JEE Main 2021 · 17 Mar 2021 · Shift 2

Q1.Which one of the following will be the output of the given circuit?

Q2.Two identical blocks $A$ and $B$ each of mass $m$ resting on the smooth horizontal floor are connected by a light spring of natural L and spring constant K.
$A$ third block $C$ of mass $m$ moving with a speed $v$ along the line joining $A$ and $B$ collides with $A$. The maximum c …

Q3.A sphere of mass 2 kg and radius 0.5 m is rolling with an initial speed of $1{\mathrm{ms}}^{-1}$ goes up an inclined plane which makes an angle of $30^0$ with the horizontal plane, without slipping. How long will the sphere take to return to the starting point $A$?

Q4.The velocity of a particle is $v=v_0+gt+{\mathrm{Ft}}^2$. Its position is $\mathrm{x}=0$ at $\mathrm{t}=0$; then its displacement after time ($\mathrm{t}=1$) is:

Q5.What happens to the inductive reactance and the current in a purely inductive circuit if the frequency is halved?

Q6.Two identical photo-cathodes receive the light of frequencies $f_1$ and $f_2$ respectively. If the velocities of the photo-electrons coming out are $v_1$ and $v_2$ respectively, then

Q7.A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to $\frac{81}{100}$ of the height through which it falls. Find the average speed of the ball. (Take $\mathrm{g}=10{\mathrm{ms}}^{-2}$)

Q8.A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point P which lies on the centre of the semicircle?

Q9.A carrier signal $\mathrm{C}(\mathrm{t})=25\sin \left(2.512\times 10^{10}\mathrm{t}\right)$ is amplitude modulated by a message signal $\mathrm{m}(\mathrm{t})=5\sin \left(1.57\times 10^8\mathrm{t}\right)$ and transmitted through an antenna. What will be the bandwidth of t …

Q10.Two cells of emf 2E and E with internal resistance $r_1$ and $r_2$ respectively are connected in series to an external resistor R (see figure). The value of R , at which the potential difference across the terminals of the first cell becomes zero is

Q11.Match List -I with List -II

Choose the most appropriate answer from the options given below :

Q12.If one mole of the polyatomic gas is having two vibrational modes and $\beta$ is the ratio of molar specific heats for polyatomic gas $\left(\beta =\frac{C_P}{C_V}\right)$ then the value of $\beta$ is:

Q13.The atomic hydrogen emits a line spectrum consisting of various series. Which series of hydrogen atomic spectra is lying in the visible region?

Q14.Which one is the correct option for the two different thermodynamic processes?

Q15.A sound wave of frequency 245 Hz travels with the speed of $300{\mathrm{ms}}^{-1}$ along the positive x -axis. Each point of the wave moves to and for through a total distance of 6 cm . What will be the mathematical expression of this traveling wave?

Q16.An object is located at 2 km beneath the surface of the water. If the fractional compression $\frac{\Delta \mathrm{V}}{\mathrm{V}}$ is $1.36\%$, the ratio of hydraulic stress to the corresponding hydraulic strain will be
[Given: density of water is $1000{\mathrm{kgm}}^{-3}$ and …

Q17.A geostationary satellite is orbiting around an arbitrary planet ' $P$ ' at a height of $11R$ above the surface of ' $P$ ', $R$ being the radius of ' $P$ '. The time period of another satellite in hours at a height of 2R from the surface of ' $P$ ' is ____ ' $P$ ' has the ti …

Q18.The four arms of a Wheatstone bridge have resistances as shown in the figure. A galvanometer of $15\Omega$ resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 V is maintained across AC.

Q19.A block of mass 1 kg attached to a spring is made to oscillate with initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion. (take $\ln 2=0.693$)

Q20.Two particles $A$ and $B$ of equal masses are suspended from two massless springs of spring constants ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of $A$ and $B$ is

Q21.A $2\mu \mathrm{F}$ capacitor ${\mathrm{C}}_1$ is first charged to a potential difference of 10 V using a battery. Then the battery is removed and the capacitor is connected to an uncharged capacitor ${\mathrm{C}}_2$ of $8\mu \mathrm{F}$. The charge in ${\mathrm{C}}_2$ on equ …

Q22.A body of mass 1 kg rests on a horizontal floor with which it has a coefficient of static friction $\frac{1}{\sqrt{3}}$. It is desired to make the body move by applying the minimum possible force F N. The value of $F$ will be ______ (Round off to Nearest Integer)
[Take …

Q23.The disc of mass M with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position $\frac{\mathrm{x}}{3}\frac{\mathrm{a}}{\pi },\frac{\mathrm{x}}{3}\frac{\mathrm{a}}{\pi }$ where $x$ is--------. (Roun …

Q24.The image of an object placed in air formed by a convex refracting surface is at a distance of 10 m behind the surface. The image is real and is at $\frac{2^{\text{rd}}}{3}$ of the distance of the object from the surface. The wavelength of light inside the surface is …

Q25.A boy of mass 4 kg is standing on a piece of wood having mass 5 kg. If the coefficient of friction between the wood and the floor is 0.5, the maximum force that boy can exert on the rope so that the piece of wood does not move from its place is ______ N. (Round off to the Nearest …

Q26.The electric field intensity produced by the radiation coming from a 100 W bulb at a distance of 3 m is E. The electric field intensity produced by the radiation coming from 60 W at the same distance is $\sqrt{\frac{x}{5}}E$. Where the value of $x=$ ______

Q27.Sea-water at a frequency $\mathrm{f}=9\times 10^2\mathrm{Hz}$, has permittivity $\epsilon =80{\epsilon }_0$ and resistivity $\rho =0.25\Omega \mathrm{m}$. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source …

Q28.Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes $0.01{\mathrm{cm}}^3$ of oleic acid per ${\mathrm{cm}}^3$ of the solution. Then you make a thin film of this solution (monomolecular thickness) of area $4{\mathrm{cm}}^2$ by co …

Q29.A particle of mass $m$ moves in a circular orbit in a central potential field $U(r)=U_0{r}^4$. If quantization conditions are applied, radii of possible orbitals $r_n$ vary with $n^{\frac{1}{\alpha }}$, where $\alpha$ is ______

Q30.The electric field in a region is given by $\vec{E}=\frac{2}{5}{E}_0\hat{i}+\frac{3}{5}{E}_0\hat{j}$ with $E_0=4.0\times 10^3\frac{N}{C}$. The flux of this field through a rectangular surface area $0.4{\mathrm{m}}^2$ parallel to the $Y-Z$ plane is ______ …

Q1.The functional groups that are responsible for the ion-exchange property of cation and anion exchange resins, respectively, are:

Q2.The set that represents the pair of neutral oxides of nitrogen is:

Q3.Which of the following statement(s) is (are) incorrect reason for eutrophication?
(A) Excess usage of fertilisers
(B) Excess usage of detergents
(C) Dense plant population in water bodies
(D) Lack of nutrients in water bodies that prevent plant growth

Choose the most appropriate …

Q4.The common positive oxidation states for an element with atomic number 24, are:

Q5.Match List - I with List - II

List - I
(a) Haematite
(b) Bauxite
(c) Magnetite
(d) Malachite

List - II
(i) ${\mathrm{Al}}_2{\mathrm{O}}_3\cdot x{\mathrm{H}}_2\mathrm{O}$
(ii) ${\mathrm{Fe}}_2{\mathrm{O}}_3$
(iii) ${\mathrm{CuCO}}_3\cdot \mathrm{Cu}(\mathrm{OH}{)}_2$
(iv) …

Q6.Fructose is an example of:

Q7.Match List - I with List - II

List - I
Chemical Compound
(a) Sucralose
(b) Glyceryl ester of stearic acid
(c) Sodium benzoate
(d) Bithionol

List - II
Used as
(i) Synthetic detergent
(ii) Artificial sweetener
(iii) Antiseptic
(iv) Food preservative

Choose the correct match:

Q8.For the coagulation of a negative sol, the species below, that has the highest flocculating power is:

Q9.Nitrogen can be estimated by Kjeldahl's method for which of the following compound?

Q10.The set of elements that differ in mutual relationship from those of the other sets is:

Q11.Choose the correct statement regarding the formation of carbocations A and B given.

Q12.One of the by-products formed during the recovery of ${\mathrm{NH}}_3$ from solvay process is:

Q13.In the above reaction, the structural formula of (A), "X" and "Y" respectively are:

Q14.During which of the following processes, does entropy decrease?
(A) Freezing of water to ice at $0^{\circ }\mathrm{C}$
(B) Freezing of water to ice at $-10^{\circ }\mathrm{C}$
(C) …

Q15.The correct pair(s) of the ambident nucleophiles is (are):
(A) $\mathrm{AgCN}/\mathrm{KCN}$
(B) $\mathrm{RCOOAg}/\mathrm{RCOOK}$
(C) ${\mathrm{AgNO}}_2/{\mathrm{KNO}}_2$
(D) $\mathrm{AgI}/\mathrm{KI}$

Q16.Match List - I with List - II:
List - I
List - II
(a) $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]\left[\mathrm{Cr}(\mathrm{CN}{)}_6\right]$
(i) Linkage isomerism
(b) $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_3{\left({\mathrm{NO}}_2\right)}_3\right]$
(ii) Solv …

Q17.Given below are two statements:
Statement I : 2-methylbutane on oxidation with ${\mathrm{KMnO}}_4$ gives 2-methylbutan-2-ol.
Statement II: n-alkanes can be easily oxidised to corresponding alcohols with ${\mathrm{KMnO}}_4$. Choose the correct option:

Q18.${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6\stackrel{\text{ Enzyme B }}{\to }2{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}+2{\mathrm{CO}}_2$

In the above reactions, the enzyme $A$ and enzyme $B$ respectively are:

Q19.Primary, secondary and tertiary amines can be separated using:

Q20.Amongst the following, the linear species is:

Q21.The reaction $2\mathrm{A}+{\mathrm{B}}_2\to 2\mathrm{AB}$ is an elementary reaction.
For a certain quantity of reactants, if the volume of the reaction vessel is reduced by a factor of 3, the rate of reaction increases by a factor of ______.
(Round off to the nearest …

Q22.In the ground state of atomic $\mathrm{Fe}(\mathrm{Z}=26)$, the spin-only magnetic moment is ______ $\times 10^{-1}\mathrm{BM}$.
(Round off to the nearest integer).
[Given $\sqrt{3}=1.73,\sqrt{2}=1.41$]

Q23.Consider the above reaction. The percentage yield of amide product is ______.
(Round off to the nearest integer).
(Given: Atomic mass: C:12.0u, H:1.0 u, N: $14.0\mathrm{u},\mathrm{O}:16.0\mathrm{u},\mathrm{Cl}:35.5\mathrm{u}$)

Q24.A KCl solution of conductivity $0.14{\mathrm{Sm}}^{-1}$ shows a resistance of $4.19\Omega$ in a conductivity cell. If the same cell is filled with an HCl solution, the resistance drops to $1.03\Omega$. The conductivity of the HCl solution is ______ …

Q25.On complete reaction of ${\mathrm{FeCl}}_3$ with oxalic acid in aqueous solution containing KOH, resulted in the formation of product A. The secondary valency of Fe in the product A is ______.
(Round off to the nearest integer).

Q26.A 1 molal ${\mathrm{K}}_4\mathrm{Fe}(\mathrm{CN}{)}_6$ solution has a degree of dissociation of 0.4. Its boiling point is equal to that of another solution which contains 18.1 weight percent of a non electrolytic solute A. The molar mass of $A$ is ______ u. (Round off to the …

Q27.The number of chlorine atoms in 20 mL of chlorine gas at STP is ______ $\times 10^{21}$ (Round off to the nearest integer).
[Assume chlorine is an ideal gas at STP …

Q28.Consider the reaction ${\mathrm{N}}_2{\mathrm{O}}_4(\mathrm{g})\rightleftharpoons 2{\mathrm{NO}}_2(\mathrm{g})$. The temperature at which ${\mathrm{K}}_{\mathrm{C}}=20.4$ and ${\mathrm{K}}_{\mathrm{p}}=600.1$, is ______ K. (Round off to the nearest integer).
[Assume all gase …

Q29.The total number of C - C sigma bond$/\mathrm{s}$ in mesityl oxide $\left({\mathrm{C}}_6{\mathrm{H}}_{10}\mathrm{O}\right)$ is ______. (Round off to the nearest integer).

Q30.KBr is doped with $10^{-5}$ mole percent of ${\mathrm{SrBr}}_2$. The number of cationic vacancies in 1 g of KBr crystal is ______ $\times 10^{14}$. (Round off to the nearest integer).
[Atomic Mass: K: $39.1\mathrm{u}$, Br: $79.9\mathrm{u}$, …

Q1.Let $S_1,S_2$ and $S_3$ be three sets defined as
$S_1=\{z\in C:|z-1|\le \sqrt{2}\}$
$S_2=\{z\in C:\mathrm{Re}((1-i)z)\ge 1\}$
$S_3=\{z\in C:\mathrm{Im}(z)\le 1\}$
Then the set $S_1\cap S_2\cap S_3$

Q2.If the Boolean expression $(p\wedge q)\odot (p\otimes q)$ is a tautology, then $\odot$ and $\otimes$ are respectively by:

Q3.Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k},x,y\in R,x>0$, be such that $|\overrightarrow{PQ}|=\sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OQ}$. I …

Q4.If the curve $y=y(x)$ is the solution of the differential equation
$2\left(x^2+x^{5/4}\right)dy-y\left(x+x^{1/4}\right)dx=2x^{9/4}dx,x>0$ which passes through the point $\left(1,1-\frac{4}{3}{\log }_{e}2\right)$, then the value of $y(16)$ is equal to:

Q5.Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac{1}{2}$ and probability of occurrence of 0 at the odd place be $\frac{1}{3}$. Then the probability that '10' is followed by 01 is …

Q6.Consider the function $f:R\to R$ defined by $f(x)=\{\begin{array}{ll}\left(2-\sin \left(\frac{1}{x}\right)\right)& |x|,x\ne 0\\ 0& ,x=0\end{array}$. Then $f$ is:

Q7.Let $y=y(x)$ be in the solution of the differential equation $\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3))dx,0\le x\le \frac{\pi }{2},y(0)=0$. Then $y\left(\frac{\pi }{3}\right)$ is equal to:

Q8.The value of the limit ${\mathrm{Lim}}_{\theta \to 0}\frac{\tan \left(\pi {\cos }^2\theta \right)}{\sin \left(2\pi {\sin }^2\theta \right)}$ is equal to:

Q9.The value of
${\lim }_{n\to \infty }\frac{[r]+[2r]+\dots +[nr]}{n^2}$,
Where $r$ is a non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$, is equal to:

Q10.The number of solutions of the equation ${\sin }^{-1}\left[x^2+\frac{1}{3}\right]+{\cos }^{-1}\left[x^2-\frac{2}{3}\right]=x^2$, for $x\in [-1,1]$ and $[x]$ denotes the greatest less than or equal to $x$, is:

Q11.Let $f:R\to R$ be defined as $f(x)=e^{-x}\sin x$. If $F:[0,1]\to R$ is a differentiable function such that $F(x)={\int }_0^{x}f(t)dt$, then the value of ${\int }_0^1\left(F^{\prime }(x)+f(x)\right)e^{x}dx$ lies in the interval

Q12.If the sides $AB,BC$ and $CA$ of triangle $ABC$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

Q13.If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is, $\alpha x+\beta y+\gamma z=24$ then …

Q14.Two tangents are drawn from a point $P$ to the circle $x^2+y^2-2x-4y+4=0$, such that the angle between these tangents is ${\tan }^{-1}\left(\frac{12}{5}\right)$, where ${\tan }^{-1}\left(\frac{12}{5}\right)\in (0,\pi )$. If the centre of the circle is denoted by $C$ a …

Q15.If $x,y,z$ are in arithmetic progression with common difference $d,x\ne 3d$, and the determinant of the matrix $\left[\begin{array}{ccc}3& 4\sqrt{2}& x\\ 4& 5\sqrt{2}& y\\ 5& k& z\end{array}\right]$ is zero, then the value of $k^2$ is:

Q16.Let $L$ be a tangent line to parabola $y^2=4x-20$ at $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^2}{2}+\frac{y^2}{b}=1$, then the value of $b$ is equal to:

Q17.The number of solution of the equation $x+2\tan x=\frac{\pi }{2}$ in the interval $[0,2\pi ]$ is:

Q18.If the integral ${\int }_0^{10}\frac{[\sin 2\pi x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$ where $\alpha ,\beta ,\gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x$, then the value of $\alpha +\beta +\gamma$ is equal …

Q19.Let the tangent to the circle $x^2+y^2=25$ at the point $R(3,4)$ meet $x$-axis and $y$-axis at points $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle OPQ , then …

Q20.The value of ${\sum }_{\mathrm{r}=0}^6\left({}^6{\mathrm{C}}_{\mathrm{r}}\cdot {}^6{\mathrm{C}}_{6-\mathrm{r}}\right)$ is equal to:

Q21.Let $I_n={\int }_1^e{x}^{19}(\log |x|{)}^{n}dx$, where $n\in N$. If (20) $l_{10}=\alpha l_9+\beta l_8$, for natural numbers $\alpha$ and $\beta$ then $\alpha -\beta$ equals to ______.

Q22.Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta \end{array}\right]\ne \left[\begin{array}{l}0\\ 0\end{array}\right]$ such that $\mathrm{AB}=\mathrm{B}$ and …

Q23.If $1,{\log }_{10}\left(4^{\mathrm{x}}-2\right)$ and ${\log }_{10}\left(4^{\mathrm{x}}+\frac{18}{5}\right)$ are in arithmetic progression for a real number x , then the value of the determinant …

Q24.Let $\tan \alpha ,\tan \beta$ and $\tan \gamma ;\alpha ,\beta ,\gamma \ne \frac{(2n-1)\pi }{2},n\in N$ be the slopes of three line segments $OA,OB$ and OC , respectively, where O is origin. If circumcentre of $△\mathrm{ABC}$ coincides with origin and its …

Q25.Let $P$ be an arbitrary point having sum of the squares of the distance from the planes $x+y+z=0$, $\ell x-nz=0$ and $x-2y+z=0$ equal to 9. If the locus of the point $P$ is $x^2+y^2+z^2=9$, then the value of $\ell -n$ is equal to ____.

Q26.Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. If the vector $\vec{x}$ is perpendicular to $(3\hat{i}+2\hat{j}-\hat{k})$ and its projection on $\vec{a}$ is $\frac{17\sqrt{6}}{2}$, then the …

Q27.Let $f:[-1,1]\to R$ be defined as $f(x)=a{x}^2+bx+c$ for all $x\in [-1,1]$, where $a,b,c\in R$ such that $f(-1)=2$, $f^{\prime }(-1)=1$ and for $x\in (-1,1)$ the maximum value of $f^{\prime \prime }(x)$ is $\frac{1}{2}$. If $f(x)\le \alpha$, $x\in [-1,1]$, then the …

Q28.Consider a set of $3n$ numbers having variance 4. In this set, the mean of first $2n$ numbers is 6 and the mean of the remaining n number is 3. A new set is constructed by adding 1 into each of first $2n$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If th …

Q29.Let $f:[-3,1]\to R$ be given as
$f(x)=\{\begin{array}{llc}\min \{(x+6),& x^2\},& -3\le x<0\\ \max \{\sqrt{x},& x^2\},& 0\le x\le 1\end{array}$
If the area bounded by $y=f(x)$ and $x$-axis is $A$, then the value of $6A$ is equal to ____.

Q30.Let the coefficients of third, fourth and fifth terms in the expansion of ${\left(x+\frac{a}{x^2}\right)}^n$, $x\ne 0$, be in the ratio 12:8:3. Then the term independent of x in the expansion is equal to ____.