the parabola with its vertex at O . Let P be a point on the parabola and A be a point on the x -axis such that $\angle \mathrm{OPA}=90^{\circ}$. Then the locus of the centroid of such triangles OPA is :

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by x Ry if and only if . Let be the number of elements in R , and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then is equal to:

The largest $\mathrm{n} \in \mathrm{N}$, for which $7^{\mathrm{n}}$ divides 101!, is :

For the matrices $\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and , if , then among the following which one is true?

If $\int_{0}^{1} 4 \cot ^{-1}\left(1-2 \mathrm{x}+4 \mathrm{x}^{2}\right) \mathrm{dx}=\mathrm{a} \tan ^{-1}(2)-\operatorname{blog}_{\mathrm{e}}(5)$, where , then is equal to .

If the area of the region $\left\{(x, y): 1-2 x \leq y \leq 4-x^{2}\right.$, is , then the value of is :