Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is

Let $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(0\right)=0, f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }\left(0\right)=1.$ If $g\left(x\right)={\int }_x^{\frac{\mathrm{\pi }}{2}} \left[f^{\prime }\left(t\right)cosect-\cot t cosect f\left(t\right)\right]dt$ , for $x\in \left(0,\frac{\pi }{2}\right]$ then $\underset{x\to 0}{\lim }g\left(x\right)=$

Match the following.

Let \(A B C D\) be a quadrilateral with area 18 , with side \(A B\) parallel to the side \(C D\) and \(A B=2 C D\). Let \(A D\) be perpendicular to \(A B\) and \(C D\). If a circle is drawn inside the quadrilateral \(A B C D\) touching all the sides, then its radius is

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form \(\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\), where each of \(a, b\) and \(c\) is either \(\omega\) or \(\omega^2\). Then, the number of distinct matrices in the set \(S\) is

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}\) determines a family of circles with

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that the trace of \(A\) is not divisible by \(p\) but det \((A)\) is divisible by \(p\) is
[Note : The trace of a matrix is the sum of its diagonal entries.]

A vibrating string of certain length \(l\) under a tension \(T\) resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length \(75 \mathrm{~cm}\) inside a tube closed at one end. The string also generates 4 beat/s when excited along with a tuning fork of frequency \(n\). Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be \(340 \mathrm{~m} / \mathrm{s}\), the frequency \(n\) of the tuning fork (in \(\mathrm{Hz}\) ) is

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
When disc \(B\) is brought in contact with disc \(A\), they acquire a common angular velocity in time \(t\). The average frictional torque on one disc by the other during this period is

Column I		Column II
$\left(\mathrm{A}\right)$	In a triangle $∆XYZ$ let $a,b$ and $\mathrm{c}$ be the lengths of the angles $X,Y$ and $\mathrm{Z}$, respectively. If $2\left(a^2-b^2\right)=c^2$ and $\lambda =\frac{\sin \left(X-Y\right)}{\mathrm{sinZ}}$then possible values of $n$ for which $\cos \left(n\pi \lambda \right)=0$ is are	$\left(\mathrm{P}\right)$	$1$
$\left(\mathrm{B}\right)$	In a triangle $\mathrm{\Delta }XYZ,$ let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $1+\cos 2x-2\cos 2y$$=2\sin x\sin y$, then possible value(s) of $\frac{a}{b}$ is (are)	$\left(\mathrm{Q}\right)$	$2$
$\left(\mathrm{C}\right)$	In ${\mathbb{R}}^2,$ let $\sqrt{3}\mathrm{ }\hat{\mathrm{i}}+\hat{\mathrm{j}},\mathrm{ }\hat{\mathrm{i}}+\sqrt{3}\mathrm{ }\hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+\left(1-\beta \right)\mathrm{ }\hat{\mathrm{j}}$ be the position vectors of $X,Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow{OX}$ with $\overrightarrow{OY}$ is $\frac{3}{\sqrt{2}}$, then possible value (s) of $\left|\beta \right|$ is (are)	$\left(\mathrm{R}\right)$	$3$
$\left(\mathrm{D}\right)$	Suppose that $F\left(\alpha \right)$ denotes the area of the region bounded by $x=0,\mathrm{ }x=2, y^2=4x$ and$y=\left|ax-1\right|+$$\left|\alpha x-2\right|+\alpha x,$  where $\alpha \in \left\{0, 1\right\}.$ Then the value (s) of $F\left(\alpha \right)+\frac{8}{3} \sqrt{2},$ when $\alpha =0$ and $\alpha =1$, is (are)	$\left(\mathrm{S}\right)$	$5$
		$\left(\mathrm{T}\right)$	$6$

 

Consider the reaction $\mathrm{A}\rightleftharpoons \mathrm{B}$ at $1000\mathrm{K}$. At time ${\mathrm{t}}^{\text{'}}$, the temperature of the system was increased to $2000\mathrm{K}$ and the system was allowed to reach equilibrium. Throughout this experiment the partial pressure of $\mathrm{A}$ was maintained at $1$ bar. Given below is the plot of the partial pressure of $\mathrm{B}$ with time. What is the ratio of the standard Gibbs energy of the reaction at $1000\mathrm{K}$ to that at $2000\mathrm{K}$?

Let $f:\left[a, b\right]\to \left[1, \infty \right)$ be a continuous function and let $g :R\to R$ be defined as
$g\left(x\right)= \left\{\begin{array}{ccc}0& if& x<a,\\ \underset{a}{\overset{x}{\int }}f\left(t\right)dt& if& a \le x \le b,\\ \underset{a}{\overset{b}{\int }}f\left(t\right)dt& if& x>b\end{array}\right.$ . Then

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The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

When \(d \approx a\) but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height \(h\) above the loop. In that case