Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is atleast one more than the number of girls ahead of her is

If the straight lines \(\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}\) and \(\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}\) are coplanar, then the plane (s) containing these two lines is (are)

Let \(S_1=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}\)
\(S_2=\{(i, j): 1 \leq i < j+2 \leq 10, i, j\) \(\in\{1,2, \ldots, 10\}\} \)
\(S_3=\{(i, j, k, l): 1 \leq i < j < k < l, i,\) \(j, k, l \in\{1,2, \ldots . ., 10\}\}\)
and \(S_4=\{(i, j, k, l): i, j, k\) and \(l\) are distint elements in \(\{1,2, \ldots, 10\}\}\) If the total number of elements in the set \(S_r\) is \(n_r, r=1,2,3,4\) then which of the following statements is (are) TRUE?

Let \(f(x)\) be a non-constant twice differentiable function defined on \((-\infty, \infty)\), such that \(f(x)=f(1-x)\) and \(f^{\prime}\left(\frac{1}{4}\right)=0\). Then,

If $f:R\to R$ is a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$ for all $x\in R$ and $f\left(\frac{1}{2}\right)=\frac{1}{2}, f\left(1\right)=1$ then

Assuming $2\mathrm{s}\mathrm{ }–\mathrm{ }2\mathrm{p}$ mixing is NOT operative, the paramagnetic species among the following is

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If \(r=5 \times 10^{-4} \mathrm{~m}, \rho=10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), \(g=10 \mathrm{~ms}^{-2}, T=0.11 \mathrm{Nm}^{-1}\), the radius of the drop when it detaches from the dropper is approximately

If \(f(x)=\min \left\{1, x^2, x^3\right\}\), then

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

Let \(S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.\) and \(\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}\). If the area of the region \(S\) is \(\alpha \sqrt{2}\), then \(\alpha\) is equal to

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is