List of practice Questions

Match the following reactions with the product obtained from them:

What are X and Y respectively in the following reaction sequence?

Arrange the following in decreasing order of their acidity:

What are X and Y in the following set of reactions?

An alkyl halide $ C_3H_7Cl $, on reaction with a reagent X, gave the major product Y ($ C_4H_7N $). Y on hydrolysis released a gas, which turns red litmus to blue. What are X and Y?

The domain of the real valued function $ f(x) = \frac{1}{\sqrt{log_{0.5}(2x-3)}} + \sqrt{4 - 9x^2} $ is:

The system of equations $ x + 2y + 3z = 6 $, $ x + 3y + 5z = 9 $, $ 2x + 5y + az = 12 $ has no solution when $ a = $:

If a complex number $ z $ is such that $ \frac{z-2i}{z-2} $ and the locus of $ z $ is a closed curve, then the area of the region bounded by that closed curve and lying in the first quadrant is:

The real part of $ \frac{\left( \cos a + i \sin a \right)^6}{\left( \sin b + i \cos b \right)^8} $ is:

If $ \alpha, \beta, \gamma $ are roots of the equation $ x^3 + ax^2 + bx + c = 0 $, then $ \alpha^{-1} + \beta^{-1} + \gamma^{-1} $ is:

The number of different permutations that can be formed by taking 4 letters at a time from the letters of the word "REPETITION" is:

If $ \frac{1}{x^4 + 1} = \frac{Ax + B}{x^2 + \sqrt{2}x + 1} + \frac{Cx + D}{x^2 - \sqrt{2}x + 1} $, then $ BD - AC = $:

The smallest positive value (in degrees) of $ \theta $ for which $ \tan(\theta + 100^\circ) = \tan(\theta + 50^\circ) \tan(\theta - 50^\circ) $ is valid, is:

Statement (S1): $ \sin 55^\circ + \sin 53^\circ - \sin 19^\circ - \sin 17^\circ = \cos 2^\circ $
Statement (S2): The range of $ \frac{1}{3 - \cos 2x} $ is $ \left[ \frac{1}{4}, \frac{1}{2} \right] $ Which one of the following is correct?

In $\Delta ABC$ if $B = 90^\circ$ then $2(r + R) = $

In a triangle ABC, if $ (a-b)(s-c) = (b-c)(s-a) $, then $ r_1 + r_3 = $:

Let ABC be an equilateral triangle of side $a$. M and N are two points on the sides AB and AC respectively such that $AN = K \cdot AC$ and $AB = 3 \cdot AM$. If the vectors $BN$ and $CM$ are perpendicular, then $K = $ ?

Let a and b be two non-collinear vectors of unit modulus. If u = a − (a · b)b and v = a × b, then ∥v∥ = ?

Find the shortest distance between the skew lines $\vec{r} = (-\hat{i} - 2\hat{j} - 3\hat{k}) + t(3\hat{i} - 2\hat{j} - 2\hat{k})$ and $\vec{r} = (7\hat{i} + 4\hat{k}) + s(\hat{i} - 2\hat{j} + 2\hat{k})$.

If 7 different balls are distributed among 4 different boxes, then the probability that the first box contains 3 balls is:

Out of the first 5 consecutive natural numbers, if two different numbers x and y are chosen at random, then the probability that x⁴- y⁴is divisible by 5 is:

If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability distribution of number of kings is:

In a consignment of 15 articles, it is found that 3 are defective. If a sample of 5 articles is chosen at random from it, then the probability of having 2 defective articles is:

The area (in square units) of the triangle formed by the lines 6x² + 13xy + 6y² = 0 and x + 2y + 3 = 0 is:

If the area of the circum-circle of the triangle formed by the line 2x + 5y + a = 0 and the positive coordinate axes is $\frac{29\pi}{4}$ sq. units, then |a| =