List of practice Questions

Evaluate the limit: $$ \lim_{x \to 1} \frac{x + x^2 + x^3 + \dots + x^n - n}{x - 1}. $$ 
If $$ y = (1 + a + a^2 + \dots)e^{nx} $$ then the relative error in $y$ is:
If the extremities of the latus recta having positive ordinate of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b) $$ lie on the parabola $$ x^2 + 2ay - 4 = 0, $$ then the points $(a, b)$ lie on the curve:
The Laplace transform of a signal $X(t)$ is $$ X(s) = \frac{4s + 1}{s^2 + 6s + 3}. $$ The initial value $X(0)$ is:
If $$ \int \frac{1}{x^4 + 8x^2 + 9} \, dx = \frac{1}{k} \left[ \frac{1}{\sqrt{14}} \tan^{-1} (f(x)) - \frac{1}{\sqrt{2}} \tan^{-1} (g(x)) \right] + c, $$ then $$ \frac{k}{\sqrt{2}} + f(\sqrt{3}) + g(1) = $$ 
The angle between the lines $$ \frac{x-1}{2} = \frac{2y+3}{4} = \frac{z+5}{-2} \quad \text{and} \quad \frac{x-3}{4} = \frac{y+1}{-4} = \frac{z+3}{-4} $$ is equal to 
The domain of the real-valued function $$ f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) $$ is:
If $$ f(x) = \min \{ x, x^2 \} $$ Which of the following is true? 
Identify the correct statements from the following:
I. For spontaneous reaction, $ \Delta S_{\text{total}} < 0 $ and $ \Delta H > 0 $
II. Pressure is an intensive property
III. Bomb calorimeter works at constant volume
The number of multipliers and delay elements required in the direct form II realization of $$ H(z) = \frac{1 + 0.5z^{-1} - 2z^{-2}}{1 + z^{-1} - 2z^{-2}} $$ 

If the curves $$ 2x^2 + ky^2 = 30 \quad \text{and} \quad 3y^2 = 28x $$ cut each other orthogonally, then \( k = \) 

The value of $ \frac{1}{1 + p^{(y-z)} + p^{(x-z)}} + \frac{1}{1 + p^{(x-y)} + p^{(z-y)}} + \frac{1}{1 + p^{(y-x)} + p^{(z-x)}}, $ is:

If the function

$ f(x) = \begin{cases} \frac{\cos ax - \cos 9x}{x^2}, & \text{if } x \neq 0 \\ 16, & \text{if } x = 0 \end{cases} $

is continuous at $ x = 0 $, then $ a = ? $

Consider the discrete-time systems $ T_1 $ and $ T_2 $ defined as follows: 
$ [T_1x][n] = x[0] + x[1] + \dots + x[n], $ 
$ [T_2x][n] = x[0] + \frac{1}{2}x[1] + \dots + \frac{1}{2^n}x[n]. $ 
Which of the following statements is true?

Choose the minimum number of op-amps required to implement the given expression. $ V_o = \left[ 1 + \frac{R_2}{R_1} \right] V_1 - \frac{R_2}{R_1} V_2 $

At $ T $ (K), the following data was obtained for the reaction: $ S_2O_8^{2-} + 3 I^- \rightarrow 2 SO_4^{2-} + I_3^- $.

From the data, the rate constant of the reaction (in $ M^{-1} s^{-1} $) is: 

If the axes are rotated through an angle \( \alpha \), then the number of values of \( \alpha \) such that the transformed equation of \( x^2 + y^2 + 2x + 2y - 5 = 0 \) contains no linear terms is:
Ategmic, Unitegmic, and Bitegmic ovules are present serially in:
The mass percentage of glucose in acetonitrile when 6 g of glucose is dissolved in 294 g of acetonitrile is:
A man can cover a distance in 1 hour 24 minutes by covering 2/3 of the distance at 4 km/h and the rest at 5 km/h. The total distance is
The total displacement of the object that exhibits the following motion is
A known positive charge is located at point P as shown above, between two unknown charges, Q1 and Q2. P is closer to Q2 than Q1. If the net electric force acting on the charge at P is zero, it may correctly be concluded that:
A square closed loop of area A, lying in the horizontal plane, is moving horizontally with velocity \( v \) in a uniform vertical magnetic field \( B \). Which one of the following statements is FALSE?
A particle moving from a fixed point on a straight line travels a distance \(S\) meters in \(t\) sec. If \(S = t^3 - t^2 + t + 3\), then the distance (in mts) travelled by the particle when it comes to rest is:
If \( P_0 \) and \( P_S \) are the vapour pressures of the solvent and solution respectively and \( X_0 \) and \( X_S \) are mole fractions of solvent and solute respectively, then