List of practice Questions

Benzene when treated with Br\(_2\) in the presence of FeBr\(_3\), gives 1,4-dibromobenzene and 1,2-dibromobenzene. Which type of reaction is this?
Alkali metal with highest first ionization enthalpy is:
A first-order reaction is 75 percent completed in 6000s. What is the half-life?
Given that \( a, \frac{3}{4} a r^2, a r^3 \) are in G.P. Product of first four terms = \( \frac{3^6}{4^3} \), then find \( a \):
Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)
Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]
If \[ \frac{x}{4} + \frac{x}{3} < 13, \text{ then } x \in \]
Find the limit: \[ \lim_{x \to 0} \frac{\sin[x]}{[x]}, \text{ where } [x] \text{ represents greatest integer function} \]
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Find the probability of getting at most 2 heads when 4 coins are tossed.
Find the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \sin^2 x)}{x^2} \]
Find the values of: \[ \cos 75^\circ, \cos 15^\circ, \cos 45^\circ \]
Evaluate the expression: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} \]
Axis of parabola is \( x = 0 \). If the vertex is at a distance of 3 from the origin above the x-axis, find the vertex.
Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Evaluate: \[ \sec \left( \cos^{-1} \left( \frac{2024}{2025} \right) \right) \]
Evaluate the integral: \[ \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx \]
Solve for \( a \) and \( b \) given the equations: \[ \sin x + \sin y = a, \quad \cos x + \cos y = b, \quad x + y = \frac{2\pi}{3} \]
Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).
Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).
If \( -5<x \leq -1 \), then \( -21 \leq 5x + 4 \leq b \). Find \( b \).
An unbiased die is tossed until a sum \( S \) is obtained. If \( X \) denotes the number of times tossed, find the ratio \( \frac{P(X = 2)}{P(X = 5)} \).
Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).
If \( f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \), find the domain of \( f(x) \).
Given the function \( F(x) = |\sin(3x)| - \cos(3x) \), for \( \frac{\pi}{6} \leq x \leq \frac{\pi}{3} \), find \( f' \left( \frac{\pi}{4} \right) \).