List of top Mathematics Questions

If the function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \), where \( a>0 \), attains its local maximum and minimum at \( p \) and \( q \), respectively, such that \( p^2 = q \), then \( f(3) \) is equal to:
The value of the real variable \( x>0 \) that minimizes the function \( f(x) = x^x e^{-x} \) is ________
The general solution of the differential equation \( x^2 y'' - xy' + 5y = 0 \) is _______
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number
Find the angle between the vectors \( \mathbf{a} = (2, 3, 1) \) and \( \mathbf{b} = (1, -1, 4) \).
A box contains 5 red balls and 3 blue balls. If two balls are drawn randomly without replacement, what is the probability that one of the balls is red and the other is blue?
Find the determinant of the matrix \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \).

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c} \text{ and } \mathbf{a} \times \mathbf{b} \neq 0 \text{ Show that } \mathbf{b} = \mathbf{c}$.

Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
Find the angle between the vectors \( \mathbf{a} = (2, -1, 3) \) and \( \mathbf{b} = (1, 4, -2) \).
If \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \), find the determinant of \( A^2 \).

Let $ a_1, a_2, a_3, \ldots $ be in an A.P. such that $$ \sum_{k=1}^{12} 2a_{2k - 1} = \frac{72}{5}, \quad \text{and} \quad \sum_{k=1}^{n} a_k = 0, $$ then $ n $ is:

The angle between vectors $ \mathbf{a} = \hat{i} + \hat{j} - 2\hat{k} $ and $ \mathbf{b} = 3\hat{i} - \hat{j} + 2\hat{k} $ is:
If \[ f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2}, & x \neq 0 \\ 1, & x = 0 \end{cases} \] is continuous at \( x = 0 \), then the value of \( a \) is:

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three vectors such that \(\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}\) and \(\mathbf{a} \times \mathbf{b} \neq 0. Show \;that \;\mathbf{b} = \mathbf{c}\).

If $y = 5 \cos x - 3 \sin x$, prove that $\frac{d^2y}{dx^2} + y = 0$.
If \( A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 3 & 0 & 5 \\ 0 & 0 & 5 \end{array} \right] \), then \( A^3 \) is:

If $y = 5 \cos x - 3 \sin x$, prove that  $\frac{d^2y}{dx^2} + y = 0$.

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$.

Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.
Find the foot of the perpendicular drawn from the point \( (1, 1, 4) \) on the line \( \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \).
Find the point on the line \( \frac{x-1}{3} = \frac{y+1}{2} = \frac{z-4}{3} \) at a distance of \( \sqrt{2} \) units from the point \( (-1, -1, 2) \).
Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]
Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]
In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.