List of top Mathematics Questions

Let \( f(x) = \sin x \), \( g(x) = \cos x \), \( h(x) = x^2 \) then
\[ \lim_{x \to 1} \frac{f(g(h(x))) - f(g(h(1)))}{x - 1} = \]

If \( A = 1 + r^a + r^{2a} + r^{3a} + \cdots \infty \) and \( B = 1 + r^b + r^{2b} + r^{3b} + \cdots \infty \), then \( a/b \) is equal to

Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}} \sin^2 x \, dx \]

Evaluate the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \]

Evaluate the integral: \[ \int \frac{4x^2 \cot^{-1}(x^3)}{1+x^6}\,dx \] (where \(C\) is a constant of integration)

If \( h(x) = \sqrt{4f(x) + 3g(x)} \), \( f(1)=4 \), \( g(1)=3 \), \( f'(1)=3 \), \( g'(1)=4 \), then \( h'(1) \) is equal to:}

Evaluate the integral: \[ \int \frac{x+1}{x(1+xe^x)^2}\,dx \]

Evaluate the integral: \[ \int_{1}^{4}\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^x}}\right)dx \]

Identify the truth value of the statement pattern \( (p \land \sim q) \rightarrow r \).

Find the area of the region bounded by the parabola \( y^2 = 4ax \) and its latus rectum.

Find the value of \( k \) if the function \( f(x) = \frac{k \cos x}{\pi - 2x} \) is continuous at \( x = \frac{\pi}{2} \).

If \(A, B, C\) are vertices of a triangle with position vectors \(\vec a, \vec b, \vec c\), find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).

Evaluate the integral: \[ \int \frac{x}{x+2}\, dx \]

For the curve \( y = 3x^3 - 3x^2 + 1 \) at \( x = 1 \), find the equation of the tangent.

If \( \vec{a} \) and \( \vec{b} \) are unit vectors and the angle between them is \(60^\circ\), find \( |\vec{a}-\vec{b}| \).

Find the general solution of the differential equation \( \dfrac{dy}{dx} + \dfrac{y}{x} = x^2 \).

Calculate the area of the region bounded by the curve \( y^2 = 4x \) and the line \( x = 3 \).

Find the value of \( k \) if the function \( f(x) = \dfrac{k\sin x + 2\cos x}{\sin x + \cos x} \) is increasing for all \( x \).

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

The coordinates of the foot of the perpendicular from the origin to the plane $2x - 3y - 6z = 4$ are

In $\triangle ABC$, if $2a^2 = b^2 + c^2$, then the value of $\frac{\cos 3A}{\cos A} + 2$ is

The value of $\tan^{-1}(\sqrt{3}) + \sec^{-1}(-2) - \sin^{-1}\left(-\frac{1}{2}\right)$ is

Evaluate the integral: \(\int \frac{2x}{x^2 - 5x + 4} \, dx\)

Evaluate the integral: \(\int \frac{\sin x}{\sin 4x} \, dx\)

For the curve \(y = 3x^3 - 3x^2 + 1\) at \(x = 1\), find the equation of the tangent.