List of practice Questions

For a particle moving in x-direction according to the relation \(x = 4t^3 - 3t\), consider the following statements:

[(a)] At \(t = 0.866\), \(x = 0\)
[(b)] Direction of velocity of particle remains same
[(c)] Direction of velocity of particle changes at \(x = -1\)
[(d)] Direction of velocity of particle changes at \(x = 0.5\,\text{m}\)
[(e)] Acceleration is non-negative
Correct statements are:
If percentage increase in Young's modulus \(Y\) is \(1%\), percentage increase in density of material is \(0.5%\) and longitudinal wave traveling in a metallic bar has wave velocity of \(400\,\text{m/s}\), then find final velocity of the wave.
A current is flowing along the surface of a long solid cylinder of radius \(R\). Select the correct statement(s):

[(A)] Magnetic field (\(B\)) is minimum along the axis of cylinder
[(B)] Magnetic field (\(B\)) is minimum at the surface of cylinder
[(C)] Magnetic field (\(B\)) is maximum at the surface of cylinder
[(D)] Magnetic field (\(B\)) is maximum along the axis of cylinder
[(E)] Magnetic field (\(B\)) is same all over the cross section of cylinder
In the given circuit, energy stored in the inductor is \(16\,\text{J}\) and power dissipated in resistance is \(32\,\text{W}\). Find the value of \(\dfrac{X_L}{R}\).
Consider the following statements:

[(A)] \(\mathrm{KMnO_4}\) is diamagnetic while \(\mathrm{K_2MnO_4}\) is paramagnetic.
[(B)] Manganate ion contains \(\mathrm{Mn^{6+}}\) while permanganate ion contains \(\mathrm{Mn^{7+}}\).
[(C)] \(\mathrm{Mn^{2+}}\) ion on reaction with \(\mathrm{S_2O_8^{2-}}\) ions gives manganate ion.
[(D)] Both \(\mathrm{MnO_4^-}\) and \(\mathrm{MnO_4^{2-}}\) are tetrahedral.
Correct statements are:
Out of \( \mathrm{N, P, S, Cl, F} \), number of valence electrons in the {least metallic element and most metallic element respectively is: }
Statement-I : \(\mathrm{KMnO_4}\) is a good reducing agent.
Statement-II : \(\mathrm{KMnO_4}\) reduces nitrite, oxalate and iodide ions.
For the reaction \( \mathrm{A \rightleftharpoons B} \), the number of moles of A and B at equilibrium in a \(1\,\text{L}\) vessel are \(0.50\) and \(0.375\), respectively. If \(0.10\) mol of A is added further, determine the number of moles of A and B at the new equilibrium.
Consider the following electromagnetic waves A, B and C: (i) The wavelength of A is \(400\,\text{nm}\).
(ii) The frequency of B is \(10^{16}\,\text{s}^{-1}\).
(iii) Wave number of C is \(10^{4}\,\text{cm}^{-1}\).
The correct order of energies is:
Two positively charged particles are accelerated by \(200\,\text{keV}\). The masses of particles are \(m_1 = 1\,\text{amu}\) and \(m_2 = 4\,\text{amu}\).
If the de-Broglie wavelength \((\lambda_d)_{m_1}\) is \(x\) times of the second particle \((\lambda_d)_{m_2}\), determine the value of \(x\).
\[ \mathrm{CaCO_3 + 2HCl \rightarrow CaCl_2 + H_2O + CO_2} \] In the above reaction, \(90\,\text{g}\) of \(\mathrm{CaCO_3}\) is added to \(300\,\text{mL}\) of \(38.55%\) (w/w) HCl solution with density \(1.13\,\text{g mL}^{-1}\). Which of the following option is correct?
Find the value of \[ \frac{6}{3^{26}} + 10\cdot\frac{1}{3^{25}} + 10\cdot\frac{2}{3^{24}} + \cdots + 10\cdot\frac{2^{24}}{3^{1}} : \]
The sum of coefficients of \(x^{499}\) and \(x^{500}\) in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:
Let matrix \[ A=\begin{pmatrix} 3 & -4\\ 1 & -1 \end{pmatrix} \] and \(A^{100} = 100B + I\). Find the sum of all the elements in \(B^{100}\).
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{12(3+[x])\,dx}{3+[\sin x]+[\cos x]} \] (where \([\,]\) denotes the greatest integer function) is:
Find the maximum distance between the two curves: \[ |z-2| = 4 \quad \text{and} \quad |z-2| + |z+2| = 5 \]
Let \[ f(x)=\lim_{\theta\to 0}\frac{\cos(\pi x)-x^{2/\theta}\sin(x-1)}{1-x^{2/\theta}(x-1)}. \] Statement 1: \(f(x)\) is discontinuous at \(x=1\).
Statement 2: \(f(x)\) is continuous at \(x=-1\).
Statement 1 : \(2^{513}+2^{013}+8^{13}+3^{13}\) is divisible by \(7\).
Statement 2 : The value of integral part of \((7+4\sqrt{3})^{25}\) is an odd number.
If \[ \sum_{r=1}^{25}\frac{r}{r^4+r^2+1}=\frac{p}{q}, \] where \(p\) and \(q\) are coprime positive integers, then \(p+q\) is equal to:
Find the value of \[ \tan\!\left[\left(2\sin^{-1}\frac{2}{\sqrt{13}}\right)-2\cos^{-1}\!\left(\frac{3}{\sqrt{10}}\right)\right] \]
Given \[ f(x)=\int \frac{dx}{x^{2/3}+2\sqrt{x}} \quad \text{and} \quad f(0)=-26+24\ln 2. \] If \(f(1)=A+B\ln 3\), then find \((A+B)\).
If the image of a point \(P(3,2,1)\) in the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] is \(Q\), then the distance of \(Q\) from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:
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 \(|p|+q\).
Consider two parabolas \(P_1,\ P_2\) and a line \(L\):
\[ P_1:\ y=4x^2,\qquad P_2:\ y=x^2+27,\qquad L:\ y=\alpha x \] If the area bounded by \(P_1\) and \(P_2\) is six times the area bounded by \(P_1\) and \(L\), find \(\alpha\).
Let \(L\) be the distance of point \(P(-1,2,5)\) from the line \[ \frac{x-1}{2}=\frac{y-3}{2}=\frac{z+1}{1} \] measured {parallel to a line having direction ratios \(4,\,3,\,-5\). Then \(L^2\) is equal to: }