In a triangle $PQR$ , let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$ . If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}.\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}.\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$, then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is _______

The equation of a plane passing through the line of intersection of the planes \(x+2 y+3 z=2\) and \(x-y+z=3\)

and at a distance \(\frac{2}{\sqrt{3}}\) from the point \((3,1,-1)\) is

Let $S=\left\{x\in \left(-\pi , \pi \right):x\ne 0, \pm \frac{\pi }{2}\right\}$ . The sum of all distinct solution of the equation $\sqrt{3}\sec x+cosec x+2\left(\tan x-\cot x\right)=0$, in the set S is equal to 

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $\mathrm{V}\mathrm{ }\mathrm{m}{\mathrm{m}}^3$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is 10mm, then the value of $\frac{\mathrm{V}}{250\mathrm{\pi }}\mathrm{ }$ is

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac\left(a-c\right)+ad\left(a-d\right)+bc\left(b-c\right)+bd\left(b-d\right)$

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Let \(A_1, G_1, H_1\) denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For \(n \geq 2\), let \(A_{n-1}\) and \(H_{n-1}\) has arithmetic, geometric and harmonic means as \(A_n, G_n, H_n\), respectively.
Question:
Which of the following statements is correct?

Match the statements of Column I with these in Column II.
[Note : Here \(z\) takes values in the complex plane and \(\operatorname{Im}(z)\) and \(\operatorname{Re}(z)\) denote respectively, the imaginary part and the real part of \(z\) ]

Column I	Column II
(A)The set of points \(z\) satisfying \(|z-i| z||=|z+i| z||\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(|z+4|+|z-4|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(|w|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(|\operatorname{Im} z| \leq 1\)
(D) If \(|w|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(|\operatorname{Re} z| \leq 2\)
	(t) the set of points \(z\) satisfying \(|z| \leq 3\)

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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?

In the following \([x]\) denotes the greatest integer less than or equal to \(x\). Match the functions in Column I with the properties Column II.

In the following reaction sequence, the major product \(\mathbf{P}\) is formed.

Glycerol reacts completely with excess \(\mathbf{P}\) in the presence of an acid catalyst to form \(\mathbf{Q}\). Reaction of \(\mathbf{Q}\) with excess \(\mathrm{NaOH}\) followed by the treatment with \(\mathrm{CaCl}_2\) yields Ca-soap \(\mathbf{R}\), quantitatively. Starting with one mole of \(\mathbf{Q}\), the amount of \(\mathbf{R}\) produced in gram is ________.
[Given, atomic weight: \(\mathrm{H}=1, \mathrm{C}=12,\) \(\mathrm{~N}=14, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{Cl}=35,\) \(\mathrm{Ca}=40\)]

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Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
If \(P\) is a point on \(C_1\) and \(Q\) is a point on \(C_2\), then \(\frac{P A^2+P B^2+P C^2+P D^2}{Q A^2+Q B^2+Q C^2+Q D^2}\) is equal to

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.

List - I	List - II
(P) \(P _2 O _3\) \(+~3 H _2 O \rightarrow\)	(1) \(P ( O )\left( OCH _3\right)\) \(Cl _2\)
(Q) \(P _4+3 NaOH\) \(+~3 H _2 O \rightarrow\)	(2) \(H _3 PO _3\)
(R) \(PCl _5+ CH _3\) \(COOH \rightarrow\)	(3) \(PH _3\)
(S) \(H _3 PO _2\) \(+~2 H _2 O ~+\) \(4 AgNO _3 \rightarrow\)	(4) \(POCL _3\)
	(5) \(H _3 PO _4\)