List of practice Questions

Given that: $M = \int_{0}^{\infty} \frac{\log t}{1 + t^3} \, dt$, and $N = \int_{-\infty}^{\infty} \frac{e^{2t} t}{1 + e^{3t}} \, dt$. Then, the relation between $M$ and $N$ is:

A closed-loop system has the characteristic equation given by: $ s^3 + k s^2 + (k+2) s + 3 = 0 $. 
For the system to be stable, the value of $ k $ is:

A digital filter with impulse response $ h[n] = 2^n u[n] $ will have a transfer function with a region of convergence.

$ \lim_{x \to -\frac{3}{2}} \frac{(4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}} $

A satellite attitude control system, as shown below, has a plant with transfer function:  $ G(s) = \frac{1}{s^2}, $ cascaded with a compensator:  $ C(s) = \frac{K(s + \alpha)}{s + 4}, $ where $ K $ and $ \alpha $ are positive real constants. In order for the closed-loop system to have poles at $ -1 \pm j\sqrt{3} $, the value of $ \alpha $ must be $\_\_\_\_\_$.

The equation of the circle passing through the origin and cutting the circles $x^2 + y^2 + 6x - 15 = 0$ and $x^2 + y^2 - 8y - 10 = 0$ orthogonally is:

The unit vector perpendicular to each of the vectors $ (2\hat{i} - \hat{j} + \hat{k}) \text{ and } (3\hat{i} + 4\hat{j}) \text{ is} $
The plane $ xoz $ divides the join of $ (1, -1, 5) $ and $ (2, 3, 5) $ in the ratio $ \lambda : 1 $, then $ \lambda $ is
Angle between the line $ \vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda (-\hat{i} + \hat{j} + \hat{k}) $ and the plane $ \vec{r} \cdot (3\hat{i} + 2\hat{j} - \hat{k}) = 4 $ is
The value of $ k $ so that $ \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} = \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} $ may be perpendicular is given by
If A and B are two events and $ P(A \cup B) = \frac{5}{6} $, $ P(A \cap B) = \frac{1}{3} $, $ P(\overline{B}) = \frac{1}{2} $, then A and B are
The order of the differential equation $ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = \frac{d^2y}{dx^2} $ is
The integral $ \int \frac{x e^x}{(1+x)^2} dx $ is equal to
If $ \tan x = \frac{m}{m+1} $, $ \tan \beta = \frac{1}{2m+1} $, then $ (\alpha + \beta) $ is equal to
The value of $ \tan^{-1} \left( \frac{x}{y} \right) - \tan^{-1} \left( \frac{x - y}{x + y} \right) \text{ is} $
The general solution of $ x $ satisfying the equation $ \sqrt{3} \sin x + \cos x = \sqrt{3} $, is given by
If $ \sin(\pi \cos \theta) = \cos(\pi \sin \theta) $, then the value of $ \cos \left( \theta + \frac{\pi}{4} \right) $

Let $f(x) = \frac{1 - \cos{P x}}{x \sin{x}}$ when $ x \neq 0 $ and $ f(0) = \frac{1}{2} $. If $ f $ is continuous at $ x = 0 $, then $ P $ is equal to

There are 15 points in a plane, no three of which are in a straight line, except 6, all of which are in a straight line. The number of straight lines which can be drawn by joining them is
If log2 = 0.30103 and log3 = 0.4771, find the number of digits in (648)\(^5\).
It is postulated that huge deposits of NaCl (rock salt) and CaCO3 (chalk and marble) are sites of erstwhile oceans, where the salts had been concentrated through weathering by rain and wind and leaching by rivers. Select the correct explanatory statement in this context.
Which of the following solutions, when mixed, will not form a buffer solution?
Asim got thrice as many sums wrong as he got right. If he attempted 60 sums in all, how many sums did he solve correctly?
Select the correct order of steps in which a working substance (gas) goes through in a refrigerator.
The area of the triangle formed by the lines represented by \( 3x + y + 15 = 0 \) and \( 3x^2 + 12xy - 13y^2 = 0 \) is: