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

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 :

An infinitely long straight wire carrying current I is bent in a planer shape as shown in the diagram. The radius of the circular part is r . The magnetic field at the centre O of the circular loop is :$\\$

Match List-I with List-II.

List-I	List-II
Pair of Compounds	Type of Isomers
A. 2-Methylpropene and but-1-ene	I. Stereoisomers
B. Cis-but-2-ene and trans-but-2-ene	II. Position isomers
C. 2-Butanol and diethyl ether	III. Chain isomers
D. But-1-ene and but-2-ene	IV. Functional group isomers

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

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 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 .

Let $\mathrm{A}=\left\{\mathrm{x}:\left|\mathrm{x}^{2}-10\right| \leq 6\right\}$ and Then

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:

Consider the following data: $\\ \Delta_{\mathrm{f}} \mathrm{H}^{\Theta}($ methane, g Enthalpy of sublimation of graphite Dissociation enthalpy of The bond enthalpy of bond is given by:

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 :

Aqueous HCl reacts with $\mathrm{MnO}_{2}(\mathrm{~s})$ to form $\mathrm{MnCl}_{2}(\mathrm{aq})$, and . What is the weight (in g ) of liberated when 8.7 g of is reacted with excess aqueous HCl solution ? (Given Molar mass in )

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

A large drum having radius R is spinning around its axis with angular velocity $\omega$, as shown in figure. The minimum value of $\omega$ so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M is $\mu$, is :

Let $f: R \rightarrow R$ be a twice differentiable function such that the quadratic equation $\mathrm{f}(\mathrm{x}) \mathrm{m}^{2}-2 \mathrm{f}^{\prime}(\mathrm{x}) \mathrm{m}+\mathrm{f}^{\prime}(\mathrm{x})=0$ in m , has two equal roots for every . If , and is the largest interval in which the function is increasing, then is equal to

The sum of all the roots of the equation $(\mathrm{x}-1)^{2}-5|\mathrm{x}-1|+6=0$, is:

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to:

Let O be the vertex of the parabola $x^{2}=4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C , which is bisected at the point $(1,2)$, is:

Let $\mathrm{a}_{1}=1$ and for $\mathrm{n} \geq 1, \mathrm{a}_{\mathrm{n}+1}$\\ . Then is equal to