If \(\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}\), then

Consider the functions $f,g:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=x^2+\frac{5}{12}$ and $g\left(x\right)=\left\{\begin{array}{cc}2\left(1-\frac{4\left|x\right|}{3}\right),& \left|x\right|\le \frac{3}{4},\\ 0,& \left|x\right|>\frac{3}{4}\end{array}\right.$
If $\alpha$ is the area of the region $\left\{\left(x,y\right)\in \mathrm{ℝ}\times \mathrm{ℝ}:\left|x\right|\le \frac{3}{4},0\le y\le \min \left\{f\left(x\right),g\left(x\right)\right\}\right\}$, then the value of $9\alpha$ is _____ .

Let \(L_1\) be the line of intersection of the planes given by the equation- \(2 x+3 y+z=4 \text { and } x+2 y+z=5 .\)
Let \(L_2\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_1\). Let \(M\) denote the plane given by the equation- \(2 x+y-2 z=6\)
Suppose that the line \(L_2\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

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 $a, b\mathbb{ }\in \mathbb{R}$ 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

Let \(f: R \rightarrow R\) be a continuous function, which satisfies \(f(x)=\int_0^x f(t) d t\). Then, the value of \(f(\ln 5)\) is

\(\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}\) equals

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

LIST-I contains reactions and LIST-II contains major products

List - I	List - II
(A)	(P)
(B)	(Q)
(C)	(R)
(D)	(S)
	(T)

Match each reaction in LIST-I with one or more products in LIST-II and choose the correct option

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Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately \(6.023 \times 10^{23}\) ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A \(4.0\) molar aqueous solution of \(\mathrm{NaCl}\) is prepared and \(500 \mathrm{~mL}\) of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: \(\mathrm{Na}=23, \mathrm{Hg}=200 ; 1\) faraday \(=96500\) coulombs).
Question:
The total number of moles of chlorine gas evolved is

A lamina is made by removing a small disc of diameter \(2 R\) from a bigger disc of uniform mass density and radius \(2 R\), as shown in the figure. The moment of inertia of this lamina about axes passing though \(O\) and \(P\) is \(I_{O}\) and \(I_{P}\) respectively. Both these axes are perpendicular to the plane of the lamina. The ratio \(I_{P} / I_{O}\) to the nearest integer is

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The maximum value of \(v_0\) for which the disk will roll without slipping is

Let $A_1,A_2,A_3,\dots ,A_8$ be the vertices of a regular octagon that lie on a circle of radius $2$. Let $P$ be a point on the circle and let $P{A}_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2,\dots ,8$. If $P$ varies over the circle, then the maximum value of the product $P{A}_1·P{A}_2\dots ·P{A}_8$, is