List of practice Questions

A uniform thin metal plate of mass \(10 \, \text{kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of the center of mass of the plate is \(\frac{n}{9}\). The value of \(n\) is ______.

Critical angle of incidence for a pair of optical media is \(45^\circ\).
The refractive indices of the first and second media are in the ratio:

Correct Bernoulli's equation is (symbols have their usual meaning):

The output Y of following circuit for given inputs is :

A clock has a 75 cm long second hand and a 60 cm long minute hand, respectively. In 30 minutes duration, the tip of the second hand will travel \(x\) distance more than the tip of the minute hand. The value of \(x\) in meters is nearly (Take \(\pi = 3.14\)):

Paramagnetic substances:
A. align themselves along the directions of external magnetic field.
B. attract strongly towards external magnetic field.
C. has susceptibility little more than zero.
D. move from a region of strong magnetic field to weak magnetic field.
Choose the most appropriate answer from the options given below:

The binding energy of a certain nucleus is \(18 \times 10^8 \, \text{J}\). How much is the difference between the total mass of all the nucleons and the nuclear mass of the given nucleus?

In an expression \(a \times 10^b\):

Three bodies A, B and C have equal kinetic energies and their masses are 400 g, 1.2 kg and 1.6 kg respectively. The ratio of their linear momenta is :

The value of \[ \lim_{x \to 0} 2 \left( \frac{1 - \cos x \sqrt{\cos 2x} \, \sqrt[3]{\cos 3x} \ldots \sqrt[10]{\cos 10x}}{x^2} \right) \] is _____.

Let the area of the region enclosed by the curve $y = \min\{\sin x, \cos x\}$ and the x-axis between $x = -\pi$ to $x = \pi$ be $A$. Then $A^2$ is equal to _____.

If the range of $f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to ____.

Let the positive integers be written in the form :

If the $k^\text{th}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number $5310$ will be, is _____

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7\bar{X} + 4\bar{Y}$ is equal to _____.

For the function $f(x) = (\cos x) - x + 1, x \in \mathbb{R}$, find the correct relationship between the following two statements
(S1) $f(x) = 0$ for only one value of x is $[0, \pi]$.
(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

If the set $R = {(a, b) : a + 5b = 42, a, b \in \mathbb{N}}$ has $m$ elements and $\sum_{n=1}^m (1 + i^n) = x + iy$, where $i = \sqrt{-1}$, then the value of $m + x + y$ is:

Let $[t]$ be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f: A \to \mathbb{Z}$ be the function $f(x) = \left[ \log_2 \left( x^2 + \left[ \frac{x^3}{5} \right] \right) \right]$. The number of one-to-one functions from A to the range of f is:

The equations of two sides AB and AC of a triangle ABC are $4x + y = 14$ and $3x - 2y = 5$, respectively. The point $\left(2, -\frac{4}{3}\right)$ divides the third side BC internally in the ratio 2 : 1. The equation of the side BC is:

Let $f(x) = 4\cos^3 x + 3\sqrt{3} \cos^2 x - 10$. The number of points of local maxima of $f$ in interval $(0, 2\pi)$ is:

The number of critical points of the function $f(x) = (x - 2)^{2/3}(2x + 1)$ is:

Let the circles $C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$ and $C_2 : (x - 8)^2 + \left( y - \frac{15}{2} \right)^2 = r_2^2$ touch each other externally at the point $(6, 6)$. If the point $(6, 6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2 : 1$, then $(\alpha + \beta) + 4\left(r_1^2 + r_2^2\right)$ equals _____.

The sum of all the solutions of the equation \[(8)^{2x} - 16 \cdot (8)^x + 48 = 0\]is:

The value of $k \in \mathbb{N}$ for which the integral \[ I_n = \int_0^1 (1 - x^k)^n \, dx, \, n \in \mathbb{N}, \] satisfies $147 \, I_{20} = 148 \, I_{21}$ is:

Wavenumber for a radiation having $5800 \, \mathring{A}$ wavelength is $x \times 10 \, \text{cm}^{-1}$. The value of $x$ is _________