List of practice Questions

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: $$ P(\bar{A} \mid B) = ? $$

If $ \sin\theta + \csc\theta = 4 $, then find: $$ \sin^2\theta + \csc^2\theta = ? $$

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 any triangle $ ABC $, evaluate: $$ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} $$

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

Given that $ \vec{a}, \vec{b}, \vec{c} $ are non-coplanar vectors and: $$ \vec{a} + 3\vec{b} + 4\vec{c} = x(\vec{a} - 2\vec{b} + 3\vec{c}) + y(\vec{a} + 5\vec{b} - 2\vec{c}) + z(6\vec{a} + 14\vec{b} + 4\vec{c}) $$ Find $ x + y + z $

Out of 205 students:
- 105 pass in English
- 70 pass in Mathematics
- 30 pass in both
How many students fail in both subjects?

In an exam:
- Three subjects have maximum marks = $ n $ each
- Fourth subject has max marks = $ 2n $
- Find the number of ways to score total $ 3n $ marks.

Given: $$ \frac{2x^2 + 1}{x^3 - 1} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + x + 1} $$ Find the value of $ 7A + 2B + C $

If $ \sin \theta = -\frac{3}{4} $, then compute $ \sin 2\theta $

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 $$ \sin^4 \theta \cos^2 \theta = \sum_{n=0}^{\infty} a_{2n} \cos 2n\theta $$ then the least value of $ n $ for which $ a_{2n} = 0 $ is:

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

The sum of the real roots of the equation: $$ x^4 - 2x^3 + x - 380 = 0 $$ is:

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

If $$ ^{10}C_2 = ^{3^{n+1}}C_3 $$ then the value of $ n $ is:

There are 10 points in a plane, out of which 6 are collinear. If $ N $ is the number of triangles that can be formed, then $ N = $ ?

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) $

By simplifying the expression: $$ i^{18} - 3i^7 + i^2(1 + i^4)(i^{22}) $$ we get:

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.