CBSE Class class-12 Mathematics — Very Short Answer Questions

(a) Check whether the function f(x) defined as $f(x) = \begin{cases} \dfrac{|x - 3|}{2(x - 3)}, & x < 3 \\ \dfrac{x-6}{6}, & x \ge 3 \end{cases}$ is continuous at $x = 3$ or not. OR (b) If $\sqrt{3} (x^2 + y^2) = 4xy$, then find $\dfrac{dy}{dx}$ at $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of $1 \text{ mm}^3/\text{min}$. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is $10 \text{ mm}$, if the semi-vertical angle of conical bottle is $\pi/6$.

Find the vector of magnitude 14 in the direction of vector QP, where P and Q are the points $(1, 3, 2)$ and $(-1, 0, 8)$ respectively.

Vectors $a = 3i - 2j + 2k$ and $b = i + 2k$ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

(a) Simplify: $\tan^{-1}\left[\dfrac{\cos 2x - \sin 2x}{\cos 2x + \sin 2x}\right]$, $0 < x < \pi/4$. OR (b) Evaluate: $\tan\left[\sin^{-1} 1 - \cos^{-1}(-1/2)\right]$.