List of practice Questions

A uniform rod $A B$ of length $\ell$ and mass $m$ is free to rotate about point $A .$ The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $\frac{m \ell^{2}}{3}$, the initial angular acceleration of the rod will be:-

$\int\limits_{0}^{8}| x -5| d x$ is equal to

If $B_{H}=\frac{1}{\sqrt{3}} B_{V}$, find angle of dip. (where symbols have their usual meanings)

If the normal to the curve $y = f(x)$ at $(3,4)$ makes an angle $\frac{3 \pi}{4}$ with the positive x-axis, then $f'(3)$ is equal to

The equation of a directrix of the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1 $ is

what is the Q-value of the reaction $p+{ }^{7} Li \rightarrow{ }^{4} He +{ }^{4} He$ The atomic masses of ${ }^{1} H ,{ }^{4} He$ and ${ }^{7} Li$ are $1.007825\, u , 4.002603\,u$ and $7.016004\, u$ respectively

f(x) is cubic polynomial with f(2) = 18 and f(1) = -1. Also f(x) has local maxima at x = -1 and f '(x) has local minima at x = 0, then

Name the complex Ni(\(PF_6\))\(_2\):

If \( x = -9 \) is a root of \( \begin{pmatrix} 2 & 3 \\ 7 & 6 \end{pmatrix} \times \begin{pmatrix} x \end{pmatrix} = 0 \), then other two roots are:

Let \( A = \begin{pmatrix} 1 & 3 & 2 \\ 4 & 2 & 5 \\ 7 & -t & -6 \end{pmatrix} \), then the values of \( t \) for which inverse of \( A \) does not exist are:

If \( f'(x) = \frac{x}{\sqrt{1 + x^2}} \) and \( f(0) = 0 \), then \( f(x) = \):

The probability density \( f(x) \) of a continuous random variable is given by \( f(x) = K e^{-|x| \) for \( -\infty<x<\infty \). Then the value of \( K \) is:}

Find the number of ways in which five large books, four medium-size books, and three small books can be placed on a shelf so that all books of the same size are together.

Consider the set \( Q \) of rational numbers. Let \( * \) be the operation \( a * b = a + b - ab \). The identity element under \( * \) is:

The statement \( p \to q \) is equivalent to:

In rolling two fair dice, what is the probability of obtaining a sum greater than 3 but not exceeding 6?

Team A has probability \( \frac{2}{3} \) of winning whenever it plays. Suppose A plays four games. What is the probability that A wins more than half of its games?

An unprepared student takes five questions of true-false type quiz and guesses every answer. What is the probability that the student will pass the quiz if at least four correct answers is the passing grade?

What is the area of a loop of the curve \( r = a \sin 30^\circ \)?

The value of the integral \( \int_1^4 \sqrt{t} \, dt \) is:

The differential equation that represents all parabolas each of which has a latus rectum \( 4a \) and whose axes are parallel to the x-axis is:

The solution of \( x \sec \left( \frac{x}{y} \right) - y \, dx + x \, dy = 0 \) is:

The particular integral of \( \frac{d^2 y}{dx^2} + 2y = x^2 \) is:

The solution of \( D^2 + 16y = \cos 4x \) is:

Determine which one of the following relations on \( X = \{1, 2, 3, 4\} \) is not transitive.