List of top Questions asked in AP EAPCET

If $$ \int \frac{x^4 + 1}{x^2 + 1} dx = Ax^3 + Bx^2 + Cx + D \tan^{-1} x + E, $$ then find $ A + B + C + D $.

If $$ \int \frac{x^2 - x + 2}{x^2 + x + 2} dx = x - \log(f(x)) + \frac{2}{\sqrt{7}} \tan^{-1}(g(x)) + c, $$ then find $$ f(-1) + \sqrt{7} g(-1). $$

If $$ \int \frac{dx}{1 - \sin^4 x} = A \tan x + B \tan^{-1}(\sqrt{2} \tan x) + C, $$ then find $ A^2 - B^2 $.

If $$ \int \frac{a \cos x + 3 \sin x}{5 \cos x + 2 \sin x} dx = \frac{26}{29} x - \frac{k}{29} \log |5 \cos x + 2 \sin x| + c, $$ then find $ |a + k| $.

Evaluate the integral: $$ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx $$

If the function $$ f(x) = \sin x - \cos^2 x $$ is defined on the interval $ [-\pi, \pi] $, then $ f $ is strictly increasing in the interval:

If the area of a square is 575 square units, then the approximate value of its side is:

If $$ y = (\log_x \sin x)^x, $$ then find $$ \frac{dy}{dx}. $$

If a real valued function $$ f(x) = \begin{cases} \frac{\sin a(x - [x])}{e^{x - [x]}}, & x<1 \\ b + 1, & x = 1 \\ \frac{|x^2 + x - 2|}{x - 1}, & x>1 \end{cases} $$ is continuous at $ x=1 $, then find $ b $. Here, $ [x] $ denotes the greatest integer function.

If $$ \sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0, $$ then find $$ \frac{dy}{dx}. $$

If $$ f(x) = 2 + |\sin^{-1} x|, $$ and $$ A = \{ x \in \mathbb{R} \mid f'(x) \text{ exists} \}, $$ then find $ A $.

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 tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$

The direction cosines of the line making angles $$ \frac{\pi}{4}, \frac{\pi}{3} $$ and $ \theta $ (where $ 0<\theta<\frac{\pi}{2} $) with X, Y, and Z axes respectively are:

Evaluate: $$ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + 4x^2} - \sqrt{x^2 - 3x} \right) $$

Evaluate: $$ \lim_{x \to \infty} \left[ x - \log(\cosh x) \right] $$

If the equation of the plane passing through point $ (3, 2, 5) $ and perpendicular to the planes $$ 2x - 3y + 5z = 7, \quad 5x + 2y - 3z = 11 $$ is $$ x + by + cz + d = 0, $$ then find $ 2b + 3c + d $.

The points $ A(-1, 2, 3), B(2, -3, 1), C(3, 1, -2) $ are:

If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.

If $ x - y - 3 = 0 $ is a normal drawn through the point $ (5, 2) $ to the parabola $ y^2 = 4x $, then the slope of the other normal that can be drawn through the same point to the parabola is?

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 the normal drawn at the point $$ P \left(\frac{\pi}{4}\right) $$ on the ellipse $$ x^2 + 4y^2 - 4 = 0 $$ meets the ellipse again at $ Q(\alpha, \beta) $, then find $ \alpha $.

If the equation of the circle having the common chord to the circles $$ x^2 + y^2 + x - 3y - 10 = 0 $$ and $$ x^2 + y^2 + 2x - y - 20 = 0 $$ as its diameter is $$ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, $$ then find $ \alpha + 2\beta + \gamma $.

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 $.