List of top Mathematics Questions asked in AP EAPCET

If the line $$ 4x - 3y + 7 = 0 $$ touches the circle $$ x^2 + y^2 - 6x + 4y - 12 = 0 $$ at $ (\alpha, \beta) $, then find $ \alpha + 2\beta $.
The slope of the common tangent drawn to the circles $$ x^2 + y^2 - 4x + 12y - 216 = 0 $$ and $$ x^2 + y^2 + 6x - 12y + 36 = 0 $$ is:
If $ r_1 $ and $ r_2 $ are radii of two circles touching all the four circles $$ (x \pm r)^2 + (y \pm r)^2 = r^2, $$ then find the value of $$ \frac{r_1 + r_2}{r}. $$
If \( \int \frac{\cos^3 x}{\sin^2 x + \sin^4 x} dx = c - \operatorname{cosec} x - f(x) \), then \( f\left(\frac{\pi}{2}\right) = \)
If \( \log_2 3 = p \), express \( \log_8 9 \) in terms of \( p \):
If \( \int \frac{\cos^3 x}{\sin^2 x + \sin^4 x} dx = c - \operatorname{cosec} x - f(x) \), then \( f\left(\frac{\pi}{2}\right) = \)
$\int (x^3 y^3) \, dx =$
Identify the correct option from the following:
Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]
Evaluate \[ \int \frac{1}{1 + x^2} \, dx = ? \]
\( \int \frac{x}{(1-x^2)\sqrt{2 - x^2} dx = \)}
\( 1 + \frac{4}{15} + \frac{4 \cdot 10}{15 \cdot 30} + \frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45} + \cdots \infty \)
\( \int \sin^{-1}\left(\frac{x}{\sqrt{a+x}}\right) \, dx = \)
Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]
\( \int_{-1}^{4} \sqrt{\frac{4-x}{x+1}} \ dx = \)
\( \int_{-1}^{4} \sqrt{\frac{4-x}{x+1}} \ dx = \)
Evaluate \[ \int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ? \]
Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]
Find the area (in square units) of the region bounded by the lines \( x=0 \), \( x=\frac{\pi}{2} \), and the curves \( f(x) = \sin x \), \( g(x) = \cos x \):
Evaluate the limit: \[ \lim_{x \to \infty} \frac{3x+4\cos^2x}{\sqrt{x^2-5\sin^2x}} \]
\( \lim_{n\to\infty} \frac{1}{n^3} \sum_{k=1}^{n} k^2 x = \)
If a real valued function \[ f(x) = \begin{cases} \frac{x^2 (a + 3) x (a + 1)}{x + 3}, & x \neq -3
-\frac{5}{2}, & x = -3 \end{cases} \] is continuous at $x = -3$, then $\lim_{x \to -3} [x^2 x + 1] =$ Identify the correct option from the following:
If the function \( f \) defined by \[ f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x<0 \\ a, & x = 0 \\ \dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x>0 \end{cases} \] is continuous at \( x = 0 \), then \( a = \)
\( \lim_{n\to\infty} \frac{1}{n^3} \sum_{k=1}^{n} k^2 x = \)
Evaluate the limit: \[ \lim_{x \to \infty} \frac{3x+4\cos^2x}{\sqrt{x^2-5\sin^2x}} \]
Find the area (in square units) of the region bounded by the lines \( x=0 \), \( x=\frac{\pi}{2} \), and the curves \( f(x) = \sin x \), \( g(x) = \cos x \):