$\vec{\mathrm{F}} = 4\mathrm{t}^{3}\hat{\mathrm{i}} - 3\mathrm{t}^{2}\hat{\mathrm{j}}, \mathrm{m} = 4\mathrm{~kg}$ at $\mathrm{t} = 0$ particle is at rest and at origin then find velocity and position at $\mathbf{t} = \mathbf{2}\mathbf{~sec}$.
A rod of mass $m$ and length $\boldsymbol{l}$ is attached to two ideal strings. Find tension in left string just after right string is cut. (A) $\frac{2}{3}\mathrm{mg}$ (B) $\frac{\mathrm{mg}}{4}$ (C) $\frac{\mathrm{mg}}{5}$ (D) $\frac{\mathrm{mg}}{2}$
Ellipse $E: \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1$, A hyperbola confocal with ellipse and eccentricity of hyperbola is equal to 5. The length of latus rectum of hyperbola is, if principle axis of hyperbola is $x$-axis? (A) $\frac{96}{\sqrt{5}}$ (B) $24\sqrt{5}$ (C) $18\sqrt{5}$ (D) $12\sqrt{5}$
If the mean and variance of observations $x, y, 12, 14, 4, 10, 2$ is 8 and 16 respectively where $\mathrm{x} > \mathrm{y}$. Then, the value of $3\mathrm{x} - \mathrm{y}$ is (A) 24 (B) 22 (C) 20 (D) 18
The value of $\int_{-\pi/6}^{\pi/6} \left(\frac{\pi + 4x^{11}}{1 - \sin(|x| + \frac{\pi}{6})}\right)dx$ is equal to (A) $8\pi$ (B) $7\pi$ (C) $5\pi$ (D) $4\pi$