List of practice Questions

Consider two statements:
Statement 1: $ \lim_{x \to 0} \frac{\tan^{-1} x + \ln \left( \frac{1+x}{1-x} \right) - 2x}{x^5} = \frac{2}{5} $
Statement 2: $ \lim_{x \to 1} x \left( \frac{2}{1-x} \right) = e^2 \; \text{and can be solved by the method} \lim_{x \to 1} \frac{f(x)}{g(x) - 1} $

Find the area of the region defined by the conditions: $ \left\{ (x, y): 0 \leq y \leq \sqrt{9x}, y^2 \geq 3 - 6x \right\} \text{(in square units)} $

Let $ f $ be defined as $ R \setminus \{0\} \to R $ such that $ f(x) = \frac{x}{3} + \frac{3}{x} + 3 $ If $ f(x) $ is strictly increasing in $ (-\infty, \alpha_1) \cup (\alpha_2, \infty) $ and strictly decreasing in $ (\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5) $, then $ \sum_{i=1}^5 (\alpha_i)^2 $ is equal to:

Let $ A = \{0, 1, 2, 3, 4, 5, 6\} $ and $ R_1 = \{(x, y): \max(x, y) \in \{3, 4 \}$. Consider the two statements:
Statement 1: Total number of elements in $ R_1 $ is 18.
Statement 2: $ R $ is symmetric but not reflexive and transitive.

Let $ I_1 = \int_{\frac{1}{2}}^{1} 2x \cdot f(2x(1 - 2x)) \, dx $
and $ I_2 = \int_{-1}^{1} f(x(1 - x)) \, dx \; \text{then} \frac{I_2}{I_1} \text{ equals to:} $

For a nucleus of mass number $A$ and radius $R$, mass density $\rho$. Then choose the correct option.

The sum of squares of roots of $ |x - 2|^2 - |x - 2| - 2 = 0 $ and $ x^2 - 2|x - 3| - 5 = 0 $ equals to:

A circle touches the parabola $ y^2 = 9x $ at $ (4, 6) $ and the positive x-axis. Find the radius of the circle.

If $ A = \begin{pmatrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2p & 8 + 3p + 2q \\ 6 & 12 + 3p & 20 + 6p + 3q \end{pmatrix} $, then the value of $ \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n $, then $ m + n $ is equal to:

If $ f(x) = x - 1 $ and $ g(x) = e^x $, and the differential equation is: $ \frac{dy}{dx} = \left( e^{-2\sqrt{x}} g(f(f(f(x)))) - \frac{y}{\sqrt{x}} \right) $ with the initial condition $ y(0) = 0 $, then $ y(1) $ equals:

The value of the following expression: $ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 2} + 1}{\tan 2} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2 2} - 1}{\tan 2} \right) $

Let $ \vec{a} = \hat{i} + 4\hat{j} + 3\hat{k} $ and $ \vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k} $, and $ \vec{c} $ is a vector perpendicular to $ \vec{a} $ and lies in the plane of $ \vec{a} $, $ \vec{b} $. Then $ \vec{c} $ is equal to:

Given: $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90}, $ $ \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \cdots = \alpha, $ $ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \cdots = \beta $ Find the value of $ \frac{\alpha}{\beta} $.

There are 12 points in a plane in which 5 are collinear such that no three of them are in a straight line. Then the number of triangles that can be formed from any 3 vertices from 12 points.

Probability of event A is 0.7 and event B is 0.4, and $ P(A \cap B^c) = 0.5 $, then the value of $ P(B | A \cup B^c) $ is equal to:

Evaluate the integral: $ \int_{-1}^{3/2} | \pi^2 x \sin(\pi x) | \, dx $

On combustion of 0.21 g of an organic compound containing C, H, and O, it gave 0.127 g of H$_2$O and 0.307 g of CO$_2$. The percentage of H and O in the given organic compound respectively are:

Match List-I with List-II and select the correct option.