List of practice Questions

Identify the end-product $ [D] $ formed when solution salicylate undergoes the following series of reactions.

For a given reaction, $ X(g) + Y(g) \to Z(g) $, the order of reaction with respect to $ X $ and $ Y $ are $ m $ and $ n $ respectively. If the concentration of $ X $ is tripled and that of $ Y $ is decreased to one third, what is the ratio between the new rate to the original rate of the reaction?
What is the nature of the function $ f(x) = x^3 - 3x^2 + 4x $ on real numbers?
Given are 4 statements related to the chemical properties of Glucose. Identify the two incorrect statements from the following.
Evaluate: $ \cos^{-1} \left( \cos \frac{35\pi}{18} \right) - \sin^{-1} \left( \sin \frac{35\pi}{18} \right) $
The maximum value of $ P = 500x + 400y $ for the given constraints $ x + y \leq 200, \quad x \geq 20, \quad y \geq 4x, \quad y \geq 0 $
If $ y = \sin^{-1} \left( \frac{5x + 12\sqrt{1 - x^2}}{13} \right) $ then $ \frac{dy}{dx} $ equals
If $ \frac{dy}{dx} = y + 3 \quad \text{and} \quad y(0) = 2, \quad \text{then} \quad y(\log 2) \text{ is equal to:} $
What is the probability of a randomly chosen 2-digit number being divisible by 3?
The area bounded by the curve $ y = \cos x $, $ x = 0 $, and $ x = \pi $ is
If $ I_n = \int_0^\pi \tan^n x \, dx $, for $ n \geq 2 $, then $ I_n + I_{n-2} = $
If $ (1 - 4i)^3 = a + ib $, then the value of $ a $ and $ b $ is
Two finite sets have $ m $ and $ n $ number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of $ m $ and $ n $ are respectively.
The integral $ \int e^x \left[ \frac{x^2 + 1}{(x+1)^2} \right] \, dx $
The co-ordinate of the foot of the perpendicular from $ P(1, 8, 4) $ on the line joining $ R(0, -1, 3) $ and $ Q(2, -3, -1) $ is
The general solution of the differential equation $ (1 + \tan y)(dx - dy) + 2x \, dx \, dy = 0 $
If a real number $ a $ is such that $ \int_0^a x \, dx \leq a + 4 $, then what is the range of $ a $?
Find the value of 'b' such that the scalar product of the vector $ \hat{i} + \hat{j} + \hat{k} $ with the unit vector parallel to the sum of the vectors $ 2\hat{i} + 4\hat{j} - 5\hat{k} $ and $ b\hat{i} + 2\hat{j} + 3\hat{k} $ is unity.
Value of $ \cos 105^\circ $
The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is
A coin is tossed until a head appears or until the coin has been tossed three times. Given that 'head' does not appear on the first toss, what is the probability that the coin is tossed thrice?
The value of $ \int \frac{dx}{\sqrt{2ax - x^2}} $
A number consists of three digits in geometric progression. The sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of left hand and middle digits is two third of the sum of the middle and right hand digits. Then the sum of digits of number is
If $ y = \sqrt{\sin x + y} $, then find $ \frac{dy}{dx} $ at $ x = 0, y = 1 $.

If $ f(x) = \begin{cases} x, & 0 \leq x \leq 1 \\ 2x - 1, & x > 1 \end{cases} $, then: