List of top Mathematics Questions asked in AP EAPCET

The function \( f(x) = x^2 + \dfrac{54}{x} \)
Evaluate: \[ \int e^{\sin x} \cdot \frac{x \cos^3 x - \sin x}{\cos^2 x} \, dx \]
If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)
If \( \int f(x)\,dx = \Psi(x) \), then \( \int x^5 f(x^3)\,dx = \):
If \[ \int \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R}, \text{ then} \] \[ \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R} \]
Let \( f(x) \) be a differentiable function such that \( f(1) = 2 \), \( f(2) = 6 \), and \[ f(x+y) = f(x) + kxy + \frac{4}{3} y^2, \quad \forall x, y \in \mathbb{R}. \] Then \( f(x) = \) ?
Let \[ f(x) = \begin{cases} \frac{5e^{|x|} + 2}{3 - e^{|x|}}, & x \ne 0 \\ 0, & x = 0 \end{cases} \] Then at \( x = 0 \), the function \( f(x) \) is:
If \( f(x) = \begin{vmatrix} 1 & 6 + x & 36 + x^2 \\ 0 & x - 3 & 3x^2 - 27 \\ 0 & 2x - 4 & 8x^2 - 32 \end{vmatrix} \), then \( \lim\limits_{x \to 1} \frac{f(x)}{f(-x)} \) is:
Let \( f(x) \) be differentiable on \( \mathbb{R} \) and \( f'(m) \ne 0, m \in \mathbb{R} \). If \( \lim\limits_{x \to m} \frac{x f(m) - m f(x)}{x - m} + f'(m) = f(m) \), then \( m = \):
The ordinates of the points on the curve \( y = \tan^{-1}(\sin(\sqrt{x})) \), \( 0 \leq x \leq 8\pi^2 \), at which the tangent is parallel to the X-axis are:
If \( u(n) = \int_0^{\frac{\pi}{2}} (1 + \sin t)^n \sin 2t \, dt,\; n \in \mathbb{N} \), then \( u(4) = \)
The integrating factor of the linear differential equation \( \frac{dy}{dx} = \frac{1}{4x + 3y} \) is
The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is
If a curve passes through (1, 2) and has the slope of its tangent \(1 - \frac{1}{x^2}\) at a point \((x, y)\), then the equation of that curve is
\(\displaystyle \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} dx =\)
\(\displaystyle \int e^x \left( \log x + \frac{1}{x^2} \right) dx =\)
If \(\displaystyle \int \frac{3x+2}{4x^2+4x+5} \, dx = A \log(4x^2+4x+5) + B \tan^{-1} \left(x + \frac{1}{2}\right) + C\), then \((A, B) =\)
\(\displaystyle \int_{0}^{\pi} \frac{\cos x}{\sqrt{1 - \sin^2 x}} \, dx =\)
Assertion (A): \(\displaystyle \int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x)\, dx\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\) Reason (R): \(\sin^6 x + \cos^6 x\) is a periodic function with period \(\dfrac{\pi}{2}\)
The area bounded by the curve \( x = \log(|y|) \), the lines \( x = -1 \) and \( x = 0 \) is:
If \( \displaystyle \int_{3}^{b} \frac{x - 1}{2x - x^2} \, dx = \frac{1}{2} \), then \( (b - 1)^2 = \)
If the slope of the tangent on a curve at any point \((x, y)\) is equal to \[ \frac{y^2 - x^2}{2xy} \] then the equation of the normal at the point \(\left(1, \frac{\sqrt{3}}{2}\right)\) is
Evaluate the integral \[ \int \frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right)} \, dx \]
The sum of the global minimum and global maximum values of the function \[ f(x) = \frac{4}{3}x^3 - 4x \quad \text{in } [0, 2] \text{ is} \]
If \(g(x)\) is the inverse of the function \(f(x)\) and \(f'(x) = \dfrac{1}{h(x)}\), then what is the value of \(g'(x)\)?