List of practice Questions

Aman has been asked to synthesise the molecule: 

Using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory.
He thought of performing ozonolysis reaction on the alkene to
produce a dicarbonyl compound followed by aldol reaction to prepare "x".
Predict the suitable alkene that can lead to the formation of "x".

For the function \( f(x) = \ln(x^2 + 1) \), what is the second derivative of \( f(x) \)?
Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).
Let \( f : \mathbb{R} - \{0\} \to \mathbb{R} \) be a function such that \[ f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}. \] If \( \lim_{x \to 0} \left( \frac{1}{\alpha x} + f(x) \right) = \beta \), then \( \alpha, \beta \in \mathbb{R} \), and \( \alpha + 2\beta \) is equal to:
If \( \alpha \) and \( \beta \) are the roots of the equation \( 2z^2 - 3z - 2i = 0 \), where \( i = \sqrt{-1} \), then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:
What is the time period of a simple pendulum of length \( L \) and acceleration due to gravity \( g \)?
In a sound wave, the displacement of the air particles follows the equation \( y = A \cos(kx - \omega t) \). What is the wave velocity?

Let circle \( C \) be the image of

\[ x^2 + y^2 - 2x + 4y - 4 = 0 \]

in the line

\[ 2x - 3y + 5 = 0 \]

and \( A \) be the point on \( C \) such that \( OA \) is parallel to the x-axis and \( A \) lies on the right-hand side of the centre \( O \) of \( C \).

If \( B(\alpha, \beta) \), with \( \beta < 4 \), lies on \( C \) such that the length of the arc \( AB \) is \( \frac{1}{6} \) of the perimeter of \( C \), then \( \beta - \sqrt{3}\alpha \) is equal to:

Given below are two statements I and II. 
Statement I: Dumas method is used for estimation of "Nitrogen" in an organic compound. 
Statement II: Dumas method involves the formation of ammonium sulfate by heating the organic compound with concentrated H\(_2\)SO\(_4\). In the light of the above statements, choose the correct answer from the options given below:

Ksp for \( \text{Cr(OH)}_3 \) is \( 1.6 \times 10^{-30} \). What is the molar solubility of this salt in water?

For a statistical data \( x_1, x_2, \dots, x_{10} \) of 10 values, a student obtained the mean as 5.5 and \[ \sum_{i=1}^{10} x_i^2 = 371. \] He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. 
The variance of the corrected data is:

A parallel plate capacitor of area \( A = 16 \, \text{cm}^2 \) and separation between the plates \( 10 \, \text{cm} \), is charged by a DC current. Consider a hypothetical plane surface of area \( A_0 = 3.2 \, \text{cm}^2 \) inside the capacitor and parallel to the plates. At an instant, the current through the circuit is 6A. At the same instant the displacement current through \( A_0 \) is \(\_\_\_\_\_ \)mA.
Let \( y = y(x) \) be the solution of the differential equation \[ \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx + (1 + x^2) dy = 0, \quad y(0) = 0. \] Then \( y(\sqrt{3}) \) is equal to:
Evaluate the limit: \[ \lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right) \] is equal to:
What is the sum of the infinite series \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \)?
What is the solution to the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) with the initial condition \( y(1) = 2 \)?
What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5 D? ('D' stands for dioptre).
Let \( S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots \) up to \( n \) terms. If the sum of the first six terms of an A.P. with first term \( -p \) and common difference \( p \) is \( \sqrt{2026 S_{2025}} \), then the absolute difference between the 20th and 15th terms of the A.P. is:
Let \( A = \{1,2,3\} \). The number of relations on \( A \), containing \( (1,2) \) and \( (2,3) \), which are reflexive and transitive but not symmetric, is ________.
Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?
Compare boiling point of given solutions:
(i) \( 10^{-4} \, \text{M NaCl} \)>(ii) \( 10^{-3} \, \text{M NaCl} \)>(iii) \( 10^{-2} \, \text{M NaCl} \)>(iv) \( 10^{-4} \, \text{M urea} \)
(iii) \( 10^{-2} \, \text{M NaCl} \)>(ii) \( 10^{-3} \, \text{M NaCl} \)>(i) \( 10^{-4} \, \text{M NaCl} \)>(iv) \( 10^{-4} \, \text{M urea} \)
(ii) \( 10^{-3} \, \text{M NaCl} \)>(i) \( 10^{-4} \, \text{M NaCl} \)>(iii) \( 10^{-2} \, \text{M NaCl} \)>(iv) \( 10^{-4} \, \text{M urea} \)
(iii) \( 10^{-2} \, \text{M NaCl} \)>(ii) \( 10^{-3} \, \text{M NaCl} \)>(i) \( 10^{-4} \, \text{M NaCl} \)>(iv) \( 10^{-4} \, \text{M urea} \)
During the transition of an electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000 Å, and it becomes 6000 Å when the electron jumps from state B to state C. Then the wavelength of the radiation emitted during the transition of electrons from state A to state B is:
For \([ \text{NiCl}_4 ]^{2-}\), what is the charge on metal and shape of complex respectively?

Consider the following statements: 
A. The junction area of a solar cell is made very narrow compared to a photodiode.
B. Solar cells are not connected with any external bias. 
C. LED is made of lightly doped p-n junction.
D. Increase of forward current results in a continuous increase in LED light intensity. 
E. LEDs have to be connected in forward bias for emission of light.

In a resistive circuit, the power dissipated by a resistor \( R \) when a current \( I \) flows through it is given by \( P = I^2 R \). What happens to the power when the current is doubled?