List of top Questions asked in AP EAPCET

Given points \( A(-3, 3), B(1, 1), C(1, -1), D(-2, -2) \), find the angle between diagonals \( AC \) and \( BD \).

Suppose a point \( P \) moves such that: \[ BP^2 - AP^2 = 121 \] where \( A = (2, 5) \), \( B = (5, 11) \). Then the locus of \( P \) is a straight line. What is the slope of this line?

A discrete random variable \( X \sim B(n, p) \) (i.e., binomial distribution). Given: \[ P(X = 2) = P(X = 3) \] Then what is the mean of \( X \)?

There are 4 hotels in a town. If 3 men check into the hotels in a day, what is the probability that each checks into a different hotel?

Two distinct points are each at a unit distance from the line \( 4x + 3y - 10 = 0 \) and lie on the line \( x + y = 4 \). If they are separated by distance \( d \), then:

If variance of a Poisson distribution is 3, find: \[ P(1<X<4) \]

From a pack of 52 cards, 3 cards are drawn at random. What is the probability that one is an ace, one is a queen, and one is a jack?

Let \( P(x, y) \) lie on one of the lines: \[ \sqrt{3}x - y + 2 = 0 \quad \text{or} \quad \sqrt{3}x + y - 2 = 0 \] and suppose it is at a distance 5 units from their point of intersection. What is the distance from \( (0, 0) \) to the foot of the perpendicular from \( P \) onto the \( y \)-axis?

Three students \( X, Y, Z \) appear for an exam. Their passing probabilities are: \[ P(X) = \frac{1}{5},\quad P(Y) = \frac{1}{4},\quad P(Z) = 1 - \frac{2}{3} = \frac{1}{3} \] What is the probability that at least two of them pass?

Two dice are thrown simultaneously. What is the probability that the product of the numbers is even?

Three vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfy: \[ \vec{a} + \vec{b} + \vec{c} = \vec{0},\quad |\vec{a}| = 1,\quad |\vec{b}| = 3,\quad |\vec{c}| = 4 \] Then the value of: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = ? \]

In parallelogram \( ABCD \), points \( M \) and \( N \) are midpoints of sides \( BC \) and \( CD \) respectively. Then the length \( AM + AN = \) ?

In \( \triangle ABC \), let \( O \) be the circumcenter and \( G \) be the centroid. Then: \[ OG^2 = ? \]

Let \( \vec{OA} = -4\hat{i} + 3\hat{k} \), \( \vec{OB} = 14\hat{i} + 2\hat{j} - 5\hat{k} \), and vector \( \vec{OD} \) bisects \( \angle AOB \), with \( |\vec{OD}| = \sqrt{6} \). Then: \[ \vec{OD} = ? \]

Let \( x \in \mathbb{R} \) and \( |x|<1 \). Then \( \tanh^{-1} x = \) ?

A farmer owns a triangular plot in Guntur. He measures the sides of the plot as 4 cm, 5 cm, and 7 cm. The area of the land (in sq. cm) is:

If the position vectors of points \( A, B, C, D \) are: \[ \vec{A} = 7\hat{i} - 4\hat{j} + 7\hat{k},\quad \vec{B} = \hat{i} - 6\hat{j} + 10\hat{k},\quad \vec{C} = -\hat{i} - 3\hat{j} + 4\hat{k},\quad \vec{D} = 5\hat{i} - \hat{j} + 5\hat{k} \] Then the quadrilateral \( ABCD \) is:

In a triangle \( ABC \), the identity holds: \[ s \left( \frac{r_1 - r}{a} + \frac{r_2 - r}{b} + \frac{r_3 - r}{c} \right) = ? \]

In a triangle, if \( b = 5 \), \( c = 6 \), and \( \tan\left(\frac{A}{2}\right) = \frac{1}{\sqrt{2}} \), then the side \( a = \) ?

Simplify the expression: \[ \sec^2 x + 5 \tan x + 5 \]

If \( \cos\theta - \sin\theta = \sqrt{5} \sin\theta \), then the value of \( \cos\theta + 4\sin\theta \) is:

If the equivalent partial fraction of \[ \frac{x^3}{(2x - 1)(x + 2)(x - 3)} \] is of the form \[ A + \frac{B}{2x - 1} + \frac{C}{x + 2} + \frac{D}{x - 3} \] then the value of \( A + B + C = \) ?

In a triangle \( ABC \), the identity holds: \[ \tan\frac{A}{2} \tan\frac{B}{2} + \tan\frac{B}{2} \tan\frac{C}{2} + \tan\frac{C}{2} \tan\frac{A}{2} = ? \]

Let \( \alpha, \beta \) be real numbers such that \( \pi<(\alpha - \beta)<3\pi \). If: \[ \sin \alpha + \sin \beta = \frac{-21}{65}, \quad \cos \alpha + \cos \beta = \frac{-2}{65} \] then the value of \( \cos\left( \frac{\beta - \alpha}{2} \right) \) is:

If \( ^{2n}C_3 : ^nC_3 = 12 : 1 \), then \( n = \) ?