List of top Mathematics Questions

If a vector makes angles \( \frac{\pi}{3}, \frac{\pi}{4} \) and \( \gamma \) with \( \hat{i}, \hat{j} \), and \( \hat{k} \), respectively, where \( \gamma \in \left( \frac{\pi}{2}, \pi \right) \), then the angle \( \gamma \) is:
The limit \( \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \) is equal to:
The general solution of \( \frac{dy}{dx} + y = 5 \) is:
Evaluate \( \int_{-\pi/2}^{\pi/2} \sin^9 x \cos^2 x \, dx \):
If \[ \int \frac{dx}{\sqrt{16 - 9x^2}} = A \sin^{-1}(Bx) + C, { where } C { is an arbitrary constant, then } A + B = \]
If three distinct numbers are chosen randomly from the first 50 natural numbers, then the probability that all of them are divisible by 2 and 3 is:
If \( f(x) = \begin{cases} x^2 & \text{for } x < 0 \\ 5x - 3 & \text{for } 0 \leq x \leq 2 \\ x^2 + 1 & \text{for } x > 2 \end{cases} \), then the positive value of \( x \) for which \( f(x) = 2 \) is:
The function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing in the interval:
Evaluate \( \lim_{x \to 0} \frac{\sin 2x + \sin 5x}{\sin 4x + \sin 6x} \):
Evaluate \( \int x e^{x^2} \, dx \):
The minimum value of \( f(x) = 7x^4 + 28x^3 + 31 \) is:
If the straight lines \( \frac{x - 3}{2} = \frac{y - 4}{3} = \frac{z - 6}{-1} \) and \( \frac{x - 2}{a} = \frac{y + 3}{b} = \frac{z + 4}{-1} \) are parallel, then \( a^2 + b^2 \) is:
If \( y = \frac{x^2}{x - 1} \), then \( \frac{dy}{dx} \) at \( x = -1 \) is:
The angle between the lines \( \frac{x}{1} = \frac{y}{1} = \frac{z}{1} \) and \( \frac{x}{0} = \frac{y}{1} = \frac{z}{-1} \) is:
A ray of light passing through the point \( (1, 2) \) is reflected on the \( x \)-axis at a point \( P \) and passes through the point \( (5, 6) \). Then the abscissa of the point \( P \) is:
If \( 3 \sin \theta + 5 \cos \theta = 5 \), then the value of \( 5 \sin \theta - 3 \cos \theta \) is:
If \( g(x) = -\sqrt{25 - x^2} \), then \( g'(1) \) is:
The differential equation \( \frac{dy}{dx} + x = A \) (where A is constant) represents:
The second term of a G.P. is 4, then the product of the first three terms is:
If the focus of a parabola is \( (0, -3) \) and its directrix is \( y = 3 \), then its equation is:
The lines \( \frac{x + 3}{-2} = \frac{y}{1} = \frac{z - 4}{3} \) and \( \frac{x - 1}{\mu} = \frac{y - 1}{\mu + 1} = \frac{z}{\mu + 2} \) are perpendicular to each other. Then the value of \( \mu \) is:
If \( \left( \frac{1 - i}{1 + i} \right)^{10} = a + ib \), then the values of \( a \) and \( b \) are, respectively:
If \( \sin \theta = \frac{b}{a} \), then \( \frac{\sqrt{a+b}} {\sqrt{a-b}} + \frac{\sqrt{a-b}} {\sqrt{a+b}} \) is equal to:
Let \( \vec{a} = \hat{i} - \hat{j} \), \( \vec{b} = \hat{j} - \hat{k} \), and \( \vec{c} = \hat{k} - \hat{i} \), then the value of \( \vec{b} \cdot (\vec{a} + \vec{c}) \) is:
The line \( \frac{x}{5} + \frac{y}{b} = 1 \) passes through the point \( (13, 32) \) and is parallel to the line \( \frac{x}{c} + \frac{y}{3} = 1 \). Then the values of \( b \) and \( c \) are, respectively: