List of top Questions asked in AP EAPCET- 2023

Let the point $ L $ lying in the first quadrant be one end of a latus rectum of the ellipse \[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \] Let $ P $ and $ Q $ be the points where the normal drawn at $ L $ meets the major and minor axes. Then the distance between $ P $ and $ Q $ is:

Let the equation \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] represent a point circle (not at the origin). Then which one of the following conditions must hold?

Let $ \ell $ be the directrix of the parabola $ 9y^2 + 12y + 9x - 14 = 0 $, and $ \ell_1 $ be the line passing through the vertex of this parabola and the origin. If $ (h, k) $ is the point of intersection of $ \ell $ and $ \ell_1 $, then $ h + k = $:

Let $ f(x) = \frac{x}{\sqrt{1- x}}$ + $\frac{\sqrt{1- x}}{x}$. If $ \lim_{x _ 1^-} f(x) = l $ and $ \lim_{x \to m} f(x) = \frac{5}{2} $, then the set of all possible finite values of $ l $ and $ m $ is

If $ \alpha $ and $ \beta $ are the roots of the equation $ ax^2 + bx + c = 0 $, then the equation whose roots are $ \alpha + \beta $ and $ \frac{1}{\alpha} + \frac{1}{\beta} $ is

If $ \alpha $ and $ \beta $ are the roots of the equation $ 2x^2 - 4x + 3 = 0 $, then $ \frac{2(\alpha^4 + \beta^4) + 3(\alpha^2 + \beta^2)}{\alpha + \beta} = $

If the roots of the equation $ 16x^3 - 44x^2 + 36x - 9 = 0 $ are in harmonic progression, then its greatest root is

If $ \frac{x^4 - 6x^3 + 9x^2 + 5x - 20}{x^2 - x - 2} = f(x) + \frac{a}{x + p} + \frac{b}{x + q} $, then $ 2(a + b) = $

If $ \triangle ABC $ is a right-angled isosceles triangle and $ \angle C = 90^\circ $, then $ r = \frac{1}{5} $ is:

The sides of a triangle are $ 3x + 2y - 6 = 0 $, $ 2x - 3y + 6 = 0 $ and $ x + 2y + 2 = 0 $. If $ P(0, b) $ lies either on the triangle or inside the triangle, then $ b $ lies in the interval

Let origin be the centroid of an equilateral triangle ABC and one of its sides is along the straight line $ x + y = 3 $. If $ R $ and $ r $ are its circumradius and inradius respectively, then $ R + r = $:

In $ \triangle ABC $, if $ a, b, c $ are 5, 12, and 13 respectively, then $ b^2 \sin 2C + c^2 \sin 2B $ is:

In $ \triangle ABC $, if $ (a - b)(s - c) = (b - c)(s - a) $, then $ r_1, r_2, r_3 $ are in:

In $ \triangle ABC $, if \[ a \cos^2 \frac{C}{2} + \cos^2 \frac{A}{2} = \frac{3b}{2}, \] then $ a + c : b $ is:

In $ \triangle ABC $, if $ r_1 = 2r_2 = 3r_3 $, then

In $ \triangle ABC $, $ \angle B = 60^\circ $ and $ \angle A = 75^\circ $. If a point $ D $ divides $ BC $ in the ratio $ 2:3 $, then $ \sin \angle BAD : \sin \angle CAD = $

Let $ A = \begin{pmatrix} 0 & 3 & 5 & -7 \\ 8 & 0 & -1 & 0 \\ 6 & -1 & 0 & 0 \end{pmatrix} $ and $ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $. If $ D = [\alpha \, \beta \, \gamma]^T $ is the solution of $ X^T B^T = A^T X $, then $ D^T A = $

The rank of the matrix \[ \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 1 & -1 & 4 \\ 2 & 2 & 8 \\ \end{pmatrix} \] is

If $ S = \begin{bmatrix} 0 & 1 \\ 0 & 1 \\ 1 & 0 \end{bmatrix} $ and $ A = \frac{1}{2} \begin{bmatrix} b + c & c - a \\ a + b & b - c \end{bmatrix} $, then $ SAS^{-1} $ is:

If $ A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}, B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, C = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} $, then the expression $ A^2 + B^2 + C^2 $ is:

Let $ A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $, where $ \theta \in \mathbb{R} $. Then $ A^n $, for any positive integer $ n $, is equal to:

If there exists a $ k^{th} $ order non-singular sub-matrix in a matrix $ P $ of order $ m \times n $, then the rank $ (\rho) $ of $ P $

Let $ A = \begin{bmatrix} 2 & 1 & 3 & -1 \\1 & -2 & 2 & -3 \end{bmatrix}, B = \begin{bmatrix} 2 & 1 & 0 & 3 \\1 & -1 & 2 & 3 \end{bmatrix} $, and the equation $ 2A + 3B - 5C = 0 $. Find the matrix $ C $.

A matrix whose elements $ a_{ij} $ are defined by $ a_{ij} = \frac{1}{3}|i - 5j|, \; i, j = 1, 2, 3 $ is

If $ P, Q $ and $ R $ are $ 3 \times 3 $ matrices such that} \[ P = \begin{bmatrix} 3x^2 + x + 3 & 2x^2 - x + 4 & 7x^2 + 8x + 5 \\ 5x^2 + 3x + 2 & 4x^2 - 2x - 1 & 7x^2 + 5x + 8 \\ 3x^2 + 2x + 5 & 4x^2 - x - 2 & 3x^2 + 8x + 7 \end{bmatrix} = Px^2 + Qx + R \] then det $ R = $ ?