Perpendiculars are drawn from points on the line $\frac{\text{x}+2}{2}=\frac{\text{y}+1}{-1}=\frac{\text{z}}{3}$  to the plane $x+y+z=3$. The feet of perpendiculars lie on the line

For $\text{a}\in \text{R}$ (the set of all real numbers), $\text{a}\ne -1$, $\underset{\text{n}\to \infty }{\text{lim}}\frac{\left(1^{\text{a}}+2^{\text{a}}+\dots +{\text{n}}^{\text{a}}\right)}{{\left(\text{n}+1\right)}^{\text{a}-1}\left[\left(\text{na}+1\right)+\left(\text{na}+2\right)+\dots +\left(\text{na}+\text{n}\right)\right]}=\frac{1}{60}$ Then a =

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

The integral \(\int \frac{\sec ^{2} x}{(\sec x+\tan x)^{\frac{9}{2}}} d x\) equals (for some arbitrary constant \(\mathrm{K}\) )

Consider the $6\times 6$ square in the figure. Let $A_1,A_2,\dots ,A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.

(There are two questions based on $\mathrm{PARAGRAPH} "\mathrm{II}"$, the question given below is one of them)
Two distinct points are chosen randomly out of the points $A_1,A_2,\dots \dots ,A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is

Let $\widehat{u}=u_1\widehat{i}+u_2\widehat{j}+u_3\widehat{k}$ be a unit vector in ${\mathbb{R}}^3$ and $\widehat{w}=\frac{1}{\sqrt{6}} \left(\widehat{i}+ \widehat{j}+2\widehat{k}\right).$ Given that there exists a vector $\overrightarrow{v}$ in ${\mathbb{R}}^3$ such that $\left|\widehat{u}\times \overrightarrow{v}\right|=1$ and $\widehat{w} . \left(\widehat{u}\times \overrightarrow{v}\right)=1.$ Which of the following statement(s) is (are) correct ?

Let $O$ be the center of the circle $x^2+y^2=r^2,$ where $r>\frac{\sqrt{5}}{2}.$ Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is  $2x+4y=5$. If the center of the circumcircle of the triangle $OPQ$ lies on the line $x+2y=4,$ then the value of $r$ is ______

Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at 25^o C. For this process, the correct statement is 

In a triangle $ABC$, let $AB=\sqrt{23},BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is

Let ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ denotes the lines
$\overrightarrow{\mathrm{r}}=\widehat{\mathrm{i}}+\mathrm{\lambda }\left(-\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\lambda }\in \mathrm{R}$ and $\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\mu }\in \mathrm{R}$
respectively. If ${\mathrm{L}}_3$ is a line which is perpendicular to both ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ and cuts both of them, then which of the following options describe(s) ${\mathrm{L}}_3$ ?

Match the following

Column I	Column II
(A) \(CH _3- CHBr - CD _3\) on treatment with alc. KOH gives \(CH _2= CH - CD _3\) as a major product.	(P) E1 reaction
(B) \(Ph - CHBr - CH _3\) reacts faster than \(Ph - CHBr - CD _3\)	(Q) E2 reaction
(C) \(Ph - CH _2- CH _2 Br\) on treatment with \(C _2 H _5 OD / C _2 H _5 O ^{-}\)gives \(Ph - CD = CH _2\) as the major product	(R) El cb reaction
(D) \(PhCH _2 CH _2 Br\) and \(PhCH _2 CH _2 Br\) react with same rate	(S) First order reaction

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question :

Which one of the following statements is correct?