List of top Questions asked in AP EAPCET

The general solution of the differential equation $$ (y^2 + x + 1) \, dy = (y + 1) \, dx $$ is:

If \[ \cos \frac{ \pi }{8} + \cos \frac{3 \pi }{8} + \cos \frac{5 \pi }{8} + \cos \frac{7 \pi }{8} = k, \] then evaluate \[ \sin^{-1} \left( \frac{k}{\sqrt{2}} \right) + \cos^{-1} \left( \frac{k}{3} \right)= \]

A line $ L $ intersects the lines $ 3x - 2y - 1 = 0 $ and $ x + 2y + 1 = 0 $ at the points $ A $ and $ B $. If the point $ (1,2) $ bisects the line segment $ AB $ and $ \frac{a}{b} x + \frac{b}{a} y = 1 $ is the equation of the line $ L $, then $ a + 2b + 1 = ? $

Evaluate the expression: \[ \frac{\sin 1^\circ + \sin 2^\circ + \dots + \sin 89^\circ}{2(\cos 1^\circ + \cos 2^\circ + \dots + \cos 44^\circ) + 1} =\]

Let $ \overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w} $ be three unit vectors. Let $ \overrightarrow{p} = \overrightarrow{u} + \overrightarrow{v} + \overrightarrow{w} $, $ \overrightarrow{q} = \overrightarrow{u} \times (\overrightarrow{p} \times \overrightarrow{w}) $. If $ \overrightarrow{p} \cdot \overrightarrow{u} = \frac{3}{2} $, $ \overrightarrow{p} \cdot \overrightarrow{v} = \frac{7}{4} $, $ |\overrightarrow{p}| = 2 $, and $ \overrightarrow{v} = K \overrightarrow{q} $, then $ K = ? $

If $ P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e $ is a polynomial such that: \[ P(0) = 1, \quad P(1) = 2, \quad P(2) = 5, \] \[ P(3) = 10, \quad P(4) = 17, \] then find the value of $ P(5) $=

Evaluate the expression: \[ \frac{\cos 10^\circ + \cos 80^\circ}{\sin 80^\circ - \sin 10^\circ}. \]

If $ 2x^2 + 3x - 2 = 0 $ and $ 3x^2 + \alpha x - 2 = 0 $ have one common root, then the sum of all possible values of $ \alpha $ is:

The distance from a point $ (1,1,1) $ to a variable plane $\pi$ is 12 units and the points of intersections of the plane with X, Y, Z-axes are $ A, B, C $ respectively. If the point of intersection of the planes through the points $ A, B, C $ and parallel to the coordinate planes is $ P $, then the equation of the locus of $ P $ is:

Two digits are selected at random from the digits 1 through 9. If their sum is even, then the probability that both are odd is:

If $ y = x - x^2 $, then the rate of change of $ y^2 $ with respect to $ x^2 $ at $ x = 2 $ is:

If a function $f(x)$ is defined as: $$ f(x) = \begin{cases} \frac{\tan(4x) + \tan 2x}{x} & \text{if } x> 0 \\ \beta & \text{if } x = 0 \\ \frac{\sin 3x - \tan 3x}{x^2} & \text{if } x < 0 \end{cases} $$ and is continuous at $x = 0$, then find $|\alpha| + |\beta|$.

If $\alpha$ and $\beta$ are two double roots of the equation: $$ x^2 + 3(a + 3)x - 9a = 0 $$ for different values of $a$ (where $\alpha > \beta$), then the minimum value of the equation: $$ x^2 + \alpha x - \beta = 0 $$ is:

If power of a point $ (4,2) $ with respect to the circle $ x^2 + y^2 - 2x + 6y + a^2 - 16 = 0 $ is 9, then the sum of the lengths of all possible intercepts made by such circles on the coordinate axes is

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:

The angle between the curves $ y^2 = 2x $ and $ x^2 + y^2 = 8 $ is

If $ A(1,0,2) $, $ B(2,1,0) $, $ C(2,-5,3) $, and $ D(0,3,2) $ are four points and the point of intersection of the lines $ AB $ and $ CD $ is $ P(a,b,c) $, then $ a + b + c = ? $

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:

Let $ f(x) = 3 + 2x $ and $ g_n(x) = (f \circ f \circ \dots \text{n times}) (x) $. If for all $ n \in \mathbb{N} $, the lines $ y = g_n(x) $ pass through a fixed point $ (a, \beta) $, then $ \alpha + \beta = ? $

Sum of the positive roots of the equation: \[ \begin{vmatrix} x^2 + 2x + 2 & x + 2 & 1 \\ 2x + 1 & x - 1 & 1 \\ x + 2 & -1 & 1 \end{vmatrix} = is \; 0. \]

If the sum of two roots of $ x^3 + px^2 + qx - 5 = 0 $ is equal to its third root, then $ p(q^2 - 4q) = $ ?

The coefficient of $ x^5 $ in the expansion of $ \left( 2x^3 - \frac{1}{3x^2} \right)^5 $ is:

Evaluate the integral: \[ \int \frac{2x^2 - 3}{(x^2 - 4)(x^2 + 1)} \,dx = A \tan^{-1} x + B \log(x - 2) + C \log(x + 2) \] Given that, \[ 64A + 7B - 5C = ? \]

Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \]

The number of ordered pairs $ (x, y) $ satisfying the equations: \[ \sin x + \sin y = \sin (x + y) \quad \text{and} \quad |x| + |y| = 1. \]