List of top Questions asked in AP EAPCET- 2022

Condition that two curves \( y^2 = 4ax \) and \( xy = c^2 \) cut orthogonally is:
If \( \int \frac{x^8+4}{x^4-2x^2+2} dx = Ax^5+Bx^3+Cx+K \), then \(5A+3B+C = \)
If \( f(x) = \int \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) dx + C \), then \( f'(a) = \)
Evaluate \( \int \frac{\cos(\sec^2 x) - 2022}{\cos^{2022} x} dx = f(x) + C \Rightarrow f\left( \frac{\pi}{4} \right) = \)
Evaluate \( \lim_{n \to \infty} \left( \frac{1^2}{n^3 + 1^3} + \frac{2^2}{n^3 + 2^3} + \cdots + \frac{n^2}{n^3 + n^3} \right) \)
If \( f(x) = \begin{cases 4 & -\infty<x<-\sqrt{5}
x^2-1 & -\sqrt{5} \le x \le \sqrt{5}
4 & \sqrt{5}<x<\infty \end{cases} \). If k is the number of points where f(x) is not differentiable then k -- 2 =}
If \( f(x) = \max \{3 - x, 3 + x, 6\} \) is not differentiable at \( x = a \) and \( x = b \), then \( |a| + |b| = \)
If \(x+y=k, x>0, y>0\) then \(x^2+y^2\) is minimum if
\( \int_0^1 \alpha^k x^k dx = \)
\( \int \frac{e^{2x}}{\sin^2 x} \left( 2 \log \csc x + \sin 2x \right) dx = \)
\( \int_{a-1}^{a} \frac{e^x(a-x)}{(x-a+1)^2} dx = \)
The curve \( y = ax^3 + bx^2 + cx + 5 \) touches the x-axis at point \( P(-2, 0) \). Then the value of \( c \) is:
If \( \int e^{\sqrt{x}} / \sqrt{x} (x + \sqrt{x}) dx = e^{\sqrt{x}}[Ax + B\sqrt{x} + C] + K \), then \( A + B + C = \):
If \( \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right] \), then \( 2h(x)-f(x)-g(x) = \)
If the curves \( y = x^3 - 3x^2 - 8x - 4 \) and \( y = 3x^2 + 7x + 4 \) touch each other at a point \( P \), then the equation of the common tangent at \( P \) is
If \( \int f(x) dx = F(x) + C \), then \( \frac{d}{dt} \int_{g(t)}^{h(t)} f(x)\,dx = \)
\( \int_0^{\pi} x \left( \sin^2(\sin x) + \cos^2(\cos x) \right) dx = \)
Let \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) be roots of \( 18x^2 - 9\pi x + \pi^2 = 0 \), \( f(x) = x^2, g(x) = \cos x \). Then
\[ \int_{\alpha}^{\beta} x (g \circ f(x)) \, dx = ? \]
If \( \int_{0}^{\pi/2} \sin^m x \cos^4 x \, dx = \frac{7\pi}{2048} \), then \( m = \)
A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is:
If \( f(x) = \sin(\tan^{-1} x) \), then \( \int_0^1 xf''(x)dx = \)
The parametric form of a curve is \( x = \frac{t^3}{t^2 - 1},\ y = \frac{t}{t^2 - 1} \), then \( \int \frac{dx}{x - 3y} = \)
The condition that \( f(x) = ax^3 + bx^2 + cx + d \) has no extreme value is
Which of the following functions is differentiable at \( x = 0 \)?
\( \lim_{x \to a} \frac{x^3+a^3}{x+a} = 7 \implies a = \)