Let $f:\mathbb{R}\to \mathbb{R}\mathbb{ }\mathrm{a}\mathrm{n}\mathrm{d} g:\mathbb{R}\to \mathbb{R}$ be two non-constant differentiable functions. If $f^{\prime }\left(x\right)=\left(e^{\left(f\left(x\right)\right)-g\left(x\right)}\right)g^{\prime }\left(x\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in \mathbb{R}$, and $f\left(1\right)=g\left(2\right)=1$ then which of the following statement(s) is (are) TRUE?

Suppose that the foci of the ellipse $\frac{x^2}{9}+ \frac{y^2}{5}=1$ are $\left(f_1, 0\right) and (f_2, 0)$ where $f_1>0 and f_2<0.$ Let $P_1 and P_2$ be two parabolas with a common vertex at (0, 0) and with foci at $\left(f_1, 0\right)$ and $\left(2f_2, 0\right),$ respectively. Let $T_1$ be a tangent to $P_1$ which passes through $\left(2f_2, 0\right)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1, 0)$ . If $m_1$ is the slope of $T_1$ and $m_2$  is the slope of $T_2,$ then the value of $\left(\frac{1}{m_1^2}+ m_2^2\right)$ is

Let $\hat{i},\hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes. Let $\overrightarrow{a}=3\hat{i}+\hat{j}-\hat{k}$ $\overrightarrow{b}=\hat{i}+b_2\hat{j}+b_3\hat{k}, b_2,b_3\in \mathrm{ℝ}$ $\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k},c_1,c_2,c_3\in \mathrm{ℝ}$ be three vectors such that $b_2{b}_3>0,\overrightarrow{a}·\overrightarrow{b}=0$ and $\left(\begin{array}{ccc}0& -c_3& c_2\\ c_3& 0& -c_1\\ -c_2& c_1& 0\end{array}\right)\left(\begin{array}{l}1\\ b_2\\ b_3\end{array}\right)=\left(\begin{array}{c}3-c_1\\ 1-c_2\\ -1-c_3\end{array}\right).$ Then, which of the following is/are TRUE?

Let \(A B C\) be a triangle such that \(\angle A C B=\frac{\pi}{6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively. The value(s) of \(x\) for which \(a=x^2+x+1, b=x^2-1\) and \(c=2 x+1\) is (are)

For all \(x>0\), let \(y_1(x), y_2(x)\), and \(y_3(x)\) be the functions satisfying
\(\begin{aligned} & \frac{d y_1}{d x}-(\sin x)^2 y_1=0, y_1(1)=5 \\ & \frac{d y_2}{d x}-(\cos x)^2 y_2=0, y_2(1)=\frac{1}{3} \\ & \frac{d y_3}{d x}-\left(\frac{2-x^3}{x^3}\right) y_3=0, y_3(1)=\frac{3}{5 e}\end{aligned}\)
respectively. Then \(\lim _{x \rightarrow 0^{+}} \frac{y_1(x) y_2(x) y_3(x)+2 x}{e^{3 x} \sin x}\) is equal to _____

Consider the lines given by
\[
L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0
\]
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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 ?

Light of wavelength ${\lambda }_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\varphi$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is ${\lambda }_e,$ which of the following statement(s) is(are) true ?

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Thermal decomposition of gaseous \(\mathrm{X}_{2}\) to gaseous \(\mathrm{X}\) at \(298 \mathrm{~K}\) takes place according to the following equation:

\(\mathrm{X}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{X}(\mathrm{g})\)

The standard reaction Gibbs energy, \(\Delta_{r} G^{\circ}\), of this reaction is positive. At the start of the reaction, there is one mole of \(\mathrm{X}_{2}\) and no \(\mathrm{X}\). As the reaction proceeds, the number of moles of \(\mathrm{X}\) formed is given by \(\beta\). Thus, \(\beta_{\text {equilibrium }}\) is the number of moles of \(\mathrm{X}\) formed at equilibrium. The reaction is carried out at a constant total pressure of \(2\) bar. Consider the gases to behave ideally. (Given : \(R=0.083 \mathrm{~L}\) bar \(\mathrm{K}^{-1} \mathrm{~mol}^{-1}\) )

Question:
The equilibrium constant \(K_{P}\) for this reaction at \(298 \mathrm{~K},\) in terms of \(\beta_{equilibrium}\) is

Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

For the circuit shown in the figure

Two ideal batteries of emf ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ and three resistances ${\mathrm{R}}_1,\mathrm{}{\mathrm{R}}_2$ and ${\mathrm{R}}_3$ are connected as shown in the figure. The current in resistance ${\mathrm{R}}_2$ would be zero if

The total number of basic groups in the following form of lysine is