List of top Mathematics Questions

The minimum value of \( f(x) = x + \frac{4}{x + 2} \) is
Given \[ \frac{d}{dx} \left( \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x} \right) = f(x) \left[ \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1 \right] \] Then find \( f(5) \).
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)
\( \lim_{x \to a} \frac{x^3+a^3}{x+a} = 7 \implies a = \)
The minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin is
\( \int_{-\pi/4}^{\pi/4} \cos^8 x \ dx = \)
Evaluate: \( \int \frac{1 + \tan x \tan(x + \alpha)}{\tan x \tan(x + \alpha)} dx \):
Equation of tangent to the curve \( y = x + \frac{4}{x^2} \) which is parallel to x-axis is
Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$
For the differential equation \( \frac{d^3 y}{dx^3} = 0 \), \( y = ax^2 + bx + c \) is:
\(\tanh(\log x) = \) (Assuming \(\log x\) is natural logarithm \(\ln x\))
If \( f(x) = \int_{x^2}^{\cos^2 x} (2x \tan^2 x - 2x - 6 \tan x) dx \), and \( f(0) = \pi \), then \( f(x) = \):
Evaluate: \[ \int_{0}^{\pi/4} e^{\tan^2 \theta} \sin^2 \theta \tan \theta \, d\theta = \]
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
If $x \neq 0$ and $f(x)$ satisfies $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$, then $\dfrac{d}{dx} \left(x^2 f(x)\right)$ at $x = 1$ is
\( \lim_{x \to 0} x^3 (\sqrt{x^2+\sqrt{x^4+1}} - \sqrt{2}x) = \)
The function \( f(x) = 2x^3 - 9ax^2 + 12a^2x + 1 \ (a>0) \) attains its maximum and minimum at \( p \) and \( q \) respectively and \( p^2 = q \). Then the value of \( a \) is:
Evaluate the integral \( \int \frac{1 - k \cos^2 x}{\sin^k x \cos^2 x} dx \).
\( \int \frac{dx}{(1+\sqrt{x})^{2022}} = \)
If \( f(x) = \max(\sin x, \cos x) \) and \( g(x) = \min(\sin x, \cos x) \), then \[ \int_0^{\pi/2} f(x) dx + \int_0^{\pi/2} g(x) dx = \]
The number of those tangents to the curve $y^2 - 2x^3 - 4y + 8 = 0$ which pass through the point $(1, 2)$ is
The perimeter of a sector is constant. If its area is to be maximum, the sectorial angle should be:
The normal to the curve y = f(x) at the points (3, 4) makes an angle \( \frac{3\pi}{4} \) with positive x-axis then \(f'(3) = \)
Assertion (A) : \( f(x)=|x| \) is differentiable at \( x=a \neq 0 \) and continuous but not differentiable at \( x=0 \) Reason (R) : If a function is differentiable at a point then it is continuous at that point. But converse is not true. .
The set of all \( x \) for which \( \sin x \leq x \) is: