List of practice Questions

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).

A plane makes intercepts \(-2, \frac{4}{3}, \frac{-4}{5}\) on the X, Y, Z axes respectively. Find the direction cosines of a normal to the plane.

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

Find the locus of the incenter of the triangle formed by: \[ xy - 4x - 4y + 16 = 0,\quad x + y = 5 \]

A circle touches the y-axis at a distance 4 units from the origin and cuts off an intercept 6 from the x-axis. Find its equation.

Find the set of all points at a distance of at least 2 units from the point \( (-3, 0) \).

A circle touches both coordinate axes and its center lies on the line \( x - 2y - 3 = 0 \). What is its equation?

A circle of diameter \( R \) touches the parabola \( x^2 + y^2 - 4y = 0 \) and passes through the point \( (4, 5) \). Which of the following is correct?

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?

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 \( X \)?

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:

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?

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: \[ OG^2 = ? \]

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: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = ? \]

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:

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

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:

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 \( A + B + C = \) ?

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

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:

There are 4 oranges, 5 apples, and 7 mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is:

If \( ^9C_3 + ^9C_5 = ^{10}C_r \) for some \( r \in \mathbb{N} \), then \( r = \) ?