Let $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{ }\underset{\mathrm{x}}{\overset{{\mathrm{x}}^2+\frac{\mathrm{\pi }}{6}}{\int }}2{\cos }^2\mathrm{t}\mathrm{ }\mathrm{d}\mathrm{t}$ for all $\mathrm{x}\mathrm{ }\in \mathbb{R}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{f}\mathrm{ }:\mathrm{ }\left[0,\frac{1}{2}\right]\to \left[0,\mathrm{ }\mathrm{\infty }\right)$ be a continuous function. For $\mathrm{a}\in \left[0,\frac{1}{2}\right],$ if ${\mathrm{F}}^{\mathrm{\prime }}\left(\mathrm{a}\right)+2$ is the area of the region bounded by $\mathrm{x}=0,\mathrm{ }\mathrm{y}=0,\mathrm{ }\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{x}=\mathrm{a},$ then $\mathrm{f}\left(0\right)$ is

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
Equations of the sides \(Q R, R P\) are

Let $f:R\to R$ be given by $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-5\right).$ Define $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt, x>0.$ Then which of the following options is/are correct?

Match the integrals in Column I with the values in Column II.

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0.
Question:
The number of matrices in \(A\) is

The value of  $I={\displaystyle \sum }_{k=1}^{98}\underset{k}{\overset{k+1}{{\displaystyle \int }}}\frac{k+1}{x(x+1)}dx,$ then

Aqueous solution of \(\text{Na} _2\text S_2\text O _3\) on reaction with \(\text{Cl} _2\) gives

The cubic unit cell structure of a compound containing cation $\mathrm{M}$ and anion $\mathrm{X}$ is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

The number of geometric isomers possible for the complex ${\left[\text{C}\text{o}{\text{L}}_2\text{C}{\text{l}}_2\right]}^{-}\left(\text{L}={\text{H}}_2\text{N}\text{C}{\text{H}}_2\text{C}{\text{H}}_2{\text{O}}^{-}\right)$ is
 

Let $\alpha$ be a positive real number. Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\left(\alpha ,\infty \right)\to \mathrm{ℝ}$ be the functions defined by $f\left(x\right)=\sin \left(\frac{\pi x}{12}\right)$ and $g\left(x\right)=\frac{2{\log }_e\left(\sqrt{x}-\sqrt{\alpha }\right)}{{\log }_e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha }}\right)}$.
Then the value of $\underset{x\to {\alpha }^{+}}{\lim }f\left(g\left(x\right)\right)$ is ______.

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

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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 1 , the energy stored in the capacitor \(E_{C}\) and heat dissipated across resistance \(E_{D}\) are related by: