Paragraph:

Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),

\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),

\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),

\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)

and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)


Question:

Which of the following statements is TRUE ?

Let \(y(x)\) be the solution of the differential equation
\(x^2 \frac{d y}{d x}+x y=x^2+y^2, x>\frac{1}{e}\)
satisfying \(y(1)=0\). Then the value of \(2 \frac{(y(e))^2}{y\left(e^2\right)}\) is _____ .

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

For $x\in \mathrm{ℝ}$, let $y\left(x\right)$ be a solution of the differential equation $\left(x^2-5\right)\frac{dy}{dx}-2xy=-2x{\left(x^2-5\right)}^2$ such that $y\left(2\right)=7$. Then the maximum value of the function $y\left(x\right)$ is

The area of the region between the curves \(y=\sqrt{\frac{1+\sin x}{\cos x}}\) and \(y=\sqrt{\frac{1-\sin x}{\cos x}}\) bounded by the lines \(x=0\) and \(x=\frac{\pi}{4}\) is

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4$ , then which of the following options is the only Correct combination?

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?

Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) \(NaOH\) \( /Br _2\)	(P) Condensation
(II) Acetophenone	(ii) \(B r_2 / h v\)	(Q) Carboxylation
(III) Benzaldehyde	(iii) \(\left( CH _3 CO \right)_2\) \(O / CH _3\) \(COOK\)	(R) Substitution
(IV) Phenol	(iv) \(NaOH\) \(/CO _2\)	(S) Haloform

The only CORRECT combination in which the reaction proceeds through radical mechanism is

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $x_0$. Consider two cases : (i) when the block is at $x_0\text{;}$ and (ii) when the block is at $x=x_0+\text{A}\text{.}$ In both the cases, a particle with mass m $\left(<\text{M}\right)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

A ball is thrown from the location \(\left(x_0, y_0\right)=(0,0)\) of a horizontal playground with an initial speed \(v_0\) at an angle \(\theta_0\) from the \(+x\)-direction. The ball is to be hit by a stone, which is thrown at the same time from the location \(\left(x_1, y_1\right)=(L, 0)\). The stone is thrown at an angle \(\left(180-\theta_1\right)\) from the \(+x\)-direction with a suitable initial speed. For a fixed \(v_0\), when \(\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)\), the stone hits the ball after time \(T_1\), and when \(\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)\), it hits the ball after time \(T_2\). In such a case, \(\left(T_1 / T_2\right)^2\) is _______ .

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and ${\mathrm{U}}_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

	Column I		Column II
A.	${\mathrm{U}}_1\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left[1-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]}^2$	P.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}\mathrm{a}$
B.	${\mathrm{U}}_2\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2$	Q.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}0$ .
C.	${\mathrm{U}}_3\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\exp \left[-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]$	R.	The force acting on the particle is zero at $\mathrm{x}=\mathrm{}-\mathrm{a}$
D.	${\mathrm{U}}_4\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}\left[\frac{\mathrm{x}}{\mathrm{a}}-\frac{1}{3}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^3\right]$	S.	The particle experiences an attractive force towards $\mathrm{x}\mathrm{}=\mathrm{}0$ in the region $\left|\mathrm{x}\right|<\mathrm{a}$
		T.	The particle with total energy $\frac{{\mathrm{U}}_0}{4}$ can oscillate about the point $\mathrm{x}=\mathrm{}-\mathrm{a}$

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true?

The activity of a freshly prepared radioactive sample is \(10^{10}\) disintegrations per second, whose mean life is \(10^{-9} \mathrm{~s}\). The mass of an atom of this radioisotope is \(10^{-25} \mathrm{~kg}\). The mass (in \(\mathrm{mg}\) ) of the radioactive sample is