List of top Mathematics Questions

If \[ y = \sin^{-1} x, \] then \[ (1 - x^2)y_2 - xy_1 = 0. \]
If \[ \frac{d}{dx} \left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) = ax + b, \] then \( (a,b) \) is:
The equation of the tangent to the curve \[ y = x^3 - 2x + 7 \] at the point \( (1,6) \) is:
If the percentage error in the radius of a circle is 3, then the percentage error in its area is:
If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:
If the function \[ f(x) = \frac{\sqrt{1+x} - 1}{x} \] is continuous at \( x = 0 \), then \( f(0) \) is:
Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}. \]
If \( \mathbf{a} = 4\hat{i} + 5\hat{j} - 3\hat{k} \) and \( \mathbf{b} = 6\hat{i} - 2\hat{j} - 2\hat{k} \) are two vectors, then the magnitude of the component of \( \mathbf{b} \) parallel to \( \mathbf{a} \) is:
The value of $\lim_{𝑥→3}\frac{(𝑥^2 −9)}{(𝑥^2−4𝑥+3 )} $ is (rounded off to the nearest integer).
The general solution of the differential equation \( (x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx \) is:
Determine the order and degree of the differential equation \( \frac{d^3y}{dx^3} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} \):
Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \):
Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \):
Evaluate the integral \( \int_{-\frac{1}{24}}^{\frac{1}{24}} \sec(x) \log\left(\frac{1-x}{1+x}\right) \, dx \):
Evaluate the integral \( \int \frac{2 - \sin x}{2 \cos x + 3} \, dx \):
Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):

Evaluate the integral \( \int \frac{dx}{\sqrt{\sin^3 x \cdot \cos (x - \alpha)}} \):

Evaluate the integral \( \int \frac{dx}{x(x^4 + 1)} \):
A value of \(c\) according to the Lagrange's mean value theorem for \(f(x) = (x - 1)(x - 2)(x - 3)\) in \([0,4]\) is:
The acute angle between the curves \(x^2 + y^2 = x + y\) and \(x^2 + y^2 = 2y\) is:
The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:
If \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) respectively. If \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), then \( k_1 + k_2 = \)
In the interval \([0, 3]\), the function \(f(x) = |x - 1| + |x - 2|\) is:
If \(y = t^2 + t^3\) and \(x = t - t^4\), then \(\frac{d^2y}{dx^2}\) at \(t = 1\) is:
If \(f(x) = \begin{cases} 2x + 3, & \text{if } x \leq 1 \\ 2ax + bx, & \text{if } x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)