List of top Questions asked in AP EAPCET- 2023

The lines \( L_1: y - x = 0 \) and \( L_2: 2x + y = 0 \) intersect the line \( L_3: y + 2 = 0 \) at points \( P \) and \( Q \) respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersects \( L_3 \) at \( R \). Statement-1: \( PR : RQ = 2\sqrt{2} : \sqrt{5} \) Statement-2: In any triangle, the bisector of an angle divides the triangle into two similar triangles.
In \( \triangle ABC \), if the midpoints of the sides \( AB, BC, CA \) are respectively \( (l, 0, 0), (0, m, 0), (0, 0, n) \), then: \[ \frac{AB^2 + BC^2 + CA^2}{l^2 + m^2 + n^2} = ? \]
The equation of the plane passing through the point \( (1, 2, 2) \) and perpendicular to the planes \[ x - y + 2z = 3 \quad \text{and} \quad 2x - 2y + z + 12 = 0 \] is:
If \( t \in \mathbb{R} \setminus \{-1\} \), then the locus of the point \[ \left( \frac{3at}{1 + t^3}, \frac{3at^2}{1 + t^3} \right) \] is:
The centre of a square of side 4 units length is \( (3, 7) \), and one of the diagonals is parallel to the line \( y = x \). If \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) are the vertices of this square, then \[ \frac{y_1 y_2 y_3 y_4}{x_1 x_2 x_3 x_4} = ? \]
An analysis of monthly wages paid to the workers of two jute mills A and B gives the following data:
\begin{tabular}{|c|c|c|} \hline & Mill - A & Mill - B
\hline No. of workers & 500 & 600
Average daily wage (in rupees) & 186 & 175
Variance of distribution of wages & 81 & 100
\hline \end{tabular} Then:
A, B, C are three horses participating in a race. The probability of horse A to win is twice that of B, and the probability of B to win is twice that of C. Then the probabilities of A, B, and C winning the race are respectively:
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 the position vectors of the points \( A \) and \( B \) are \( 2i + 3j - k \) and \( i - j + 2k \) respectively, then the unit vector along \( \overrightarrow{BA} \) and in the direction of \( AB \) is:
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 \( r_1 - r = \frac{a}{3} \) and \( r_2 - r = \frac{b}{3} \), then \( r_1 + r_2 - r \) is:
If \( 10 \sin^4 \alpha + 15 \cos^4 \alpha = 6 \), then \( 16 \tan^6 \alpha + 27 \cot^6 \alpha \) is:
In \( \triangle ABC \), if \( \cos^2 A + \cos^2 B + \cos^2 C = 1 \), then \( \triangle ABC \) is:
If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^3 - 3(2x^2) + 32 = 0 \) with \( \beta<1 \), then \( 2\alpha + 3\beta \) is:
If \( \alpha \), \( \beta \), and \( \gamma \) are the roots of the equation \( x^3 - ax^2 + bx - c = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is:
If \( C_j \) stands for \( ^nC_j \), then \[ \frac{C_0}{2} + \frac{C_1}{2.2^2} + \frac{C_2}{3.2^3} + \dots + C_n = \frac{3^n}{2^{n+1} (n+1)} \]
If \( \alpha_1, \alpha_2, \alpha_3 \) are the roots of the equation \( x^3 + 3x + 2 = 0 \), then \( \alpha_1^5 + \alpha_2^5 + \alpha_3^5 \) is:
Find the value of \( (3 + \sqrt{8})^5 + (3 - \sqrt{8})^5 \):
If \( z_1 = 2 + 5i \), \( z_2 = -1 + 4i \) and \( z_3 = i \), then
\[ \frac{|z_1 - z_3|}{|z_3 - z_2|} = ? \]
The locus of the variable point \( z = x + iy \) whose amplitude is always equal to \( \theta \), is:
For \( x \in \mathbb{R} \), the minimum value of \( \frac{x^2 + 2x + 5}{x^2 + 4x + 10} \) is:
If \( f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x} \) for \( x \in \mathbb{R} \), then find \( f(2023) \).
The rank of the matrix \( A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \end{bmatrix} \) is:
If \( f(0) = 0, f(1) = 1, f(2) = 2 \) and \( f(x) = f(x-2) + f(x-3) \) for \( x = 3, 4, 5, \dots \), then find \( f(10) \).