List of top Mathematics Questions asked in AP EAPCET

If $ S_n = \int_0^{\frac{\pi}{2}} \frac{\sin((2n-1)x)}{\sin x} \, dx $, and $ n $ is an integer, then $ S_{n+1} - S_n = $:
If $ a = 2n $ and $ b = 2m+1 $ for all $ m, n \in \mathbb{N} $, then \[ \int_{-\pi}^{\pi} e^{\sin^2 x} \cot^{b}(2n+1) x \, dx = \]
Let $ f : \mathbb{R} \to \mathbb{R} $ be a continuous function. If $ px + my + n = 0 $ is a tangent drawn to the curve $ y = f(x) $ at $ x = \alpha $, then at $ x = 0 $, the value of \[ \frac{d}{dx} \left( f(\alpha e^{2x}) \right) \]
In the interval $ \left( \frac{1}{e}, e \right) $, a decreasing function among the following functions is:
If the height of a cone of greatest volume that can be inscribed in a sphere of radius $ R $ is $ kR $, then the ratio of the volume of the cone to the volume of the sphere is:
If $ f(x) = \frac{x}{(1 + nx^n)^{1/n}} $ for $ n \geq 2 $, then \[ \int x^{n-2} f(x) \, dx = \]
The integral \[ \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Options:
$\int \frac{dx}{(x-1)\sqrt{x+2}} =$
The maximum value of $ a $ such that the second derivative of $ x^4 + ax^3 + \frac{3x^2}{2} + 1 $ is positive for all real $ x $ is
If a function $ f(x) $ defined on $ [a, b] $ is discontinuous at $ x = \alpha \in (a, b) $, then:
Assertion (A): If $ f(x) $ is not continuous at $ x = a $, then it is not differentiable at $ x = a $. Reason (R): If $ f(x) $ is differentiable at a point, then it is continuous at that point.
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:
Let $ S $ be a circle concentric with the circle $ 3x^2 + 3y^2 + x + y - 1 = 0 $. If the length of the tangent drawn from a point $ (2, -2) $ to the given circle is the radius of the circle $ S $, then the power of the point $ (2, 1) $ with respect to the circle $ S $ is
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:
Tangents are drawn from point $ (1, 1) $ to the ellipse $ x^2 + y^2 + 10x + 8y - 23 = 0 $. If $ m_1, m_2 $ (with $ m_1>m_2 $) are the slopes of these tangents, then with respect to the given ellipse, the point $ P(m_1, m_2) $ lies:
The line $ x + y + 2 = 0 $ intersects the circle $ x^2 + y^2 + 4x - 4y - 4 = 0 $ in two points $ A $ and $ B $. Let $ S = x^2 + y^2 + 2gx + 2fy + c = 0 $ be a different circle passing through the points $ A $ and $ B $. If the distance of the centre of $ S $ from $ AB $ is $ \sqrt{2} $, then $ g + f + c = $:
If the focal chord drawn through the point $ P(5, 5) $ to the parabola $ y^2 = 5x $ meets the parabola again at the point $ Q $, then the tangent drawn to this parabola at $ Q $ meets the axis of the parabola at the point:
If the planes $ 2x + 3y + 4z + 7 = 0 $ and $ 4x + ky + 8z + 1 = 0 $ are parallel, then the equation of the plane passing through the point $ (k, k, k) $ and having the direction ratios of its normal as $ (k-1, k, k+1) $ is
If the lines joining the origin to the points of intersection of $ 2x + 3y = k $ and $ 3x^2 - xy + 3y^2 + 2x - 3y - 4 = 0 $ are at right angles, then:
Let $ PQR $ be a right-angled isosceles triangle, right-angled at $ P(2,1) $. If the equation of the side $ QR $ is $ 2x + y = 3 $, then the equation of one of its sides other than $ QR $ is:
The coordinates of the point which divides the line joining the points $ (2, 3, 4) $ and $ (3, -4, 7) $ in the ratio 2 : 4 externally is:
Let $ M \left( \frac{-7}{2}, \frac{-5}{2} \right) $ be the midpoint of the chord $ AB $ of the circle. The equation of the circle is $ x^2 + y^2 + 10x + 8y - 23 = 0 $. If $ ax + by + 1 = 0 $ is the equation of $ AB $, then $ 3a + 3b = $:
If the inverse point of the point $ (3, 2) $ with respect to the circle $ x^2 + y^2 - 2x + 4y - 4 = 0 $ is $ (l, m) $, then $ 2l + 19m = $:
The number of persons joining a cinema ticket counter in a minute follows a Poisson distribution with parameter 6, then the probability that at least one and at most five persons join the queue in a particular minute is
If $ X $ is a random variable with the probability distribution \[ P(X = k) = \frac{(k+1)c}{2^k}, \quad k = 0, 1, 2, ..., \] then $ P(X \geq 3) $ is: