List of top Questions asked in AP EAPCET- 2023

If eight coins are tossed simultaneously, then the probability of getting at least six heads is:
If \( \alpha \) and \( \beta \) are the roots of \( x^2 - ax - b = 0 \), and \( \alpha^2 + \beta^2 \) and \( \alpha^3 + \beta^3 \) are the roots of \( Ax^2 + Bx + C = 0 \), then \( C = \ldots \)
The minimum value of \( f(x) = \frac{x^2 - 2x + 3}{x^2 - 4x + 7} \) is:
If $X_1, X_2, ..., X_n$ are $n$ independent events such that $P(X_r) = \frac{1}{r+1}$, for $r=1, 2, ..., n$, then the probability that none of the $n$ events occur is
If the circles $x^2+y^2 - 4x + 2fy + 1 = 0$ and $x^2+y^2 + 2gx - 4y - 1 = 0$ cut orthogonally, then $r_1^2 + r_2^2 - 8=$
The number of ordered pairs $ (x, y) $ for which $ A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 2 & x \\ y & 1 & 2 \end{pmatrix} $ is a singular and symmetric matrix is:
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 \( P \)
If \( f(x) = \begin{cases} x^2, & 0 \le x<1 \\ \sqrt{x}, & 1 \le x \le 2 \end{cases} \), then \( \int_0^2 f(x) dx = \)
If \( \alpha \) is the real root and \( \beta \), \( \gamma \) are the other roots of the equation \( x^3 - a^3 = 0 \, (a>0) \), then the number of common points of the curves given by \( |z - \beta| = \frac{\sqrt{3a}}{2} \) and \( |z - \gamma| = \frac{\sqrt{3a}}{2} \) is:
If \[ f(t) = \frac{t}{2} - \frac{1}{4} \log(2t - 1), \] then \[ f'\left( \frac{t + 1}{2t + 1} \right) = ? \]
Let X-axis be the transverse axis and Y-axis be the conjugate axis of a hyperbola H. Let \( x^2 + y^2 = 16 \) be the director circle of H. If the perpendicular distance from the centre of H to its latus rectum is \( \sqrt{34} \), then \( a + b = \):
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular drawn from the origin to this line makes an angle of 60° with the line \( x + y = 0 \). Then the equation of the line L is:
If the coordinates of the point of contact of the circles \( x^2 + y^2 - 4x + 8y + 4 = 0 \) and \( x^2 + y^2 + 2x = 0 \) is \( (a, b) \), then \( a + 2b \) is:
If A and B are events of a random experiment with \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.3 \), then the probability that neither A nor B occurs is:
If \( P \left( \frac{\pi}{3} \right) \) and \( Q \left( \frac{2\pi}{3} \right) \) represent two points on the circle \( x^2 + y^2 - 4x - 6y - 12 = 0 \) in parametric form, then the length of the chord PQ is:
Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin \left( \frac{\pi}{4} + x \right) + \sin \left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \]

If $\int \sqrt{x}(1-x^3)^{-1/2} dx = \frac{2}{3}g(f(x))+c$, then

If $ N(n) = n \prod_{r=1}^{2023} (n^2 - r^2) $ where $ n > 2023 $, then the value of $ {}^{n}C_{N-1} $ when $ n = 2024 $ is:

Let $ \bar{a}, \bar{b}, \bar{c} $ be three non-coplanar vectors. Then the point of intersection of the line joining the points $ \mathbf{a} + \bar{b} + \bar{c} $, $ \bar{a} - \bar{b} + 3\bar{c} $ and the line joining the points $ 2\bar{a} - \bar{b} + \bar{c} $, $ \bar{a} - 2\bar{b} + 4\bar{c} $ is
Let $ M \left( \frac{-7}{2}, \frac{-5}{2} \right) $ be the midpoint of the chord $ AB $ of the circle. The equation of the circle is $ x^2 + y^2 + 10x + 8y - 23 = 0 $. If $ ax + by + 1 = 0 $ is the equation of $ AB $, then $ 3a + 3b = $:
The coefficient of \( x^r \) in the expansion of \( \frac{1}{\sqrt{(1 - 2x)^3}} \) is
If one ticket is selected at random from 30 tickets each with a distinct number from 1 to 30, then the probability that the number on the selected ticket is a multiple of 3 or 5 is
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 \( R + r = \):
If A and B are among 20 persons who sit at random along a round table, then the probability that there are exactly six persons between A and B is: