Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of ${\lambda }^2$ is

Let $\mathrm{A}\mathrm{P}\mathrm{ }\left(\mathrm{a};\mathrm{d}\right)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0.$ If $\mathrm{A}\mathrm{P}\left(1;3\right)\cap \mathrm{A}\mathrm{P}\left(2;5\right)\cap \mathrm{A}\mathrm{P}\left(3;7\right)=\mathrm{A}\mathrm{P}\left(\mathrm{a};\mathrm{d}\right)$ then $\mathrm{a}+\mathrm{d}$ equals ____

Let $f:\left(0, \pi \right)\to \mathbb{R}$ be a twice differentiable function such that $\underset{t\to x}{\lim }\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}={\sin }^{2}x$ for all $x\in \left(0, \pi \right)$. If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE ?

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

For the function \(f(x)=x \cos \frac{1}{x}, x \geq 1\),

Let \(\omega=e^{i \pi / 3}\) and \(a, b, c, x, y, z\) be non-zero complex numbers such that \(a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z\).
Then, the value of \(\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}\) is

Let $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

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Consider a simple \(R C\) circuit as shown in Figure \(1\).

Process 1: In the circuit the switch \(S\) is closed at \(t=0\) and the capacitor is fully charged to voltage \(V_{0}\) (i.e., charging continues for time \(T>>R C\) ). In the process some dissipation \(\left(E_{D}\right)\) occurs across the resistance \(R\). The amount of energy finally stored in the fully charged capacitor is \(E_{C}\).

Process 2: In a different process the voltage is first set to \(\frac{V_{0}}{3}\) and maintained for a charging time \(T>>R C\). Then the voltage is raised to \(\frac{2 V_{0}}{3}\) without discharging the capacitor and again maintained for a time \(T>>R C\). The process is repeated one more time by raising the voltage to \(V_{0}\) and the capacitor is charged to the same final voltage \(V_{0}\) as in Process 1.

These two processes are depicted in Figure \(2 .\)







Question:

In Process 2, total energy dissipated across the resistance \(E_{D}\) is:

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.

Two parallel wires in the plane of the paper are distance ${\mathrm{X}}_0$ apart. A point charge is moving with speed u between the wires in the same plane at a distance ${\mathrm{X}}_1$ from one of the wires. When the wires carry current of magnitude $\mathrm{ }\mathrm{I}$  in the same direction, the radius of curvature of the path of the point charge is ${\mathrm{R}}_1$ . In contrast, if the currents $\mathrm{ }\mathrm{I}$ in the two wires have directions opposite to each other, the radius of curvature of the path is ${\mathrm{R}}_2$ . If $\frac{{\mathrm{X}}_0}{{\mathrm{X}}_1}=3$ , value of $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$ is

Consider a concave mirror and a convex lens (refractive index = $1.5$) of focal length $10$ $\mathrm{cm}$ each, separated by a distance of $50$ $\mathrm{cm}$ in air (refractive index = $1$), as shown in the figure. An object is placed at a distance of $15$ $\mathrm{cm}$ from the mirror. Its erect image, formed by this combination, has magnification ${\mathrm{M}}_1$ . When the set-up is kept in a medium of refractive index $\frac{7}{6}$ , the magnification becomes ${\mathrm{M}}_2$. The magnitude $\left|\frac{{\mathrm{M}}_2}{{\mathrm{M}}_1}\right|$ is

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

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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?