Let $f^{\prime }\left(x\right)= \frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{ }\mathbb{R}$ with $f\left(\frac{1}{2}\right)=0.$ If $m \le \underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx \le M,$ then the possible values of m and M are

Let $f:\left(0, 1\right)\to \mathrm{ℝ}$ be the function defined as $f\left(x\right)=\sqrt{n}$ if $x\in [\frac{1}{n+1}, \frac{1}{n})$ where $n\in \mathrm{\mathbb{N}}$. Let $g:\left(0, 1\right)\to \mathrm{ℝ}$ be a function such that ${\int }_{x^2}^x\sqrt{\frac{1-t}{t}}dt<g\left(x\right)<2\sqrt{x}$ for all $x\in \left(0, 1\right)$. Then $\underset{x\to 0}{\lim }f\left(x\right) g\left(x\right)$

Consider the two curves
\(C_1: y^2=4 x\)
\(C_2: x^2+y^2-6 x+1=0\), then

Let \((x, y, z)\) be points with integer coordinates satisfying the system of homogeneous equations \(3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0\). Then, the number of such points for which \(x^2+y^2+z^2 \leq 100\) is

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The area bounded by the curve \(y=f(x)\) and the lines \(x=0, y=0\) and \(x=t\), lies in the interval

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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 one of the following statements is correct?

Match List I with List II and select the correct answer using the code given below the lists :
 

	List - I		List - II
A.	${\left(\frac{1}{{\text{y}}^2}\left(\frac{\text{cos}\left({\text{tan}}^{-1}\text{y}\right)+\text{y sin}\left({\text{tan}}^{-1}\text{y}\right)}{\text{cot}\left({\text{sin}}^{-1}\text{y}\right)+\text{tan}\left({\text{sin}}^{-1}\text{y}\right)}\right)+{\text{y}}^4\right)}^{1/2}takes\; value$	P.	$\frac{1}{2}\sqrt{\frac{5}{3}}$
B.	​If cosx + cosy + cosz = 0 = sinx + siny + sinz then possible value of cos $\frac{\text{x}-\text{y}}{2}is$	Q.	$\sqrt{2}$
C.	If cos $\left(\frac{\pi }{4}-\text{x}\right)$ cos2x + sinx sin 2x secx = cosx sin2x secx + cos $\left(\frac{\pi }{4}+\text{x}\right)$ cos2x then possible value of secx is	R.	$\frac{1}{2}$
D.	If cot $\left({\text{sin}}^{-1}\sqrt{1-{\text{x}}^2}\right)=\text{sin}\left({\text{tan}}^{-1}\left(\text{x}\sqrt{6}\right)\right)\text{,}x\ne 0\text{,}$  then possible value of x is	S.	1

Consider a body of mass $1.0 kg$ at rest at the origin at time $t=0$. A force $\overrightarrow{F}=\left(\alpha t\widehat{i}+\beta \widehat{j}\right)$ is applied on the body, where $\alpha =1.0N{s}^{-1}and \beta =1.0N$ . The torque acting on the body about the origin at time $=1.0 s \mathrm{i}\mathrm{s} \overrightarrow{\tau }$ . Which of the following statements is (are) true?

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p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Z\), is

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $xy$-plane with the center of the loop always on the axis $\hat{\mathrm{n}}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counter clockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{\mathrm{n}}$ direction. List-I gives time dependences of $\hat{\mathrm{n}}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi }{6\omega }$, Let $\alpha =\frac{A^2{\mu }_0^2{m}^2{I}^2\omega }{2R}$.

	List-I		List-II
(i)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{j}}+\cos \omega t\hat{\mathrm{k}}\right)$	(p)	$0$
(ii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{j}}\right)$	(q)	$-\frac{\alpha }{4}\hat{\mathrm{i}}$
(iii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{k}}\right)$	(r)	$\frac{3\alpha }{4}\hat{\mathrm{i}}$
(iv)	$\frac{1}{\sqrt{2}}\left(\cos \omega t\hat{\mathrm{j}}+\sin \omega t\hat{\mathrm{k}}\right)$	(s)	$\frac{\alpha }{4}\hat{\mathrm{j}}$
		(t)	$-\frac{3\alpha }{4}\hat{\mathrm{i}}$

Which one of the following options is correct?

Column II shows five systems in which two objects are labelled as \(X\) and \(Y\). Also in each case a point \(P\) is shown. Column-I gives some statements about \(X\) and/or \(Y\). Match these statements to the appropriate system(s) from Column-II.

Let $a, b\mathbb{ }\in \mathbb{R}$ and $a^2+b^2\ne 0$ . Suppose $S=\left\{z\in \mathbb{C}\mathbb{ }:z=\frac{1}{a+ibt}, t\mathbb{ }\in \mathbb{R},\mathbb{ }t \ne 0\right\},$ where $i= \sqrt{-1}.$ If $z=x+iy$ and $z\in S,$ then $\left(x, y\right)$ lies on

The pair(s) of diamagnetic ions is(are)