List of top Questions asked in AP EAPCET- 2023

What is ‘X’ in the following reaction?
2–Bromo–2–methyl butane $\xrightarrow{X}$ 2–methyl–2–butanol

If two subsets $ A $ and $ B $ are selected at random from a set $ S $ containing $ n $ elements, then the probability that $ A \cap B = \emptyset $ and $ A \cup B = S $ is:

If $ \tan \left( \frac{\alpha + \beta}{2} \right) $, $ \cos (\alpha + \beta) $, $ \sin (\alpha + \beta) $, and $ \tan (\alpha + \beta) $ are matched with their values, then the correct matching 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:

Let the circle $ S: x^2 + y^2 + 2gx + 2fy + c = 0 $ cut the circles $ x^2 + y^2 - 2x + 2y - 2 = 0 $ and $ x^2 + y^2 + 4x - 6y + 9 = 0 $ orthogonally. If the centre of the circle $ S = 0 $ lies on the line $ 2x + 3y - 2 = 0 $, then $ 2g + f = $

In $ \triangle ABC $, the coordinates of the vertices are $ A(1,8,4) $, $ B(0,-11,4) $, and $ C(2,-3,1) $. If $ D $ is the foot of the perpendicular drawn from $ A $ to $ BC $, then the coordinates of $ D $ are:

The distance between two parallel planes $ ax + by + cz + d_1 = 0 $ and $ ax + by + cz + d_2 = 0 $ is given by $ \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} $. If the plane $ 2x - y + 2z + 3 = 0 $ has the distances $ \frac{1}{3} $ units from the planes $ 4x - 2y + 4z + \lambda = 0 $ and $ 2x - y + 2z + \mu = 0 $ respectively, then the maximum value of $ \lambda + \mu $ is:

Let $ S $ be the circumcircle of the triangle formed by the line $ x - 2y - 4 = 0 $ with the coordinate axes. If $ P(-2, -4) $ is a point in the plane of the circle $ S $ and $ Q $ is a point on $ S $ such that the distance between $ P $ and $ Q $ is the least, then $ PQ = $

If the chord of contact of the point $ P(h, k) $ with respect to the circle $ x^2 + y^2 - 4x - 4y + 8 = 0 $ meets the circle in two distinct points and it also makes an angle $ 45^\circ $ with the positive X-axis in the positive direction, then $ (h, k) $ cannot be:

The equation of the pair of tangents drawn from the point $ (1, 1) $ to the circle $ x^2 + y^2 + 2x + 2y + 1 = 0 $ is:

The combined equation of the lines passing through the point $ (3, 4) $ and each making an angle of 45° with the line $ x + y + 1 = 0 $ is:

The equal sides of an isosceles triangle are given by equations $ 7x - y + 3 = 0 $ and $ x + y - 3 = 0 $. If the slope $ m $ of the third side is an integer, then $ m = $:

If the lines given by $ (x^2 + y^2) \sin^2 \alpha = (x \cos \alpha - y \sin \alpha)^2 $ are perpendicular to each other, then $ \sin^2 \alpha + \tan^2 \alpha = $:

If the area of the triangle formed by the lines $ y = x + c $ and $ 2x^2 + 5xy + 3y^2 = 0 $ is $ \frac{1}{20} $ sq. units, then $ c = $

Let $ \overline{OA} = 2\overline{i} - 3\overline{j} + \overline{k} $, $ \overline{OB} = \overline{i} - 4\overline{j} - 3\overline{k} $, and $ \overline{OC} = -3\overline{i} + \overline{j} + 2\overline{k} $ be the position vectors of three points A, B, C respectively. If G is the centroid of triangle ABC, then find: \[ BC^2 + CA^2 + AB^2 + 9(OG)^2 \]

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:

Bag $ B_1 $ contains 4 white and 2 black balls. Bag $ B_2 $ contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is:

If \[ \frac{2x^2 + 5x + 6}{(x + 2)^3} = \frac{a}{x + 2} + \frac{b}{(x + 2)^2} + \frac{c}{(x + 2)^3}, \] then $ a \cdot b + b \cdot c + c \cdot a = $?

If $ \mathbf{a} $ and $ \mathbf{b} $ are two vectors such that $ \mathbf{a} $ is not parallel to $ \mathbf{b} $, and if \[ p = (x + 2y + 3)\mathbf{a} + (5x - y + 2)\mathbf{b}, \quad q = (2x + 3y + 5)\mathbf{a} + (x - 5y - 2)\mathbf{b} \] are two vectors such that $ p = 2q $, then find $ x - 2y $.

$3\mathbf{\overline{i}} - 2\mathbf{\overline{j}} - \mathbf{\overline{k}}, -2\mathbf{\overline{i}} - \mathbf{\overline{j}} + 3\mathbf{\overline{k}}, -\mathbf{\overline{i}} + 3\mathbf{\overline{j}} - 2\mathbf{\overline{k}}$ are the position vectors of the vertices $ A $, $ B $, and $ C $ of a triangle $ ABC $respectively. If $ H $ is its orthocenter, then find $ \overline{HA} + \overline{HB} + \overline{HC} $.

Let $ \pi_1 $ be the plane determined by the vectors $ \overline{\mathbf{i}} + 2\overline{\mathbf{j}} - 2\overline{\mathbf{k}} $ and $ \overline{\mathbf{i}} + 3\overline{\mathbf{j}} - 2\overline{\mathbf{k}} $. Let $ \pi_2 $ be the plane determined by the vectors $ \overline{\mathbf{j}} + 2\overline{\mathbf{k}} $ and $ 3\overline{\mathbf{k}} - 2\overline{\mathbf{i}} $. If $ \theta $ is the angle between $ \pi_1 $ and $ \pi_2 $, then $ \cos \theta = $

If $ \sec(\theta + \alpha) $, $ \sec(\theta) $, and $ \sec(\theta - \alpha) $ are in arithmetic progression, then $ \sin^2 \theta = $

If $ \alpha = \log_e(2 + \sqrt{3}) $, then evaluate: \[ \frac{\cosh \alpha }{1 - \tanh \alpha} + \frac{\sinh \alpha}{1 - \coth \alpha} \]

Evaluate the following expression: \[ \sin 21^\circ \cos 9^\circ - \cos 84^\circ \cos 6^\circ \]

The number of all possible solutions of the equation $ z^3 + z = 0 $ is: