List of practice Questions

Suppose that a book of 600 pages contains 40 print mistakes. Assume that these errors are randomly distributed throughout the book and the number of errors per page follows a Poisson distribution. The probability that all the 10 pages selected at random have no print mistakes is:

The equation of the circle passing through the origin and cutting orthogonally the circles $ x^2 + y^2 + 6x - 15 = 0 $ and $ x^2 + y^2 - 8y - 10 = 0 $ is:

If $ y = \tan(3\tan^{-1} x) $, then: \[ (1 - 3x^2)\frac{d^2y}{dx^2} - 12x \frac{dy}{dx} = \]

A large tank open to the atmosphere at the top and filled with water develops a small hole in the side at a point 20 m below the water level. If the rate of flow of water from the hole is $3 \times 10^{-3}\,\text{m}^3/\text{min}$, then the area of the hole is: (Take $ g = 10\,\text{m/s}^2 $)

In a U-tube with radii 2 mm and 4 mm, surface tension = 0.03 N/m, density = 1500 kg/m$^3$, contact angle = 0. Height difference in limbs is:

The pair of elements that form both oxides and nitrides, when burnt in air are

Identify the correct statements about the anomalous behavior of Boron.

Which of the following solids is not a molecular solid?

In an exam:
- Three subjects have maximum marks = $ n $ each
- Fourth subject has max marks = $ 2n $
- Find the number of ways to score total $ 3n $ marks.

Find the locus of midpoints of points of intersection of the line $$ x \cos \theta + y \sin \theta = 1 $$ with the coordinate axes.

Find the locus of the point of intersection of tangents drawn at the ends of a **normal chord** of the hyperbola: $$ x^2 - y^2 = a^2 $$

The wavelength associated with the electron moving in the first orbit of hydrogen atom with velocity $ 2.2 \times 10^6\, \text{m/s} $ is (in nm):
(Given: $ m_e = 9.0 \times 10^{-31}\, \text{kg},\ h = 6.6 \times 10^{-34}\, \text{Js} $)

In which of the following, oxides of three elements X, Y, and Z are arranged in increasing order of acidic nature? Their electronic configurations are:
X: $[Ne] 3s^2 3p^1$, \quad Y: $[Ne] 3s^2 3p^3$, \quad Z: $[Ne] 3s^2$

Let z and w be two distinct non-zero complex numbers if $|z|^2w - |w|^2z = z - w$, then

$\tanh(\log x) = $ (Assuming $\log x$ is natural logarithm $\ln x$)

The standard deviation of first 10 multiples of 4 is

From a well shuffled pack of 52 cards, two cards are drawn at random. Then, the probability of both the cards being kings is

$ \lim_{x \to 0} x^3 (\sqrt{x^2+\sqrt{x^4+1}} - \sqrt{2}x) = $

If the general solution of $ \frac{dy}{dx} = \frac{-y^2}{xy-y^2-x^2} $ is $ \tan^{-1}\left(\frac{x}{y}\right) = f(y) + C $, then $ f(e^3) = $

Let the mirror image of the point (a, b, c) with respect to the plane 3x – 4y + 12z + 19 = 0 be (a – 6, β, γ). If a + b + c = 5, then 7β – 9γ is equal to __________.

Let
$\vec{a}=\hat{i} - 2\hat{j} + 3\hat{k}, \vec{b}=\hat{i} - \hat{j} + \hat{k} $ and $\vec{c}$
be a vector such that
$\vec{a} + (\vec{b}×\vec{c}) = \vec{0}$ and $\vec{b}.\vec{c} = 5.$
Then the value of 3($\vec{c}.\vec{a}$) is equal to