List of practice Questions

If \( A = \left[\begin{array}{ccc} 83 & 74 & 41 \\ 93 & 96 & 31 \\ 24 & 15 & 79 \end{array} \right] \), then \(\text{det} (A - A^T) = \): 
Consider the linear system of non homogeneous equation in three variables AX = B. $\rho(A)$ and $\rho([A:B])$ are roots of the equation $x^2+bx+c=0$. AX = B has infinite number of solutions, then the option containing two possible pairs of values of (b,c) is (Note: $\rho$ denotes rank.)
Which one of the following is an autoimmune disease?
Mitochondria known as the "powerhouse" of the cell, generates ATP via oxidative phosphorylation complexes. Which among the following is known as complex I?

If the Laplace transform of $ \int_0^t \frac{((1+2t)^2-1)e^{3t}}{t} dt = \frac{A}{S-3} + \frac{B}{(S-3)^2} + \frac{C}{(S-3)} $ then $3(A+B+C)=$

Consider the following problem of vibration of a string. A tightly stretched string with fixed end points $x=0$ and $x=L$ is initially in a position given by $y(x,0) = f(x)$. It is released from this rest position and allowed to vibrate. The mathematical representation of this problem which depicts the displacement of the string $y(x,t)$ at different times for different values of $x, 0 \le x \le L$ is

If $x = f(t)$ changes the interval $ \alpha \le t \le \alpha + 2C $ to $ \beta \le x \le \beta + 2\pi $, then $\frac{f(t)}{\beta} = $

Which of the following is a convergent series?
A and B are two nonsingular matrices. If the characteristic equation of A is $ a_0\lambda^3 + a_1\lambda^2 + a_2\lambda + a_3 = 0 $ and characteristic equation of $ B^{-1}AB $ is $ b_0\lambda^3 + b_1\lambda^2 + b_2\lambda + b_3 = 0 $, then
If $ x=\alpha, y=\beta, z=\gamma $ is a positive integral solution of the equation $ x+y+z=12 $, the probability that it is such that $ \alpha<\beta<\gamma $ is
Which of the following pairs of numbers is possible to be the regression coefficients for a data of two variables
The order of convergence of the Newton's Raphson's method is
Which of the following numerical method is a multistep method to solve IVP or BVP's (IVP: Initial Value Problem, BVP: Boundary Value Problem. Multistep methods are typically for IVPs of ODEs.)
In a given semiconductor, the ratio of the number density of electron to number density of hole is 2 : 1. If $ \frac{1}{7} $th of the total current is due to the hole and the remaining is due to the electrons, the ratio of the drift velocity of holes to the drift velocity of electrons is :

If the inverse point of the point \( (-1, 1) \) with respect to the circle \( x^2 + y^2 - 2x + 2y - 1 = 0 \) is \( (p, q) \), then \( p^2 + q^2 = \) 

A transformer which steps down 330 V to 33 V is to operate a device having impedance 110 $\Omega$. The current drawn by the primary coil of the transformer is:

The density of \(\beta\)-Fe is 7.6 g/cm\(^3\). It crystallizes in a cubic lattice with \( a = 290 \) pm. 
What is the value of \( Z \)? (\( Fe = 56 \) g/mol, \( N_A = 6.022 \times 10^{23} \) mol\(^{-1}\))

The maximum value of \[ 12 \sin x - 5 \cos x + 3 \] is:
If \(x^2 + 5ax + 6 = 0\) and \(x^2 + 3ax + 2 = 0\) have a common root, then that common root is:

If \[ A = \begin{bmatrix} 1 & 0 & 2\\ 2 & 1 & 3 \\3 & 2 & 4 \end{bmatrix}, \] then evaluate \( A^2 - 5A + 6I \)=

Find the equation of a line passing through the intersection of \( 3x + y - 4 = 0 \) and \( x - y = 0 \), and making a \( 45^\circ \) angle with \( x - 3y + 5 = 0 \).
A spring of spring constant \( 200 \, {N/m} \) is initially stretched by \( 10 \) cm from the unstretched position. The work to be done to stretch the spring further by another \( 10 \) cm is:
In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):
If \( l, m, n \) are the direction cosines of a line that is perpendicular to the lines having the direction ratios \( (1,2,-1) \) and \( (-2,1,1) \), then \( (l+m+n)^2 \) =
If \( y = f(x) \) is a thrice differentiable function and a bijection, then \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = ? \]