Prove that the height of the cylinder of maximum volume inscribed in a sphere of radius \( R \) is \( \frac{2R}{\sqrt{3}} \).
Solve the differential equation \[ (x + y) \, dy + (x - y) \, dx = 0, \quad \text{if} \quad y = 1 \text{ when } x = 1. \]
Show that the vectors \( 2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}, 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle.
There are 10 black and 5 white balls in a bag. Two balls are taken out, one after another, and the first ball is not placed back before the second is taken out. Assume that the drawing of each ball from the bag is equally likely. What is the probability that both the balls drawn are black?
Find the angle between the lines \[ \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{-5} \quad \text{and} \quad \frac{x+3}{-3} = \frac{y-1}{2} = \frac{z-5}{5}. \]
Solve the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \quad \text{where} \quad x \neq 0. \]
Find the shortest distance between the lines \[ \mathbf{r}_1 = (i + 2j + k) + \lambda(i - j + k) \quad \text{and} \quad \mathbf{r}_2 = (2i - j - k) + \mu(2i + j + 2k). \]
If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \) for \( -1<x<1 \), then prove that \[ \frac{dy}{dx} = -\frac{1}{(1 + x^2)^2}. \]
Prove that the function \( f(x) = |x| \) is continuous at \( x = 0 \) but not differentiable.
A die is thrown twice. Let us represent the event 'obtaining an odd number on the first throw' by A and the event 'obtaining an odd number on the second throw' by B. Test the independency of the events A and B.
Solve the differential equation \[ (x - y) \frac{dy}{dx} = x + 2y. \]
Find the intervals in which the function \( f(x) = x^2 - 4x + 6 \) is
Find the value of \[ \int e^x \left( \tan^{-1} x + \frac{1}{1 + x^2} \right) dx. \]
Prove that the number of equivalence relations in the set \( \{1, 2, 3\} \) including \( \{(1, 2)\} \) and \( \{(2, 1)\} \) is 2.
If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \]
The given events \( A \) and \( B \) are such that \( P(A) = \frac{1{4} \), \( P(B) = \frac{1}{2} \) and \( P(A \cap B) = \frac{1}{8} \); then find \( P(A \text{ not} \cap B \text{ not}) \).
Integrate \[ \int \frac{\sin(\tan^{-1} x)}{1 + x^2} \, dx. \]
Solve: \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \]
If \( e^y(x + 1) = 1 \), show that \[ \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2. \]
A relation \( R = \{(a, b) : a = b - 2, b \geq 6 \} \) is defined on the set \( \mathbb{N} \). Then the correct answer will be: