List of top Questions asked in AP EAPCET

The positive value of $ x $ satisfying the equation \[ \int_0^x (1 - t) \, dt = \frac{1}{2} \] is:

If $ \sin \theta = \frac{3}{5} $ and $ \theta $ is not in the first quadrant, then $ 15 \sin 2\theta - 20 \cos 2\theta - 7 \tan 2\theta $ is:

The equation of the pair of asymptotes of the hyperbola $x^2 - 2y^2 - 8x + 8y + 4 = 0$ is

Three boxes $ B_1, B_2, B_3 $ contain white, black, and red balls as follows: \[ \begin{array}{|c|c|c|c|} \hline & \textbf{White} & \textbf{Black} & \textbf{Red} \\ \hline B_1 & 2 & 1 & 2 \\ B_2 & 3 & 2 & 4 \\ B_3 & 4 & 3 & 2 \\ \hline \end{array} \] A die is thrown:
- Box $ B_1 $ is chosen if the die shows 1 or 2
- Box $ B_2 $ is chosen if the die shows 3 or 4
- Box $ B_3 $ is chosen if the die shows 5 or 6
A ball is drawn at random from the selected box. the ball drawn is red, find the probability it came from box $ B_2 $:

If $ f(x) $ is differentiable on $ \mathbb{R} $, $ f(x)f'(-x) - f(-x)f'(x) = 0 $, $ f(0) = 3 $, and $ f(3) = 9 $, then $ (1 + f(-3))^3 + 1 $ is:

If $ f(x) = \frac{x^3 + 5}{\sqrt{12 + x}} $ and } \[ \int_{-5}^{5}f(x) \, dx = \int_{0}^{5} \left( f(x) + g(x) \right) \, dx, \text{ then } g(x) = \]

If $ \theta $ is the acute angle between the tangents drawn from the point $ (2, 3) $ to the hyperbola $ 5x^2 - 6y^2 - 30 = 0 $, then $ \tan \theta $ is:

Let $ A = \begin{bmatrix} 2 & 1 & 3 & -1 \\1 & -2 & 2 & -3 \end{bmatrix}, B = \begin{bmatrix} 2 & 1 & 0 & 3 \\1 & -1 & 2 & 3 \end{bmatrix} $, and the equation $ 2A + 3B - 5C = 0 $. Find the matrix $ C $.

Let $ A = \begin{pmatrix} 0 & 3 & 5 & -7 \\ 8 & 0 & -1 & 0 \\ 6 & -1 & 0 & 0 \end{pmatrix} $ and $ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $. If $ D = [\alpha \, \beta \, \gamma]^T $ is the solution of $ X^T B^T = A^T X $, then $ D^T A = $

If \[ S = 2x^2 + 2y^2 - 8x + 8y - 7 = 0 \] is the circle passing through the points of intersection of the circles \[ x^2 + y^2 - kx - ky + 1 = 0 \quad \text{and} \quad x^2 + y^2 - kx + ky - 2 = 0, \] then the length of the tangent drawn from the point $ (k, k) $ to the circle $ S $ is:

Let ‘O’ be the origin. A and B be two points with position vectors $ -3\vec{i} - 3\vec{j} + 4\vec{k} $ and $ 4\vec{i} - 4\vec{j} - 3\vec{k} $ respectively. Let $ P $ be a point such that the line drawn through $ P $ parallel to $ OB $ meets $ OA $ in $ L $, and another line through $ P $ parallel to $ OA $ meets $ OB $ in $ M $. If $ L $ divides $ OA $ in the ratio 2:3 and $ M $ divides $ OB $ in the ratio 3:2, then the distance from $ O $ to $ P $ is:

Which one of the following functions is discontinuous at $x=1$?

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 variance of 20 observations is 5. If each one of the observations is multiplied by 2, then the variance of the resulting observations is:

If \[ \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}, \] then the value of $ A + B = $:

If \[ \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} \, dx = Ax + B \log \left( 9e^{2x} - 4 \right) + C, \text{ then } (A, B) = \]

Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]

The sum of the fourth powers of the roots of the equation $ 16x^2 - 10x + 1 = 0 $ is:

Let $ \vec{a} = 3\vec{i} + \vec{j} - 2\vec{k} $, $ \vec{b} = -5\vec{i} + 7\vec{j} $, and $ \vec{c} = 3\vec{i} + y\vec{j} $ be three vectors such that $ |\vec{a} - \vec{b} + \vec{c}| = \sqrt{141} $. If $ y_1 $ and $ y_2 $ are the values of $ y $ satisfying the given condition, then $ |y_1 - y_2| = $

In $ \triangle ABC $, if $ \sin B = \sin C $ and $ 3 \cos B = 2 \cos C $, then $ \triangle ABC $ is:

If $ \alpha $ and $ \beta $ are the roots of the equation $ x^2 + x + 1 = 0 $, then the quadratic equation whose roots are $ \alpha^{2023} $ and $ \beta^{2012} $ is

Given independent events A, B, and C with rtain intersection probabilities, find $P(A)$, $P(B)$, and $P(C)$.

Arg \[ \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \] lies in the interval \[ \text{Arg} \left( \frac{4 + 2i}{1 - 2i} + \frac{3 + 4i}{2 + 3i} \right) \]

If $ S $ is the set of all real values of $ a $ such that a plane passing through the points $ (-a^2, 1, 1), (1, -a^2, 1), (1, 1, -a^2) $ also passes through the point $ (-1, -1, 1) $, then $ S = \dots $

Let $ A \subseteq \mathbb{R}, B \subseteq \mathbb{R} $ and $ f : A \to B $ be defined by $ f(x) = x^2 - 3x + 2 $. If $ f $ is a bijection, then