List of practice Questions

Series Expansion $ n^6 + \frac{1}{2} n^4 + \frac{1}{3} n^2 + \cdots + \frac{1}{n} C_n + 1 \quad n \to \infty $
The integral $ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx $ is:
Find the area of a circle whose radius is 7 cm.
Solve the system of equations: \[ x + y = 10 \] \[ 3x - y = 5 \]
In the word "UNIVERSITY", find the probability that the two "I"s do not come together.
A die is rolled once. What is the probability of rolling a number greater than 4?
Evaluate the integral: \[ \int \frac{1}{\sin^2 2x \cdot \cos^2 2x} \, dx \]
Principal Solution of $ (5 \sin \theta)(2 \cos \theta + 1) = 0 $
Find the value of $ x $ that satisfies the equation $ 2x + 3 = 11 $.
Find the value of \( x \) if \( \sin(2x) = 1 \).
The feasible region of the linear programming problem is determined by the system of inequalities: \[ x + y \leq 6, \quad x \geq 0, \quad y \geq 0. \] What is the maximum value of \( x + y \) in the feasible region?
Evaluate the integral: \( \int \sin^5 x \, dx \)
Find the value of \( \frac{5}{6} + \frac{3}{4} \).
Find the area of a triangle with vertices \( A(2, 3) \), \( B(5, 11) \), and \( C(8, 7) \).
The distance between the points \( A(3, 4) \) and \( B(-1, -2) \) is:
Find the value of the following expression: \[ \sin^2(30^\circ) + \cos^2(60^\circ) \]
Given the vectors: \[ \mathbf{a} = i + 3j - k, \quad \mathbf{b} = 3i - j + 2k, \quad \mathbf{c} = i + 2j - 2k \] and the following information: \[ \frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{c}|} = \frac{10}{3} \] Find the value of \( \alpha + \beta \) and the projection of \( \mathbf{a} \) on \( \mathbf{c} \).
If the roots of the quadratic equation \( x^2 - 7x + 12 = 0 \) are \( \alpha \) and \( \beta \), then the value of \( \alpha + \beta \) is:
The maximum value of the function \( f(x) = -2x^2 + 4x + 1 \) occurs at:
Find the area of a triangle with base 8 cm and height 6 cm.
Find the smallest angle of the triangle whose sides are \( 6 + \sqrt{12}, \sqrt{48}, \sqrt{24} \).
Evaluate the integral: \[ \int \frac{x^2 + 2x}{\sqrt{x^2 + 1}} \, dx \]
Which of the following expressions represents the Principal Solution
In $ \triangle ABC $, with usual notations, $ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \quad \text{and} \quad 2s \text{ is the perimeter of the triangle. Find the value of } s. $ Then the value of $ s $ is:
If \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), then the dot product \( \vec{A} \cdot \vec{B} \) is: