List of top Mathematics Questions

Let \[ f(x) = \begin{cases} \frac{1 - \sin^3 x}{3 \cos^2 x}, & x<\frac{\pi}{2} \\ \alpha, & x = \frac{\pi}{2} \\ \frac{\beta(1 - \sin x)}{(\pi - 2x)^2}, & x>\frac{\pi}{2} \end{cases} \] If \( f(x) \) is continuous at \( x = \frac{\pi}{2} \), find \( \alpha \beta \).

The derivative of the function $f(x) = \sin(x^2)$ at $x = \sqrt{\pi}$ is

The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is a solution of the differential equation

If the slope of the tangent at a point $(x,y)$ on a curve is $\frac{y-4}{x}$ and the curve passes through $(4,3)$, then the point where it cuts the line $y=x$ is

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$, the left side of the equation takes the form $\frac{d}{dx} (y f(x))$, then $f(x) =$

Evaluate the integral: \[ \int_{-\pi}^{\pi} \frac{2x(1 + \sin x)}{1 + \cos^2 x}\, dx \]

Let \( f(x) = \int \frac{e^{3x}}{4 + 8e^{2x} + e^{4x}} \, dx \), and \( g(x) = \int \frac{2\, dx}{e^{3x} + 8e^x + 4e^{-x}} \), then \( f(x) - g(x) = \) ?

Evaluate the integral \( \int \frac{x^2 - 2}{x^3 \sqrt{x^2 - 1}} \, dx \)

If \( a, b, c, d \ne 0 \) and \( \int \frac{at + b}{ct + d} dt = t \), then find \( \int f(x) dx \) where \( f(x) = \frac{ax + b}{cx + d} \)

\([ \cdot ]\) represents the greatest integer function. If \( \int_{\sqrt{\frac{18}{3}}}^{[x]} dx = a + b\sqrt{2} + c\sqrt{3} \), then find the value of \( a + b + c \).

Evaluate: \( \int_{0}^{4} \frac{x + 2}{\sqrt{4x - x^2}}\, dx \)

Evaluate \( \int \frac{dx}{1 + a \cos x} \), given \( a>|\sec \theta| \)

Two curves \( y = a^x \) and \( y = b^x \) intersect at some angle \( \alpha \). Find \( \tan \alpha \).

Find the area of the triangle formed by the tangent to the curve \( xy = a^2 \) at \( (x_1, y_1) \) and the coordinate axes.

If the slope of the line through \( (0, 0) \) which is tangent to the curve \( y = x^2 + x + 16 \) is \( m \), then find the value of \( m - 4 \).

If the line \( ax + by + c = 0 \) is a normal to the curve \( xy = 1 \), then which of the following is true?

In the interval \( (7, \infty) \), the function \( f(x) = |x - 5| + 2|x - 7| \) is:

Evaluate the limit: \[ \lim_{x \to \infty} \frac{A + e^{mx}}{x + Ae^{mx}} \]

Evaluate: \[ \lim_{n \to \infty} \sqrt{2} \left[ \frac{ \left(2 + \sqrt{2} \right)^n + \left(2 - \sqrt{2} \right)^n }{ \left(2 + \sqrt{2} \right)^n - \left(2 - \sqrt{2} \right)^n } \right] \]

If \( \frac{d^n y}{dx^n} = y_n \) and \( y = e^{\sqrt{x}} + e^{-\sqrt{x}} \), then find the value of \( 4x y_2 + 2 y_1 \).

Find the distance between the point \( (6, 4\sqrt{3}) \) and the focus of the parabola \( y^2 = 8x \)

Find the distance between the directrices of the ellipse: \[ \frac{x^2}{36} + \frac{y^2}{20} = 1 \]

For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the locus of the point of intersection of the tangents at the endpoints of normal chords is:

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).