List of top Questions asked in AP EAPCET- 2023

If \( \alpha = \log_e(2 + \sqrt{3}) \), then evaluate: \[ \frac{\cosh \alpha }{1 - \tanh \alpha} + \frac{\sinh \alpha}{1 - \coth \alpha} \]
If \( \cosh x = \frac{5}{4} \), then \( \tanh 3x = \)
If \( \sinh x = \frac{5}{12} \), then \( \cosh \frac{x}{2} \) is:
The difference between the focal distances of any point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ is } 6. \text{ If } (\sqrt{13}, k) \text{ is an end point of a latus rectum of this hyperbola, then } k = \]
Let $ e $ be the eccentricity of the ellipse $ \frac{x^2}{4} + \frac{y^2}{9} = 1 $. If $ \frac{1}{e} $ is the eccentricity of a hyperbola, then the eccentricity of its conjugate hyperbola is:
Let $ x + y + 1 = 0 $ and $ x - y + 4 = 0 $ be the asymptotes of a hyperbola $ H $. If $ (1, 1) $ is a point on $ H $, then the length of the latus rectum of $ H $ is
In \( \Delta ABC \), \( A, B, C \) are in arithmetic progression and \( a : c = 1 : 2 \). If \( b = 4\sqrt{3} \) cm, then the area of \( \Delta ABC \) (in sq. cm) is:
If \( f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x} \) for \( x \in \mathbb{R} \), then find \( f(2023) \).
In \( \Delta ABC \), if \( \angle C = 90^\circ \), then \( \left( \frac{I_1 - I_3}{I_1} \right) \left( \frac{I_2 - I_3}{I_2} \right) = \):
If \( \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = \):
If \( \alpha, \beta \) are acute angles such that \( \sin \beta = 2 \sin \alpha \) and \( 3 \cos \beta = 2 \cos \alpha \), then \( \sec (\alpha + \beta) \) is:
If \(\int \frac{dx}{1 + \sin x} = \tan \left( \frac{x}{2} - \theta \right) + C\), then \(\theta =\):
In \( \Delta ABC \), \( \left( \tan \frac{A}{2} + \tan \frac{B}{2} \right) \tan \frac{C}{2} = \):
If the graph of the anti derivative \( g(x) \) of \( f(x) = \log(\log x) + (\log x)^{-2} \) passes through \( (e, 2023 - e) \) and the term independent of \( x \) in \( g(x) \) is \( k \), then the sum of all the digits of \( k \) is
If \( g(x) \) is an antiderivative of \( f(x) = 1 + 2 \log 2 \) and the graph of \( y = g(x) \) passes through the point \( \left( -1, -\frac{1}{2} \right) \), then the curve meets the Y-axis at
If \[ \int \frac{x^3}{\sqrt{1 + x^2}} \, dx = A(1 + x^2)^{\frac{3}{2}} + B(1 + x^2)^{\frac{1}{2}} + C, \text{ then } A + B = \]
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
Evaluate the integral: \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\cot x}} \, dx = \]
If \( 5f(x) + 3f\left( \frac{1}{x} \right) = 2 - \frac{1}{x}, x \ne 0 \), then \( \int_1^2 f\left( \frac{1}{x} \right) dx = \)
\(\int \frac{8^{1+x} + 4^{1+x}}{2^{2x}} dx =\):
Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]
If \[ f(x) = \lim_{n \to \infty} n^2 \left( \frac{1}{x^n} - \frac{1}{x^{n+1}} \right), x>0, \] then \[ \int x f(x) \, dx = ? \]
Evaluate the following integral: \[ \int \frac{1 - \cos x}{\cos x(1 + \cos x)} \, dx \]
The area bounded by the curves \( y = x^2 \) and \( y - 6 = -|x| \) is:
If \( m \in \mathbb{Z}^+, n = 2m \) and \[ \int_0^{\frac{\pi}{2}} \sin^n x \cos^m x \, dx = K(m) \int_0^{\frac{\pi}{2}} \sin^m x \, dx, \text{ then } \frac{2^{n-1}(m - 1)!}{(2m - 1)!} \cdot K(m) = \]