List of practice Questions

Find: \[ \int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}} \, dx \]
A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up is 0.10, member B does not turn up is 0.20 and member C does not turn up is 0.05. The probability of the meeting being cancelled is:
Evaluate the integral: \[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]
Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
Find the values of \( a \) for which \( f(x) = \sin x - ax + b \) is increasing on \( \mathbb{R} \).
(a) Differentiate $ \left( \frac{5x}{x^5} \right)$ with respect to $x$.
From the given options, choose the correct one that will replace the question mark (?) in the following series: 2, 0, 3, 2, 4, 6, 5, 12, 6, ?, 7, 30

Preet and Saral were partners sharing profits and losses in the ratio of 3:2. On 31st March, 2024 they decided to change their profit sharing ratio to 1:1. On the date of reconstitution goodwill of the firm was valued at Rs 1,00,000. The journal entry for treatment of goodwill on account of change in profit-sharing ratio will be:

A battery of emf \( E \) and internal resistance \( r \) is connected to a rheostat. When a current of 2A is drawn from the battery, the potential difference across the rheostat is 5V. The potential difference becomes 4V when a current of 4A is drawn from the battery. Calculate the value of \( E \) and \( r \).
Determine the focal length of the lenses used in the spectacles.
Differentiate between the peak value and root mean square value of an alternating current. Derive the expression for the root mean square value of alternating current, in terms of its peak value.
If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?
Differentiate \( \log(x^2 + \csc^2 x) \) with respect to \( x \).
Find the domain of the function $f(x) = \sin^{-1}(-x^2)$.
(b) If $2x^2 - 5xy + y^3 = 76$, then find $\frac{dy}{dx}$.
Population of Town A and B was 20,000 in 1985. In 1989, the population of Town A was 25,000, and Town B had 28,000. What will be the difference in population between the two towns in 1993?
In education policy, the concept of inclusive education advocates for integrating students with disabilities into mainstream classrooms to promote diversity and equal opportunities. However, critics claim that focusing too much on inclusion can strain resources and may lead to a decline in educational quality for students both with and without disabilities.
Which of the following arguments would most strengthen the critics position?

"जितेंद्र नार्गे जैसे गाइड के साथ किसी भी पर्यटन स्थल का भ्रमण अधिक आनंददायक और यादगार हो सकता है।" इस कथन के समर्थन में 'साना साना हाथ जोड़ि .......' पाठ के आधार पर तर्कसंगत उत्तर दीजिए।

Three infinitely long wires with linear charge density \( \lambda \) are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?
In a container of volume 16.62 m$^3$ at 0°C temperature, 2 moles of oxygen, 5 moles of nitrogen and 3 moles of hydrogen are present, then the pressure in the container is (Universal gas constant = 8.31 J/mol K)

Complete and balance the following chemical equations: (a) \[ 2MnO_4^-(aq) + 10I^-(aq) + 16H^+(aq) \rightarrow \] (b) \[ Cr_2O_7^{2-}(aq) + 6Fe^{2+}(aq) + 14H^+(aq) \rightarrow \] 

How many molecules of carbon dioxide are formed when 0.6 g carbon is burnt in air?
Pulkit and Ravinder were partners in a firm sharing profits and losses in the ratio of 3 : 2. Sikander was admitted as a new partner for \( \frac{1}{5} \)th share in the profits of the firm. Pulkit, Ravinder and Sikander decided to share future profits in the ratio of 2 : 2 : 1. Sikander brought Rs 5,00,000 as his capital and Rs 10,00,000 as his share of premium for goodwill. The amount of premium for goodwill that will be credited to the old partners' capital accounts will be :
What is the term used for the milk produced during the initial days of lactation?
Find: \[ I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx. \]