List of practice Questions

If $\vec{a} = 2\hat{i} - t\hat{j} - 2\hat{k}$ and $\vec{b} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ are two vectors such that $\vec{a} \cdot \vec{b}$ is minimum, then $t=$

Some statements about Leclanche ll are:
Anode is zinc metal
Cathode is graphite rod surrounded by powdered {MnO2} and carbon
Electrolyte is a moist paste of ZnO and KOH
Oxidation product is ZnO

To generate the high magnetic fields, the synchrotron uses

The greatest term in the expansion of $ (1+x)^{15} $, when $ x = \frac{1}{2} $ is

The value of the following expression is: \[ \cos \left( \frac{\pi}{2^2} \right) \cdot \cos \left( \frac{\pi}{2^3} \right) \cdot \cos \left( \frac{\pi}{2^4} \right) ... \cdot \cos \left( \frac{\pi}{2^{10}} \right) \]

Match the items of List - A with those of the entries of List - B.

The probability that a non-leap year contains 53 Sundays is

Which of the following statements regarding the damping force of a damped oscillator is NOT correct? Options:

Among the following the correct statements are:
I. Carbon black is obtained by burning hydrocarbons in a limited supply of air
II. Carbon black is obtained by burning hydrocarbons in absence of air
III. Coke is obtained by burning coal at high temperatures in a limited supply of air
IV. Coke is obtained by burning coal at high temperatures in absence of air

If $ f(x) = \begin{cases} x \left( 1 + \frac{1}{2} \sin(\log x^2) \right), & x \neq 0 \\ 0, & x = 0 \end{cases} $, then $ \lim_{x \to 0} \frac{f(x) - f(0)}{x} $

If $ \frac{6x^3 + 7x^2 - 14x + 11}{6x^3 + x^2 - 10x + 3} = \frac{a}{x + p} + \frac{b}{qx + 3} + \frac{c}{3x + p} $, then $ a + b $ is:

If $ \alpha \in \mathbb{R} \setminus \{-1\} $ and \[ f(x) = |x| + \alpha |x|(|x| - 1), \] then the number of points at which $ f(x) $ is not differentiable is:

If a straight line $ L $ perpendicular to the line $ 3x - 4y = 6 $ forms a triangle of area 6 square units with coordinate axes, then the minimum perpendicular distance from the point $ (1,1) $ to the line $ L $ is:

Let $[ \cdot ]$ denote the greatest integer function.
Assertion (A): $\lim_{x \to \infty} \frac{[x]}{x} = 1$
Reason (R): $f(x) = x - 1$, $g(x) = [x]$, $h(x) = x$ and $\lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{h(x)}{x} = 1$:

The bulk modulus $ B $ and compressibility $ k $ are related by:

The lengths of the intercepts made by a circle $S$ on the $X$ and $Y$-axes are $\frac{2\sqrt{13}}{3}$ and $\frac{2\sqrt{17}}{3}$ respectively. Then the equation of the circle is:

If a circle $S$ passing through the points $A(1, 2)$ and $B(2, 1)$ has its centre $C$ located in the third quadrant at a distance of $\frac{7}{\sqrt{2}}$ units from $AB$, then the point $P(1, -2)$

Which of the following statements are correct? i. $ [\text{Mn(CN)}_6]^{3-},\ [\text{MnCl}_6]^{3-} $ both are paramagnetic
ii. Both $ [\text{NiCl}_4]^{2-},\ [\text{Ni(CO)}_4] $ are tetrahedral and diamagnetic
iii. $ [\text{Co(NH}_3)_6]^{3+} $ is inner orbital complex, but $ [\text{Ni(NH}_3)_6]^{2+} $ is outer orbital complex

Deficiency of which vitamin causes convulsions?

If $ f(x) = \begin{vmatrix} 1 & 6 + x & 36 + x^2 \\ 0 & x - 3 & 3x^2 - 27 \\ 0 & 2x - 4 & 8x^2 - 32 \end{vmatrix} $, then $ \lim\limits_{x \to 1} \frac{f(x)}{f(-x)} $ is:

Which one of the following is a homogeneous differential equation?

At any instant $ t $, the vertical distance $ Y $ and horizontal distance $ X $ of a projectile are given by \[ 2Y = 6t - g t^2 \quad \text{and} \quad X = 4t, \] The initial velocity of the projectile is: