Let $f(x) = x^3 + ax^2 + 2bx + c$, $x \in \mathbb{R}$. If $f'(1) = a$, $f''(2) = b$, and $f'''(3) = c$, then the value of $f'(5)$ is (A) $\frac{117}{5}$ (B) $\frac{62}{5}$ (C) $\frac{675}{5}$ (D) $\frac{117}{5}$
Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to: (A) 7 (B) 5 (C) 22 (D) 16
If three vectors are given as shown. If angle between vector $\vec{p}$ and $\vec{q}$ is $\theta$ where $\cos\theta = \frac{1}{\sqrt{3}}$ and $|\vec{p}| = 2\sqrt{3}$, $|\vec{q}| = 2$. Then the value of $|\vec{p} \times (\vec{q} - 3\vec{r})|^2 - 3|\vec{r}|^2$ is
Let $O$ be the vertex of the parabola $y^2 = 16x$. The locus of centroid of $\triangle OPA$ when $P$ lies on parabola and $A$ lies on $x$-axis and $\angle OPA = 90°$ is (A) $y^2 = 8(3x - 16)$ (B) $9y^2 = 8(3x - 16)$ (C) $y^2 = 8(3x + 16)$ (D) $9y^2 = 8(3x + 16)$
Let $A = \{2, 3, 5, 7, 9\}$. Consider a relation defined as $R = \{(x, y) : 2x \leq 3y, x \in A, y \in A\}$ $l =$ total number of elements in relation R $m =$ Number of elements required in R to make it symmetric. Find $l + m$. (A) 18 (B) 25 (C) 27 (D) 30
A block of mass m is at rest w.r.t. hollow cylinder which is rotating with angular speed $\omega$. Radius of cylinder is R. Find minimum coefficient of friction between block and cylinder. (A) $\frac{g}{4\omega^2 R}$ (B) $\frac{3g}{2\omega^2 R}$ (C) $\frac{g}{\omega^2 R}$ (D) $\frac{2g}{\omega^2 R}$