List of top Questions asked in AP EAPCET

A car covers a distance at a speed of 60 km/h. It returns to the starting point at a speed of \( V \). If the average speed for the entire trip is 48 km/h, then what is the magnitude of \( V \)?
A car travels with a speed of 40 km/h. Rain drops fall vertically but the traces on the side window make an angle of \( 30^\circ \) with the vertical. Find the magnitude of rain’s velocity with respect to the car.
\( \lim_{n \to \infty} \left( \frac{1}{1 + n^5} + \frac{2^4}{2^5 + n^5} + \frac{3^4}{3^5 + n^5} + \ldots + \frac{n^4}{n^5 + n^5} \right) = \)
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)
The parametric form of a curve is \( x = \frac{t^3}{t^2 - 1},\ y = \frac{t}{t^2 - 1} \), then \( \int \frac{dx}{x - 3y} = \)
\( \int_0^1 \alpha^k x^k dx = \)
\( \int_0^{\pi} x \left( \sin^2(\sin x) + \cos^2(\cos x) \right) dx = \)
\( \int \frac{e^{2x}}{\sin^2 x} \left( 2 \log \csc x + \sin 2x \right) dx = \)
Let \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) be roots of \( 18x^2 - 9\pi x + \pi^2 = 0 \), \( f(x) = x^2, g(x) = \cos x \). Then
\[ \int_{\alpha}^{\beta} x (g \circ f(x)) \, dx = ? \]
Assertion (A): \( I_n = \int \cot^n x \, dx \), then \( I_6 + I_4 = -\frac{\cot^5 x}{5} \)
Reason (R): \( \int \cot^n x \, dx = \frac{-\cot^{n-1} x}{n - 1} - \int \cot^{n - 2} x \, dx \)
The condition that \( f(x) = ax^3 + bx^2 + cx + d \) has no extreme value is
If \( ax + by = 1 \) is a normal to the parabola \( y^2 = 4px \), then the condition is
If the curves \( y = x^3 - 3x^2 - 8x - 4 \) and \( y = 3x^2 + 7x + 4 \) touch each other at a point \( P \), then the equation of the common tangent at \( P \) is
If \( x^3 - 2x^2y + 5x + y - 5 = 0 \), then at (1,1), \( y'(1) = \)
If $(a, b, c)$ are the direction ratios of a line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$, then the point $P = (a, 3b, 2c)$ lies on the plane
If $\displaystyle \lim_{x \to 0} \dfrac{|x|}{\sqrt{x^4 + 4x^2 + 5}} = k$, $\displaystyle \lim_{x \to 0} x^4 \sin\left(\dfrac{1}{3\sqrt{x}}\right) = l$, Then $k + l = $
If $\displaystyle \lim_{x \to \infty} x^n \log_e x = 0$, then $\log_e 12 =$
If $f(x) = \cot^{-1}\left( \dfrac{x^n + x^{-n}}{2} \right)$, then $f'(1) = $
The x-intercept of a plane $\pi$ passing through the point $(1, 1, 1)$ is $\dfrac{5}{2}$ and the perpendicular distance from the origin to the plane $\pi$ is $\dfrac{5}{7}$. If the y-intercept of the plane $\pi$ is negative and the z-intercept is positive, then its y-intercept is
If the point $(a, 8, -2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m:n$, then $\dfrac{2m}{n} - \dfrac{a}{3} =$
Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function satisfying $f(x) - x = \lambda$ (constant), $\forall x \in \mathbb{R}^+$ and $f(xf(y)) = f(y) + x$, $\forall x, y \in \mathbb{R}^+$. Then $\displaystyle \lim_{x \to 0} \dfrac{(f(x))^{\frac{5}{3}} - 1}{(f(x))^{\frac{2}{3}} - 1}$ is
Suppose a parabola with focus at $(0,0)$ has $x - y + 1 = 0$ as its tangent at the vertex. Then the equation of its directrix is
The eccentric angle of a point on the ellipse $x^2 + 3y^2 = 6$ lying at a distance of 2 units from its centre is
The length of the intercept on the line $4x - 3y - 10 = 0$ by the circle $x^2 + y^2 - 2x + 4y - 26 = 0$ is
The locus of a variable point whose chord of contact w.r.t. the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ subtends a right angle at the origin is