List of top Mathematics Questions asked in AP EAPCET

Evaluate: $$ \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = ? $$

Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?

Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$

If $$ \int_1^3 x^n \sqrt{x^2 - 1} \, dx = 6 $$ Then find the value of $ n $.

The line joining the points $ (0, 3) $ and $ (5, -2) $ is a tangent to the curve: $$ y = \frac{C}{x + 1} $$ Then find $ C $.

The point on the curve $ y = x^2 + 4x + 3 $ that is closest to the line $ y = 3x + 2 $ is:

If $ x = f(\theta) $, $ y = g(\theta) $, then find: $$ \frac{d^2 y}{dx^2} $$

If the normal drawn at a point $ P $ on the curve $ 3y = 6x - 5x^3 $ passes through the origin $(0, 0)$, then the positive integral value of the abscissa of point $ P $ is:

The function $ y = x^3 - ax^2 + 48x + 7 $ is increasing for all real values of $ x $. Find the interval in which $ a $ lies:

If $ a, b>0 $, then the minimum value of: $$ y = \frac{b^2}{a - x} + \frac{a^2}{x},\quad \text{for } 0<x<a $$

Given: $$ \int \frac{3x + 4}{x^3 - 2x + 4} \, dx = \log f(x) + C $$ Then: $$ f(3) = ? $$

Let plane $ \pi $ pass through $ (1, 0, 1) $ and be perpendicular to both: - $ 2x + 3y - z = 2 $ - $ x - y + 2z = 1 $ Let another plane pass through point $ (11, 7, 5) $, be parallel to $ \pi $, and have the form: $$ ax + by - z + d = 0 $$ Then evaluate: $$ \frac{a}{b} + \frac{b}{d} $$

Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$

Given: $$ 3f(\cos x) + 2f(\sin x) = 5x $$ Find: $$ f'(\cos x) + f'(\sin x) $$

Evaluate: $$ \lim_{x \to \infty} \log_e(\cosh x) + x $$

If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?

Evaluate: $$ \lim_{x \to \infty} \frac{3|x| - x}{|x| - 2x} - \lim_{x \to 0} \frac{\log(1 + x^3)}{\sin^3 x} $$

Find the radical centre of the three circles: $$ C_1: x^2 + y^2 - 1 = 0,\quad C_2: x^2 + y^2 - 8x + 15 = 0,\quad C_3: x^2 + y^2 + 10y + 24 = 0 $$

Let point $ P $ divide the line segment joining $ A(1, 2, -1) $ and $ B(-1, 0, 1) $ externally in the ratio $ 1:2 $. Given point $ Q = (1, 3, -1) $, find $ PQ $.

Which of the following equations represents a **parabola**?

If the angle between the lines joining the foci and the ends of the minor axis of the ellipse $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $ 90^\circ $, then its **eccentricity** is:

If the direction cosines of a line are: $$ \left( \frac{a}{\sqrt{83}},\ \frac{5}{\sqrt{83}},\ \frac{c}{\sqrt{83}} \right) $$ and it is given that $ c - a = 4 $, find $ ca $.

Find the locus of the point of intersection of tangents drawn at the ends of a **normal chord** of the hyperbola: $$ x^2 - y^2 = a^2 $$

If $ e_1 $ and $ e_2 $ are the eccentricities of the hyperbola $$ 16x^2 - 9y^2 = 1 $$ and its conjugate respectively, then find $ 3e_1 $ in terms of $ e_2 $.

Three points $ A, B, C $ in the plane have: - x-coordinates in G.P. - y-coordinates in the same G.P. Then, the points $ A, B, C $ lie on: