List of top Questions asked in AP EAPCET

Two bodies of masses 12 kg and 6 kg are projected simultaneously with velocities 15 ms\(^{-1}\) and 20 ms\(^{-1}\) respectively from the top of a tower of height 25 m. The body of mass 12 kg is projected vertically upwards and the body of mass 6 kg is projected horizontally. The maximum height reached by the centre of mass of the system of two bodies from the ground is:
A motor boat is moving in a river with velocity $\vec{v} = 7\hat{i} + 2\hat{j} - 5\hat{k}$ kms$^{-1}$. If the flow water offers resistive force $\vec{F} = 9\hat{i} + 3\hat{j} - 3\hat{k}$ N, then the power of the boat is
A truck of mass \( M \) and a car of mass \( \frac{M}{10} \) moving with the same momentum are brought to halt by the application of the same breaking force. The ratio of the distances travelled by the truck and car before they come to stop is:
The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):
If $y = y(x)$ is a particular solution of $\sqrt{1 - x^2} \frac{dy}{dx} + \frac{2x}{\sqrt{1 - x^2}} y = x$, $y(0) = 1$, then $y\left(\frac{1}{2}\right) =$
A car is travelling at \(30 \, \text{ms}^{-1}\) speed on a circular road of radius \(300 \, \text{m}\). If its speed is increasing at the rate of \(4 \, \text{ms}^{-2}\), then its acceleration is:
The degree of the differential equation $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$ is
The density of a substance is 4 g/cc. In a system in which the unit of length is 5 cm and the unit of mass is 20 g, the density of the substance is:
If $[\cdot]$ denotes the greatest integer function, then $\int_{0}^{1000} e^{x - \lfloor x \rfloor} dx =$
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
The area bounded by \( y - 1 = -|x| \) and \( y + 1 = |x| \) is:
Given that $\frac{d}{dx} \int_{0}^{\phi(x)} f(t) dt = f(\phi(x)) \phi'(x)$. For all $x \in (0, \frac{\pi}{2})$, if $\int_{1}^{\cos x} t^2 f(t) dt = \cos 2x$, then $f\left(\frac{1}{\sqrt{2}}\right) =$
\( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = \)
\(\int \frac{8^{1+x} + 4^{1+x}}{2^{2x}} dx =\):
If \( f(x) \) is a function such that \( f'(x) = \sqrt{f^2(x) - 1} \) and \( f(0) = 1 \), then \( f(1) = \):
If \(\int \frac{dx}{1 + \sin x} = \tan \left( \frac{x}{2} - \theta \right) + C\), then \(\theta =\):
The equation of the normal to the curve \( y = \cosh x \) drawn at the point nearest to the origin is:
Let \( n \in (0, \infty) \). If for all the curves \( y = x^n \log x \) for distinct values of \( n \), we have \( y = x - 1 \) as the tangent at a fixed point \((\alpha, \beta)\), then \(\alpha + \beta = \):
If all the normals drawn to the curve \( y = \frac{1 + 3x^2}{3 + x^2} \) at the points of intersection of \( y = \frac{1 + 3x^2}{3 + x^2} \) and \( y = 1 \) pass through the point \( (\alpha, \beta) \), then \( 3\alpha + 2\beta = \):
The locus of the point on the curve \( y = \sin x \) where the tangent drawn at that point always passes through the point \( (0, \pi) \) is:
\( f(x) \) and \( g(x) \) are differentiable functions such that \( \frac{f(x)}{g(x)} = \) a non-zero constant. If \( \frac{f'(x)}{g'(x)} = \alpha(x) \) and \( \left( \frac{f(x)}{g(x)} \right)' = \beta(x) \), then \( \frac{\alpha(x) - \beta(x)}{\alpha(x) + \beta(x)} = \):
Let \([ \cdot ]\) denote the greatest integer function.
Assertion (A): \(\lim_{x \to \infty} \frac{[x]}{x} = 1\)
Reason (R): \(f(x) = x - 1\), \(g(x) = [x]\), \(h(x) = x\) and \(\lim_{x \to \infty} \frac{f(x)}{x} = \lim_{x \to \infty} \frac{h(x)}{x} = 1\):
Let \( S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots + n \text{ terms} -\frac{1}{3}<x<\frac{1}{3} \). If \( f(x) = S_n \), then \( f(x) \) is discontinuous at the point \( x = \):
\(A(-2, 9)\) and \(B(1, 6)\) are two points on the curve \(y = x^2 + 5\). The coordinates of the point \(C\) on the curve such that the tangent drawn at \(A\) is parallel to the chord \(BC\) is:
If A and B are the points of intersection of the circles \( x^2 + y^2 - 4x + 6y - 3 = 0 \) and \( x^2 + y^2 + 2x - 2y - 2 = 0 \), then the distance between A and B is: