List of top Questions asked in AP EAPCET- 2025

\[ \lim_{y \to 0} \frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt{2}}{y^4} = \ ? \]
If the tangent to the curve \( y = ax^2 + bx + c \) at the point \( (1,1) \) is parallel to the x-axis, and the curve passes through the point \( (2,3) \), then find the value of \( a + b + c \):
If Lagrange's mean value theorem is applied to the function $$ f(x) = e^x $$ defined on the interval $ [1, 2] $ and the value of $ c \in (1, 2) $ is $ k $, then find $ e^{k-1} $.
Evaluate: \[ \int \frac{dx}{(x+1)\sqrt{x^2+1}} \]
The function \( f(x) = xe^{-x} \) for all \( x \in \mathbb{R} \) attains a maximum value at \( x = k \), then \( k = \)
Evaluate the integral \( \displaystyle \int_{1/5}^{1/2} \frac{\sqrt{x - x^2}}{x^3} \, dx \):
If the tangent of the curve $$ 4y^3 = 3ax^2 + x^3 $$ drawn at the point $ (a, a) $ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinate axes, then find $ a $.
The interval in which the function \( f(x) = \tan^{-1}(\sin x + \cos x) \) is an increasing function, is:
The slope of a tangent drawn at the point \( P(\alpha, \beta) \) lying on the curve \( y = \frac{1}{2x - 5} \) is \( -2 \). If \( P \) lies in the fourth quadrant, then \( \alpha - \beta = \)
Evaluate the integral: \[ \int_0^{\pi} \frac{x \sin x}{\sin^2 x + 2\cos^2 x} \, dx \]
Evaluate the integral: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx =\ ? \]
If \( \int \frac{dx}{2\cos x + 3\sin x + 4} = \frac{2}{\sqrt{3}}f(x) + c \), then \( f\left(\frac{2\pi}{3}\right) = \)
Evaluate: \( \int_0^{400\pi} \sqrt{1 - \cos 2x} \, dx \)
If \(\int \frac{1}{((x+4)^3 (x+1)^5)^{1/4}} \, dx = A \cdot \left(\frac{x+4}{x+1}\right)^n + c\), then
Evaluate the integral: \[ \left| \int_{-\pi/4}^{\pi/3} \tan\left(x - \frac{\pi}{6}\right) dx \right| \]
\( \int_{-\pi/2}^{\pi/2} \sin^2 x \cos^2 x (\sin x + \cos x) \, dx = \)
If \( \int \left( x^6 + x^4 + x^2 \right) \sqrt{2x^4 + 3x^2 + 6} \, dx = f(x) + c \), then \( f(3) = \)
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{1^2 + n^2} + \frac{2}{2^2 + n^2} + \frac{3}{3^2 + n^2} + \cdots + \frac{n}{n^2 + n^2} \right) \]
Area of the region (in sq. units) bounded by the curve $ y = x^2 - 5x + 4 $, $ x = 0 $, $ x = 2 $, and the X-axis is
The interval in which the curve represented by \( f(x) = 2x + \log\left(\frac{x}{2 + x}\right) \) is increasing is
The number of solutions of \[ \sin 2x + \cos 4x = 2 \quad \text{in the interval } [-\pi, \pi] \text{ is:} \]
In \( \triangle ABC \), if \[ \angle ABC = \delta, \delta = \cos^{-1} \left( \sqrt{ \frac{r_2}{r_3 r_1} } \right), \] then the expression \[ \angle ABC = \delta = \cos^{-1} \left( \sqrt{ \frac{r_2}{r_3 r_1} } \right) \]
The coefficient of \(x^3\) in the expansion of \(\frac{x^4 + 1}{(x^2 + 1)(x - 1)}\) when it is expressed in terms of positive integral powers of \(x\), is
If \(2\sin x - \cos 2x = 1\), then \( (3 - 2\sin^2x) = \)

If

\[ \cos x + \sin x = \frac{1}{2} \]

and

\[ 0 < x < \pi, \text{ then } \tan x = \ ? \]