List of practice Questions

If \( \sqrt{3} \cos \theta + \sin \theta > 0 \), then the range of \( \theta \) is:
\( \csc 48^\circ + \csc 96^\circ + \csc 192^\circ + \csc 384^\circ = \)
If \( \frac{3x+1}{(x-1)(x^2+2)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+2} \), then \( 5(A-B) = \)?
\( 1 + \frac{4}{15} + \frac{4 \cdot 10}{15 \cdot 30} + \frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45} + \cdots \infty \)
If \(C_0, C_1, C_2, \dots, C_n\) are the binomial coefficients in the expansion of \((1 + x)^n\), then \((C_0 + C_1) - (C_2 + C_3) + (C_4 + C_5) - (C_6 + C_7) + \dots = \)
The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is:
The number of ways of forming the ordered pairs (p, q) such that p>q by choosing p and q from the first 50 natural numbers is:
If all possible 4-digit numbers are formed by choosing 4 different digits from the given digits $ 1, 2, 3, 5, 8 $, then the sum of all such 4-digit numbers is:
The cubic equation whose roots are the squares of the roots of the equation \( x^3 - 2x^2 + 3x - 4 = 0 \) is
If the roots of the equation \(x^2 + 2ax + b = 0\) are real, distinct and differ utmost by \(2m\), then b lies in the interval
Let \((a-3)x^2 + 12x + (a+6) > 0, \forall x \in \mathbb{R} \text{ and } a \in (t, \infty)\). If \(\alpha\) is the least positive integral value of \(a\), then the roots of \((\alpha-3)x^2 + 12x + (\alpha+2) = 0\) are:
The sum of the squares of the imaginary roots of the equation $ z^8 - 20z^4 + 64 = 0 $ is:
If the least positive integer \( n \) satisfying the equation \(\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n = -1\) is \( p \) and the least positive integer \( m \) satisfying the equation \(\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^m = \text{cis}\left(\frac{2\pi}{3}\right)\) is \( q \), then \(\sqrt{p^2 + q^2}\) is equal to:
If \( z \) is a complex number such that \( \frac{z-1}{z-i} \) is purely imaginary and the locus of \( z \) represents a circle with center \( (\alpha, \beta) \) and radius \( r \), then the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) is:
The rank of the matrix \( \begin{bmatrix} 2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4 \end{bmatrix} \) is
If \( \alpha \) is a real root of the equation \( x^3 + 6x^2 + 5x - 42 = 0 \), then the determinant of the matrix \[ \begin{bmatrix} \alpha - 1 & \alpha + 1 & \alpha + 2 \\ \alpha - 2 & \alpha + 3 & \alpha - 3 \\ \alpha + 4 & \alpha - 4 & \alpha + 5 \end{bmatrix} \] is Options:
If \( P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( \det(A) = 4 \), then the value of \( \alpha \) is:
If \( 11^{12} - 11^2 = k(5 \times 10^9 + 6 \times 10^9 + 33 \times 10^8 + 110 \times 10^7 + \ldots + 33) \), then find the value of \( k \).
If \( f(x) = (x+1)^2 - 1, x \ge -1 \), then \( \{x \mid f(x) = f^{-1}(x)\} \) is:
Let \( [t] \) denote the greatest integer function and \( [t - m] = [t] - m \) when \( m \in \mathbb{Z} \). If \( k = 2[2x - 1] - 1 \) and \( 3[2x - 2] + 1 = 2[2x - 1] - 1 \), then the range of \( f(x) = [k + 5x] \) is:
In Young's double-slit experiment, if the distance between the slits is increased to 3 times its initial distance, then the ratio of initial and final fringe widths is.
In a Carnot engine, if the absolute temperature of the source is \( 25% \) more than the absolute temperature of the sink, then the efficiency of the engine is
If a heat engine and a refrigerator are working between the same two temperatures \(T_1\) and \(T_2\) (\(T_1>T_2\)), then the ratio of efficiency of heat engine to coefficient of performance of refrigerator is
A force \( F \) is applied on a body of mass \( m \) so that the body starts moving from rest. The power delivered by the force at time \( t \) is proportional to:
The equations for the displacements of two particles in simple harmonic motion are \( y_1 = 0.1\sin\left(100\pi t + \frac{\pi}{3}\right) \) and \( y_2 = 0.1\cos(\pi t) \) respectively. The phase difference between the velocities of the two particles at a time \( t = 0 \) is