List of practice Questions

If the direction cosines of a line are: $$ \left( \frac{a}{\sqrt{83}},\ \frac{5}{\sqrt{83}},\ \frac{c}{\sqrt{83}} \right) $$ and it is given that $ c - a = 4 $, find $ ca $.

A circle is such that it is tangent to every line of the form: $$ (x - 2)\cos\theta + (y - 2)\sin\theta = 1 $$ for all $ \theta $. Find the equation of the circle.

Let the coordinates of points A and B satisfy:
- $ x^2 + 2x - a^2 = 0 $
- $ y^2 + 4y - b^2 = 0 $ Find the equation of the circle having AB as diameter.

There are two bags:
- Bag 1: 4 red, 3 black → total = 7 balls
- Bag 2: 2 red, 3 black → total = 5 balls
A bag is chosen at random and one ball is drawn. What is the probability that the ball is red?

In a binomial distribution, if:
- Number of trials $ n $
- Mean $ = 4 $
- Variance $ = 3 $
Then find: $$ 2^{32} \cdot P\left(X = \frac{n}{2}\right) $$

Points $ P $ and $ Q $ lie on the lines: $$ 3x + 4y - 4 = 0 \quad \text{and} \quad 5x - y - 4 = 0 $$ The midpoint of segment $ PQ $ is $ (1, 5) $. Find the slope of the line $ PQ $.

If $ \vec{a} $ is collinear with $ \vec{b} = 3\hat{i} + 6\hat{j} + 6\hat{k} $ and $ \vec{a} \cdot \vec{b} = 27 $, then find $ |\vec{a}| $.

Given: $$ \vec{F} = 2\hat{i} + 2\hat{j} + 5\hat{k},\quad A = (1, 2, 5),\quad B = (-1, -2, -3) $$ and: $$ \overrightarrow{BA} \times \vec{F} = 4\hat{i} + 6\hat{j} + 2\lambda \hat{k} $$ Find the value of $ \lambda $.

If the mean deviation of the data: $$ 1, 1 + d, 1 + 2d, \ldots, 1 + 100d $$ from their mean is 255, then find $ d $.

What is the probability of getting a sum of 9 when two dice are thrown?

If $ A $ and $ B $ are two events such that $ P(B) \ne 0 $ and $ P(B) \ne 1 $, then: $$ P(\bar{A} \mid B) = ? $$

Evaluate: $$ \sin^2\left(\frac{2\pi}{3}\right) + \cos^2\left(\frac{5\pi}{6}\right) - \tan^2\left(\frac{3\pi}{4}\right) $$

Solve for $ x $ given: $$ 2 \cosh(2x) + 10 \sinh(2x) = 5 $$

In triangle $ ABC $, if: $$ r_1 = 36,\quad r_2 = 18,\quad r_3 = 12 $$ then find the semi-perimeter $ s $.

In triangle $ ABC $, given: $$ a = 6,\quad b = 5,\quad c = 4,\quad \text{find } \cos 2A $$

Evaluate: $$ \frac{1}{\sin 1^\circ \sin 2^\circ} + \frac{1}{\sin 2^\circ \sin 3^\circ} + \cdots + \frac{1}{\sin 89^\circ \sin 90^\circ} $$

Which of the following trigonometric values are negative?

• I) $ \sin(-292^\circ) $
• II) $ \tan(-193^\circ) $
• III) $ \cos(-207^\circ) $
• IV) $ \cot(-222^\circ) $

If $$ S = \left\{ m \in \mathbb{R} : x^2 - 2(1 + 3m)x + 7(3 + 2m) = 0 \text{ has distinct roots} \right\} $$ then the number of elements in $ S $ is:

Solve: $$ 4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1} $$

If one root of the cubic equation: $$ x^3 + 36 = 7x^2 $$ is double of another, then the number of negative roots is:

Let $$ G(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ If $ x + y = 0 $, then evaluate $ G(x)G(y) $

Find the values of $ x $ for which the expressions $$ \sin x + i\cos 2x \quad \text{and} \quad \cos x - i\sin 2x $$ are conjugate to each other.

The locus of a point $ z $ satisfying: $$ |z|^2 = \text{Re}(z) $$ is a circle with centre: