List of top Questions asked in AP EAPCET- 2024

The descending order of magnitude of the eccentricities of the following hyperbolas is: A. A hyperbola whose distance between foci is three times the distance between its directrices. B. Hyperbola in which the transverse axis is twice the conjugate axis. C. Hyperbola with asymptotes $ x + y + 1 = 0, x - y + 3 = 0 $.

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of numbers $ \{1, 2, 3, \ldots, 10\} $, then the probability that $ |x^2 - y^2| $ is divisible by 6 is:

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B.

Then the correct match is:

If $$ y = \sin^{-1} x, $$ then $$ (1 - x^2)y_2 - xy_1 = 0. $$

If \[ \cos x \frac{dy}{dx} - y \sin x = 6x, \quad (0 < x < \frac{\pi}{2}) \quad \text{and} \quad y(\frac{\pi}{3}) = 0, \quad \text{then} \quad y(\frac{\pi}{6}) = \]

Evaluate the integral: \[ I = \int \frac{\cos x + x \sin x}{x (x + \cos x)} dx =\]

Find the area of the region (in square units) enclosed by the curves: \[ y^2 = 8(x+2), \quad y^2 = 4(1-x) \] and the Y-axis.

The general solution of the differential equation \[ (x + y)y \,dx + (y - x)x \,dy = 0 \] is:

Let $ \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} $ be non-coplanar vectors. If $ \alpha \overrightarrow{d} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} $, $ \beta \overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c} + \overrightarrow{d} $, then $ | \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} + \overrightarrow{d} | = ? $

Let $ a>1 $ and $ 0<b<1 $. -∞$ f : \mathbb{R} \to [0, 1] $ is defined by $ f(x) = \begin{cases} a^x & \text{if } x<0 b^x & \text{if } 0 \leq x \leq 1 \end{cases} $, then $ f(x) $ is:

The radical axis of the circles \[ x^2 + y^2 + 2gx + 2fy + c = 0 \] and \[ 2x^2 + 2y^2 + 3x + 8y + 2c = 0 \] touches the circle \[ x^2 + y^2 + 2x + 2y + 1 = 0. \] Then:

If the equation of the circle whose radius is 3 units and which touches internally the circle $$ x^2 + y^2 - 4x - 6y - 12 = 0 $$ at the point $ (-1, -1) $ is $$ x^2 + y^2 + px + qy + r = 0, $$ then $ p + q - r $ is:

If $ A, B, C $ are the angles of a triangle, then \[ \sin 2A - \sin 2B + \sin 2C = \]

If $3A = \begin{bmatrix} 1 & 2 & 2 \\[0.3em] 2 & 1 & -2 \\[0.3em] a & 2 & b \end{bmatrix}$ and $AA^T = I$, then$\frac{a}{b} + \frac{b}{a} =$:

Evaluate the expression \[ 2 \cot h^{-1}(4) + \sec h^{-1}\left( \frac{3}{5} \right). \]

If the equation of the pair of straight lines passing through the point $ (1, 1) $ and perpendicular to the pair of lines $ 3x^2 + 11xy - 4y^2 = 0 $ is $ ax^2 + 2hxy + by^2 + 2gx + 2fy + 12 = 0 $, then find $ 2(a + h - b - g + f - 12) = ? $

Equation of the circle having its centre on the line $ 2x + y + 3 = 0 $ and having the lines $ 3x + 4y - 18 = 0 $ and $ 3x + 4y + 2 = 0 $ as tangents is:

If the function f(x) = $\sqrt{x^2 - 4}$ satisfies the Lagrange’s Mean Value Theorem on $[2, 4]$, then the value of $ C $ is}

Evaluate the integral: $$ I = \int_{\frac{1}{\sqrt[5]{32}}}^{\frac{1}{\sqrt[5]{31}}} \frac{1}{\sqrt[5]{x^{30} + x^{25}}} \, dx. $$

Evaluate the integral $$ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. $$

If $ x $ is real and $ \alpha, \beta $ are maximum and minimum values of $ \frac{x^2 - x + 1}{x^2 + x + 1} $ respectively, then $ \alpha + \beta = $:

For $ x<0 $, $ \frac{d}{dx} [|x|^x] $ is given by:

Evaluate the integral \[ I = \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx \]

Assertion (A): If $ B $ is a $ 3 \times 3 $ matrix and $ |B| = 6 $, then $ | \text{Adj}(B) | = 36 $. Reason (R): If $ B $ is a square matrix of order $ n $, then $ |\text{Adj}(B)| = |B|^n $.