List of top Questions asked in AP EAPCET

Let ‘O’ be the origin. A and B be two points with position vectors \( -3\vec{i} - 3\vec{j} + 4\vec{k} \) and \( 4\vec{i} - 4\vec{j} - 3\vec{k} \) respectively. Let \( P \) be a point such that the line drawn through \( P \) parallel to \( OB \) meets \( OA \) in \( L \), and another line through \( P \) parallel to \( OA \) meets \( OB \) in \( M \). If \( L \) divides \( OA \) in the ratio 2:3 and \( M \) divides \( OB \) in the ratio 3:2, then the distance from \( O \) to \( P \) is:
If \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \), then \( 16 \tan^6 \alpha + 27 \cot^6 \alpha \) is:
Which one of the following functions is discontinuous at $x=1$?
If \[ \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} \, dx = Ax + B \log \left( 9e^{2x} - 4 \right) + C, \text{ then } (A, B) = \]
If \[ \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}, \] then the value of \( A + B = \):
Assertion (A): \(\displaystyle \int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x)\, dx\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\) Reason (R): \(\sin^6 x + \cos^6 x\) is a periodic function with period \(\dfrac{\pi}{2}\)
Given independent events A, B, and C with rtain intersection probabilities, find $P(A)$, $P(B)$, and $P(C)$.
Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]
The sum of the fourth powers of the roots of the equation \( 16x^2 - 10x + 1 = 0 \) is:
Let \( \vec{a} = 3\vec{i} + \vec{j} - 2\vec{k} \), \( \vec{b} = -5\vec{i} + 7\vec{j} \), and \( \vec{c} = 3\vec{i} + y\vec{j} \) be three vectors such that \( |\vec{a} - \vec{b} + \vec{c}| = \sqrt{141} \). If \( y_1 \) and \( y_2 \) are the values of \( y \) satisfying the given condition, then \( |y_1 - y_2| = \)
The variance of 20 observations is 5. If each one of the observations is multiplied by 2, then the variance of the resulting observations is:
In \( \triangle ABC \), if \( \sin B = \sin C \) and \( 3 \cos B = 2 \cos C \), then \( \triangle ABC \) is:
If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + x + 1 = 0 \), then the quadratic equation whose roots are \( \alpha^{2023} \) and \( \beta^{2012} \) is
Given that the solid obtained by rotating a rectangle about one of its sides is a cylinder. If the perimeter of a rectangle is 48 cm and the volume of the cylinder formed by rotating it is maximum, then the dimensions of that rectangle is
If the angle between the pair of lines \( x^2 + 2\sqrt{2} xy + ky^2 = 0, k>0 \) is \( 45^\circ \), then the area (in square units) of the triangle formed by the pair of bisectors of the angles between these lines and the line \( x + 2y + 1 = 0 \) is
If \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + B ] + K \] where \( K \) is the constant of integration, then \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + C ] + K. \]
If \( S \) is the set of all real values of \( a \) such that a plane passing through the points \( (-a^2, 1, 1), (1, -a^2, 1), (1, 1, -a^2) \) also passes through the point \( (-1, -1, 1) \), then \( S = \dots \)
The mean deviation from the mean for the data $ 6, 7, 10, 12, 13, 4, 12, 16 $ is:
$\int_0^\pi x \sin^3 x \cos^2 x\ dx =$
Let \[ f(x) = \begin{cases} \frac{5e^{|x|} + 2}{3 - e^{|x|}}, & x \ne 0 \\ 0, & x = 0 \end{cases} \] Then at \( x = 0 \), the function \( f(x) \) is:
If \( S \) is the sample space of a random experiment \( \xi \) and \( P \) is a probability function defined on the power set \( P(S) \) of \( S \), then which one of the following is not satisfied by \( P \)?
If \( A = \frac{\pi}{24} \), then \[ \frac{\cos A + \cos 3A + \cos 5A + \cos 7A}{\sin A + \sin 3A + \sin 5A + \sin 7A} \]
Let \( f(x) = \max\{\cos x, \sin x, 0\} \). If the number of points at which \( f(x) \) is not differentiable in \( (0, 2024\pi) \) is \( 1012k \), then \( k = \)
The equation of the normal to the curve \( y = \cosh x \) drawn at the point nearest to the origin is:
Let a focal chord \( 12x + 5y - 27 = 0 \) of the parabola \( y^2 = kx \) intersect the parabola at points \( P \) and \( P' \). If \( S \) is the focus of the parabola, then \( 9(SP + SP') \) = ?