List of practice Questions

If the function $f(x) = \left(\frac{1}{x}\right)^{2x}; \, x>0$ attains the maximum value at $x = \frac{1}{c}$, then:

Explain, why the speaker mentions God in the poem "For Anne Gregory".

Let $I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx$. If $I(0) = 3$, then $I\left(\frac{\pi}{12}\right)$ is equal to:

“There exists a positive correlation between human capital formation and economic growth of a nation.” Justify the given statement with valid explanation.

The displacement of a particle executing SHM is given by \( x = 10 \sin\left(\omega t + \frac{\pi}{3}\right) \, \text{m} \). The time period of motion is \( 3.14 \, \text{s} \). The velocity of the particle at \( t = 0 \) is ______ m/s.

A hypothetical electromagnetic wave is show below.

The frequency of the wave is x × 10¹⁹ Hz. x = ______ (nearest integer)

Identify A and B in the given chemical reaction sequence : -

A particle of mass m is projected from ground with speed u at an angle of 30° with the horizontal. Find its angular momentum about the point of projection when it reaches its maximum height.

Following gates section is connected in a complete suitable circuit.

For which of the following combination, bulb will glow (ON):

The charge accumulated on the capacitor connected in the following circuit is ____ µC
(Given \( C = 150 \, \mu \text{F} \))

Manu, Sonu, and Rahul were partners in a firm sharing profits and losses in the ratio of \( 4 : 3 : 2 \). With effect from 1\textsuperscript{st April, 2023, they decided to share profits and losses in the future in the ratio of \( 3 : 2 : 1 \). Their Balance Sheet showed Workmen Compensation Reserve of ₹ 84,000. The claim on account of Workmen Compensation is estimated at ₹ 75,000. The journal entry to give effect to the above transaction will be:} \includegraphics[width=0.5\linewidth]{Screenshot 2025-01-17 200658.jpg} \label{fig:enter-label} \end{figure}

Solve the following linear programming problem graphically: \[ {Maximise } z = 4x + 3y, \quad {subject to the constraints:} \] \[ x + y \leq 800, \quad 2x + y \leq 1000, \quad x \leq 400, \quad x, y \geq 0. \]

If the system of equations
\( 2x + 3y - z = 5 \)  
\( x + \alpha y + 3z = -4 \)  
\( 3x - y + \beta z = 7 \)  
has infinitely many solutions, then \( 13 \alpha \beta \) is equal to:  

The ionization energy of sodium in \(kJ \;mol^{−1}\). If electromagnetic radiation of wavelength 242 nm is just sufficient to ionize sodium atom is ______.

To maintain consistent business through out the year, an entrepreneur may

Read the following excerpt from a newspaper on the problem of car parking and its impact.
A survey conducted on the residents showed that car parking in colonies has risen due to an increase in the number of cars per family. This has resulted in ugly incidents of road rage. Most of the families revealed that they are facing multiple issues due to this situation. There is no place for visitors to park and residents themselves park their cars haphazardly. In addition there is an increase in commercial vehicles parked outside colony gates, thus blocking exits.
Write a paragraph in about 120 words to analyze the news clipping.

Let \( A = \{ n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of 3 nor a multiple of 4} \} \).
Then the number of elements in \( A \) is:

“Improvement in technology has stimulated the globalisation process.” Explain the statement with examples.

What happens when D-glucose is treated with the following reagents? 
(a) Br$_2$ water 
(b) HCN

Write a factual description, in not more than 100 words, of an English Notebook that you have to submit to your teacher.

Given \( \frac{d}{dx}F(x) = \frac{1}{\sqrt{2x - x^2}} \) and \( F(1) = 0 \), find \( F(x) \).