Let $f:\left[a, b\right]\to \left[1, \infty \right)$ be a continuous function and let $g :R\to R$ be defined as
$g\left(x\right)= \left\{\begin{array}{ccc}0& if& x<a,\\ \underset{a}{\overset{x}{\int }}f\left(t\right)dt& if& a \le x \le b,\\ \underset{a}{\overset{b}{\int }}f\left(t\right)dt& if& x>b\end{array}\right.$ . Then

The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on

Let \(z=x+i y\) be a complex number, where \(x\) and \(y\) are integers. Then, the area of the rectangle whose vertices are the roots of the equation \(z \bar{z}^3+\bar{z} z^3=350\) is

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

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Let \(f:[0,1] \rightarrow \mathbb{R}\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0)=f(1)=0\) and satisfies \(f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .\)


Question:

If the function \(\mathrm{e}^{-x} f(x)\) assumes its minimum in the interval \([0,1]\) at \(x=\frac{1}{4}\), which of the following is true?

Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha\)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is

Let $f:\left[0, \infty \right)\to R$ be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t}f\left(t\right)dt$ for all $x\in \left[0,\infty \right).$ Then, which of the following statement(s) is (are) TRUE?

If \(f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t\) for all \(x \in(0, \infty)\), then

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1=300\mathrm{K}\mathrm{a}\mathrm{n}\mathrm{d}{T}_2=100\mathrm{K}$as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1\mathrm{a}\mathrm{n}\mathrm{d}{K}_2$, respectively. If the temperature at the junction of the two cylinders in the steady state is $200\mathrm{}\mathrm{K}$, then $K_1/K_2$is

A ball moves over a fixed track as shown in the figure. From \(A\) to \(B\) the ball rolls without slipping. If surface \(B C\) is frictionless and \(K_A, K_B\) and \(K_C\) are kinetic energies of the ball at \(A, B\) and \(C\) respectively, then

Among PbS, CuS, HgS, MnS, $\mathrm{A}{\mathrm{g}}_2\mathrm{S}$ , NiS, CoS, $\mathrm{B}{\mathrm{i}}_2{\mathrm{S}}_3$ and $\mathrm{S}\mathrm{n}{\mathrm{S}}_2$ , the total number of black coloured sulphides is

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The speed of the particle that can take discrete values is proportional to

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is