List of practice Questions

If \[ f(x)=1-2x+\int_{0}^{x} e^{x-t} f(t)\,dt \] and \[ g(x)=\int_{0}^{x} (f(t)+2)^{11}(t+12)^{17}(t-4)^4\,dt, \] If local minima and local maxima of \(g(x)\) occur at \(x=p\) and \(x=q\) respectively, find \(|p|+q\).
The largest \( n \in \mathbb{N} \), for which \( 7^n \) divides \( 101! \), is :
Let \(Q(a,b,c)\) be the image of the point \(P(3,2,1)\) in the line \[ \frac{x-1}{1}=\frac{y-2}{2}=\frac{z-1}{1}. \] The distance of \(Q\) from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:
Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(2x) - 2\sin x}{x^3} \]
Mean deviation about median for \( k, 2k, 3k, \ldots, 1000k \) is 500, then the value of \( k^2 \) is:
Let \[ A=\{z\in\mathbb{C}:|z-2|\le 4\} \quad\text{and}\quad B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}. \] Then the maximum value of \(|z_1-z_2|\), where \(z_1\in A\) and \(z_2\in B\), is:

If xdy - ydx = \(\sqrt{x^2 + y^2}\) dx. If y = y(x) \(\&\) y(1) = 0 then y(3) is :

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \sqrt{5\beta}}{2} \] then the value of \( (\alpha + \beta) \) is:
An ellipse has its centre at \((1,-2)\), one focus at \((3,-2)\) and one vertex at \((5,-2)\). Then the length of its latus rectum is:
What is the value of the line integral of a magnetic field around a closed loop according to Ampère’s Circuital Law?
An organic compound (A) undergoes the following reactions:
Which of the following is compound (A)?
Consider the structure of HNO$_3$:
Select the correct option having formal charge of I, II, III and IV respectively.
Find out the correct graph for the variation of \( E_{\text{cell}} \) with temperature.
In the following reaction sequence, identify compound \( (R) \).
For an ideal gas undergo isothermal reversible process from 0.5Mpa, 20dm$^3$ to 0.2Mpa at 600K. Calculate correct option. [Given $\log 5 = 0.6989, \log 2 = 0.3010$]
Which of the following statements are TRUE about Haloform reaction?:
A. Sodium hypochlorite reacts with KI to give KOI.
B. KOI is a reducing agent.
C. $\alpha, \beta$-unsaturated methylketone ($\text{CH}_3-\text{CH}=\text{CH}-\text{C=O}-\text{CH}_3$) will give iodoform reaction.
D. Isopropyl alcohol will not give iodoform test.
E. Methanoic acid will give positive iodoform test.
What is the physical significance of the "Quality Factor" (\(Q\)-factor) in an LCR resonant circuit?
Which statistical distribution describes the behavior of identical, indistinguishable particles with half-integral spin?
According to the Special Theory of Relativity, what happens to the length of an object as its velocity approaches the speed of light?
What is the relation between the group velocity and phase velocity in a non-dispersive medium?
Which phenomenon confirms the transverse nature of electromagnetic waves?
What is the Boolean expression for a two-input NAND gate?
In a p-n junction diode, what happens to the width of the depletion layer under reverse bias?
What is the energy of the ground state of a hydrogen atom according to the Bohr model?
For a thermodynamic system undergoing an adiabatic process, what is the relationship between pressure \(P\) and volume \(V\)?