List of practice Questions

Let f(x) = -55x (x<-5), 2x³ - 3x² - 120x (-5 ≤ x ≤ 4), 2x³ - 3x² - 36x - 336 (x>4). Let A = {x ∈ ℝ : f is increasing}. Then A is equal to :
A possible value of \( \tan\!\left(\frac{1}{4}\sin^{-1}\!\left(\frac{\sqrt{63}}{8}\right)\right) \) is :
The angle of elevation of a jet plane from a point A on the ground is 60°. After 20 s at speed 432 km/h, the angle changes to 30°. If the jet plane is at constant height, then its height is :
The vector equation of the plane passing through the intersection of r · (i + j + k) = 1 and r · (i - 2j) = -2, and the point (1, 0, 2) is :
For p and q, consider: (a) (~ q ∧ (p → q)) → ~ p, (b) ((p ∨ q) ∧ ~ p) → q. Which is correct ?
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :
The area of the region : R = {(x, y) : 5x² ≤ y ≤ 2x² + 9} is :
If the variance of 10 natural numbers 1, 1, 1, ......., 1, k is less than 10, then the maximum possible value of k is ______
Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (-5, 0). If the locus of the point P is a circle of radius r, then 4r² is equal to ________
If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle \( (x-2)^2 + (y-3)^2 = 25 \) at the point \( (5, 7) \) is \( A \), then \( 24A \) is equal to _______
Let i = √-1. If $\frac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \frac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$, and $n = \lfloor |k| \rfloor$. Then $\sum_{j=0}^{n+5} (j + 5)^2 - \sum_{j=0}^{n+5} (j + 5)$ is equal to __________.
The students S1, S2, ......., S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is _________
The maximum value of k for which the sum $\sum_{i=0}^k \binom{10}{i} \binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i} \binom{13}{k+1-i}$ exists, is equal to __________.
If a + α = 1, b + β = 2 and a f(x) + α f(1/x) = b x + β / x, then [f(x) + f(1/x)] / [x + 1/x] is ________
If the shortest distance between x - λ = 2y - 1 = -2z and x = y + 2λ = z - λ is √7 / 2√2, then |λ| is _______
If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots .+{ }^{n} C_{2}\right)$ is:
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is \(0.9\) and that of the second unit is \(0.8\). The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is \(p\), then \(98p\) is equal to \dots\dots.

For the closed-loop system with \(G_p(s) = \frac{14.4}{s(1 + 0.1s)}\) and \(G_c(s) = 1\), the unit-step response shows damped oscillations. The damped natural frequency is \(\underline{\hspace{2cm}}\) rad/s. (Round off to 2 decimal places.)

In the given figure, plant \(G_p(s)=\dfrac{2.2}{(1+0.1s)(1+0.4s)(1+1.2s)}\) and compensator \(G_c(s)=K \left\{ \dfrac{1+T_1 s}{1+T_2 s} \right\}\). The disturbance input is \(D(s)\). The disturbance is a unit step, and the steady-state error must not exceed 0.1 unit. Find the minimum value of \(K\). (Round off to 2 decimal places.)

A full-wave rectified sinusoid is clipped at \(\omega t = \frac{\pi}{4}\) and \(\frac{3\pi}{4}\). The ratio of the RMS value of the full-wave rectified waveform to the RMS value of the clipped waveform is \(\underline{\hspace{2cm}}\). (Round off to 2 decimal places.) 

The state space representation of a first-order system is \[ \dot{X} = -X + U, Y = X \] where \(X\) is the state variable, \(u\) is the control input and \(y\) is the controlled output. Let \(u = -KX\) be the control law. To place a closed-loop pole at \(-2\), the value of \(K\) is \(\underline{\hspace{1cm}}\). 
 

An air-core RF transformer has a primary and secondary winding. At 100 kHz, the primary sees 7.3 V\(_{p-p}\) and the secondary sees 5.0 V\(_{p-p}\). The load is 22\(\Omega\). The mutual inductance \(M\) is \(\underline{\hspace{1cm}}\) \(\mu H\). (Round off to 2 decimal places.) 

A 100 Hz square wave (0–5 V) is applied to a CR high-pass filter. The resistor voltage waveform has 6.2 V peak-to-peak. If \(R = 820\Omega\), the value of \(C\) is \(\underline{\hspace{1cm}}\) \(\mu F\). (Round off to 2 decimal places.) 

A CMOS Schmitt-trigger inverter has thresholds of 1.6 V (low-to-high) and 2.4 V (high-to-low). The capacitor is 47 nF and the resistor is 10 k\(\Omega\). The frequency of the oscillator is \(\underline{\hspace{2cm}}\) Hz. (Round off to 2 decimal places.) 

A boost converter operates at 25 kHz with a duty cycle 0.6. Input is 15 V, load is 10 \(\Omega\). Assuming ideal components, compute the equivalent input resistance \(R_{in}\) seen by the source. (Round off to 2 decimal places.)