JEE Main 2021 · 18 Mar 2021 · Shift 2

Q1.The velocity -displacement graph of a particle is shown in the figure. The acceleration - displacement graph of the same particle is represented by :

Q2.Which of the following statements are correct ?
(A) Electric monopoles do not exist whereas magnetic monopoles exist .
(B) Magnetic field lines due to a solenoid at its ends and outside can not be completely straight and confined
(C) Magnetic field lines are completely confined w …

Q3.The speed of electrons in a scanning electron microscope is $1\times 10^7\mathrm{m}/\mathrm{s}$. If protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of

Q4.A proton and an $\alpha$-particle, having kinetic energy ${\mathrm{K}}_{\mathrm{p}}$ and ${\mathrm{K}}_{\alpha }$ respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of $\alpha$-particle is $2:1$. The ratio of …

Q5.The time taken for the magnetic energy to reach $25\%$ of its maximum value, when solenoid of resistance R , inductance L is connected to a battery, is:

Q6.A particle of mass m moves in a circular orbit under the central potential field, $V(r)=-\frac{C}{r}$, where C is a constant. The correct radius - velocity graph of the particle's motion is ;

Q7.For an adiabatic expansion of an ideal gas , the fractional change in its pressure is equal to ( where $\gamma$ is the ratio of specific heats):

Q8.Three rays of light, red ( R ), green( G) and blue ( B) are incident on the face PQ of a right angled prism PQR as shown in the figure.
The refractive indices of material of the prism for red , green and blue wavelength are 1.27, 1.42 and 1.49 respectively. The colour of ray(s) e …

Q9.The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is $\vec{L}$. The magnitude of the areal velocity of the planet is ;

Q10.The decay of a proton to neutron is :

Q11.The function of time representing a simple harmonic motion with time period of $\frac{\pi }{\omega }$ is :

Q12.If the angular velocity of earth's spin is increased such that the bodies at the equator start floating , the duration of the day would be approximately :
[Take $g=10{\mathrm{ms}}^{-2}$, the radius of earth, $R=6400\times 10^3\mathrm{m}$, Take $\pi =3.14$ ]

Q13.The correct relation between $\alpha$ ( ratio of collector current to emitter current ) and $\beta$ ( ratio of collector current to base current) of a transistor is :

Q14.An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $S_1$ and that of the other part is $S_2$. Given that $S_1>S_2$. If the piston is removed then the total entropy of the system will be :

Q15.An object of mass $m_1$ collides with another object of mass $m_2$, which is at rest. After collision the objects move with equal speeds in the opposite direction. The ratio of the masses $m_2$ : $m_1$ is :

Q16.In a series LCR circuit, the inductive reactance ( ${\mathrm{X}}_{\mathrm{L}}$ ) is $10\Omega$, and the capacitive reactance ( ${\mathrm{X}}_{\mathrm{C}}$ ) is $4\Omega$. The resistance (R) in the circuit is $6\Omega$. The power factor of the circuit is :

Q17.Consider a uniform wire of mass M and length L . It is bent into a semi-circle. Its moment of inertia about a line perpendicular to the plane of the wire passing through centre is :

Q18.A plane electromagnetic wave propagating along y-direction, can have the following pair of electric field ($\vec{E}$) and magnetic field ($\vec{B}$) components :

Q19.A solid cylinder of mass $m$ is wrapped with an inextensible light string and is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:
[the coefficient of static friction, ${\mu }_{\mathrm{s}}$ is 0.4 …

Q20.Consider a sample of oxygen behaving like an ideal gas. At 300 K, the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be :
(Molecular weight of oxygen is $32\mathrm{g}/\mathrm{mol}$; $R=8.3{\mathrm{JK}}^{-1}{\mathrm{mol}}^{-1}$)

Q21.The typical output characteristics curve for a transistor working in the common-emitter configuration is shown in the figure.

The estimated current gain from the figure is ____

Q22.A TV transmission tower antenna is at a height of 20 m. Suppose that the receiving antenna is at
(i) ground level
(ii) a height of 5 m

The increase in the antenna range in case (ii) relative to case (i) is $n\%$.
The value of n, to the nearest integer, is ____

Q23.Consider a 72 cm long wire AB as shown in the figure. The galvanometer jockey is placed at P on AB at a distance x cm from A. The galvanometer shows zero deflection.

The value of x, to nearest integer, is ____

Q24.Consider a water tank as shown in the figure. Its cross-sectional area is $0.4{\mathrm{m}}^2$. The tank has an opening B near the bottom whose cross-sectional area is $1{\mathrm{cm}}^2$. A load of 24 kg is applied on the water at the top when the height of the water level …

Q25.Two wires of same length and thickness having specific resistance $6\Omega \mathrm{cm}$ and $3\Omega \mathrm{cm}$ respectively are connected in parallel. The effective resistivity is $\rho \Omega \mathrm{cm}$. The value of $\rho$, to the nearest integer, is ____

Q26.The radius of a sphere is measured to be $(7.50\pm 0.85)\mathrm{cm}$. Suppose the percentage error in its volume is $x$. The value of $x$, to the nearest integer, is ____

Q27.The projectile motion of a particle of mass $5g$ is shown in the figure. The initial velocity of the projectile is $5\sqrt{2}{\mathrm{ms}}^{-1}$ and air resistance is assumed to be negligible. The magnitude of change in momentum between the points A to B is …

Q28.A galaxy is moving away from the earth at a speed $286km/s$. The shift in wavelength of a redline at 630 nm is $x\times 10^{-10}\mathrm{m}$. The value of $x$, to the nearest integer, is ____
[Take the value of speed of light c, as …

Q29.A ball of mass 4 kg, moving with a velocity $10{\mathrm{ms}}^{-1}$, collides with a spring of length 8 m and force constant $100{\mathrm{Nm}}^{-1}$. The length of compressed spring is x m. The value of x, to the nearest integer, is ____

Q30.An infinite number of point charges, each carrying $1\mu \mathrm{C}$ charge, are placed along y-axis at $\mathrm{y}=1\mathrm{m},2\mathrm{m},4\mathrm{m},8\mathrm{m}$, ______. The total force on a 1 C point charge, placed at the origin is $x\times 10^{3}N$. The …

Q1.The secondary valency and the number of hydrogen bonded water molecule(s) in ${\mathrm{CuSO}}_4.5{\mathrm{H}}_2\mathrm{O}$ respectively, are:

Q2.Deficiency of vitamin K causes:

Q3.In the reaction of hypobromite with amide, the carbonyl carbon is lost as:

Q4.Consider the given reaction, percentage yield of:

Q5.Consider the above reaction, the product 'X' and 'Y' respectively are:

Q6.Match List - I with List - II:

List - I
(a) Mercury
(b) Copper
(c) Silicon
(d) Nickel

List - II
(i) Vapour phase refining
(ii) Distillation Refining
(iii) Electrolytic Refining
(iv) Zone Refining

Choose the most appropriate answer from the option given below:

Q7.The first ionization energy of magnesium is smaller as compared to that of elements $X$ and $Y$, but higher than that of $Z$. The elements $X,Y$ and $Z$, respectively are:

Q8.Main products formed during a reaction of 1-methoxy naphthalene with hydroiodic acid are:

Q9.The charges on the colloidal CdS sol and ${\mathrm{TiO}}_2$ sol are, respectively:

Q10.An organic compound "A" on treatment with benzene sulphonyl chloride gives compound B. B is soluble in dil. NaOH solution. Compound A is:

Q11.A solute A dimerizes in water. The boiling point of a 2 molal solution of A is $100.52^{\circ }\mathrm{C}$. The percentage association of $A$ is ____. (Round off to the nearest integer).
[Use: ${\mathrm{K}}_{\mathrm{b}}$ for water …

Q12.The number of species below that have two lone pairs of electrons in their central atom is ____. (Round off to the nearest integer). …

Q13.A xenon compound 'A' upon partial hydrolysis gives ${\mathrm{XeO}}_2{\mathrm{F}}_2$. The number of lone pair of electrons present in compound $A$ is ____. (Round off to the nearest integer).

Q14.In Tollen's test for aldehyde, the overall number of electron(s) transferred to the Tollen's reagent formula ${\left[\mathrm{Ag}{\left({\mathrm{NH}}_3\right)}_2\right]}^{+}$ per aldehyde group to form silver mirror is ______.
(Round off to the nearest integer).

Q15.10.0 mL of ${\mathrm{Na}}_2{\mathrm{CO}}_3$ solution is titrated against 0.2 M HCl solution.
The following titre values were obtained in 5 readings:
$4.8\mathrm{mL},4.9\mathrm{mL}5.0\mathrm{mL}5.0\mathrm{mL}$ and 5.0 mL
Based on these readings, and convention of titri …

Q16.The molar conductivities at infinite dilution of barium chloride, sulphuric acid and hydrochloric acid are 280, 860 and $426{\mathrm{Scm}}^2{\mathrm{mol}}^{-1}$ respectively. The molar conductivity at infinite dilution of barium sulphate is ______ …

Q17.A reaction has a half life of 1 min. The time required for $99.9\%$ completion of the reaction is ______ min. (Round off to the nearest integer).
[Use $\ln 2=0.69;\ln 10=2.3$]

Q18.Consider the above reaction where 6.1 g of Benzoic acid is used to get 7.8 g of m - bromo benzoic acid. The percentage yield of the product is ______.
(Round off to the nearest integer).
[Given : Atomic masses : C : 12.0u, H:1.0u, O:16.0u, Br:80.0 u]

Q19.The gas phase reaction
$2\mathrm{A}(\mathrm{g})\rightleftharpoons {\mathrm{A}}_2(\mathrm{g})$
At 400 K has $\Delta {\mathrm{G}}^{\circ }=+25.2\mathrm{kJ}{\mathrm{mol}}^{-1}$.
The equilibrium constant ${\mathrm{K}}_{\mathrm{C}}$ for this reaction is ______ $\times 10^{-2}$.
(Rou …

Q20.The solubility of ${\mathrm{CdSO}}_4$ in water is $8.0\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$. Its solubility in $0.01\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ solution is
______ $\times 10^{-6}\mathrm{mol}{\mathrm{L}}^{-1}$.
(Round off to the nearest integer).
(Assum …

Q21.Given below are two statements:
Statement I: Bohr's theory accounts for the stability and line spectrum of Li⁺ ion.
Statement II: Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field.
In the light of the above statements, choo …

Q22.Match List - I with List - II:
List - I: (a) Be (b) Mg (c) Ca (d) Ra
List - II: (i) Treatment of cancer (ii) Extraction of metals (iii) Incendiary bombs and signals (iv) Windows of X-ray tubes (v) Bearings for motor engines.
Choose the most appropriate answer from the o …

Q23.Given below are two statements:
Statement I: C₂H₅OH and AgCN both can generate nucleophile.
Statement II: KCN and AgCN both will generate nitrile nucleophile with all reaction conditions.
Choose the most appropriate option:

Q24.The oxide that shows magnetic property is:

Q25.In basic medium H₂O₂ exhibits which of the following reactions?
(A) Mn²⁺ → Mn⁴⁺
(B) I₂ → I⁻
(C) PbS → PbSO₄
Choose the most appropriate answer from the options given below:

Q26.A hard substance melts at high temperature and is an insulator in both solid and in molten state. This solid is most likely to be a/an:

Q27.Given below are two statements:
Statement I: Non-biodegradable wastes are generated by the thermal power plants.
Statement II: Bio-degradable detergents leads to eutrophication.
In the light of the above statements, choose the most appropriate answer from the options given bel …

Q28.In the given molecule (CH₃-CH=C=O type; carbons labelled a, b, c shown in source), hybridisation of Carbon a, b and c respectively are:

Q29.Match List - I (Class of Chemicals) with List - II (Example):
List - I: (a) Antifertility drug (b) Antibiotic (c) Tranquilizer (d) Artificial Sweetener
List - II: (i) Meprobamate (ii) Alitame (iii) Norethindrone (iv) Salvarsan
Choose the most appropriate match:

Q30.The oxidation states of nitrogen in NO, NO₂, N₂O and NO₃⁻ are in the order of:

Q1.Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$. If $R$ and $r$ be the radius of circumcircle and incircle respectively of $△ABC$, then $(R+r)$ is equal to :

Q2.A pole stands vertically inside a triangular park ABC. Let the angel of elevation of the top of the pole from each corner of the park be $\frac{\pi }{3}$. If the radius of the circumcircle of $△ABC$ is 2, then the height of the pole is equal to :

Q3.Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "ARB iff there exists a nonsingular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true?

Q4.Consider a hyperbola $\mathrm{H}:{\mathrm{x}}^2-2{\mathrm{y}}^2=4$. Let the tangent at a point $\mathrm{P}(4,\sqrt{6})$ meet the x-axis at Q and latus rectum at $\mathrm{R}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right),{\mathrm{x}}_1>0$. If F is a focus of H which is neare …

Q5.Let $f:R-\{3\}\to R-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$.
Let $g:R\to R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$ is equal to :

Q6.Let in a series of $2n$ observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $a^2+b^2$ is eq …

Q7.Let $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are two non-zero vectors perpendicular to each other and $|\overrightarrow{\mathrm{a}}|=|\overrightarrow{\mathrm{b}}|$. If …

Q8.If $15{\sin }^4\alpha +10{\cos }^4\alpha =6$, for some $\alpha \in R$, then the value of $27{\sec }^6\alpha +8{\mathrm{cosec}}^6\alpha$ is equal to :

Q9.If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

Q10.Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^2/2}-x\right),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to:

Q11.Let $S_1$ be the sum of first $2n$ terms of an arithmetic progression. Let $S_2$ be the sum of first $4n$ terms of the same arithmetic progression. If $(S_2-S_1)$ is 1000, then the sum of the first $6n$ terms of the arithmetic progression is equal to :

Q12.Let $f:R\to R$ be a function defined as
$f(x)=\{\begin{array}{cc}\frac{\sin (a+1)x+\sin 2x}{2x},& \text{if }x<0\\ b& ,\text{if }x=0\\ \frac{\sqrt{x+b{x}^3}-\sqrt{x}}{b{x}^{5/2}}& ,\text{if }x>0\end{array}$
If $f$ is continuous at $x=0$, then …

Q13.Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3\sqrt{3}\cos \theta ,\sin \theta )$ where $\theta \in \left(0,\frac{\pi }{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Q14.Let the system of linear equations
$4x+\lambda y+2z=0$
$2x-y+z=0$
$\mu x+2y+3z=0,\lambda ,\mu \in R$
has a non-trivial solution. Then which of the following is true?

Q15.Let $g(x)={\int }_0^{x}f(t)dt$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\le f(t)\le 1$ for all $t\in [0,1]$ and $0\le f(t)\le \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which $g(3)$ lies is :

Q16.In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to :

Q17.Let a complex number be $\mathrm{w}=1-\sqrt{3}\mathrm{i}$. Let another complex number z be such that $|\mathrm{zw}|=1$ and $\arg (z)-\arg (w)=\frac{\pi }{2}$. Then the area of the triangle with vertices origin, $z$ and $w$ is equal to :

Q18.The area bounded by the curve $4y^2=x^2(4-x)(x-2)$ is equal to :

Q19.Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :

Q20.Let $S_1:x^2+y^2=9$ and $S_2:(x-2{)}^2+y^2=1$. Then the locus of center of a variable circle $S$ which touches $S_1$ internally and $S_2$ externally always passes through the points :

Q21.Let ${}^n{C}_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x{)}^n$. If ${\sum }_{k=0}^{10}{\left(2^2+3k\right)}^n{C}_k=\alpha \cdot 3^{10}+\beta \cdot 2^{10},\alpha ,\beta \in R$, then $\alpha +\beta$ is equal to

Q22.Let I be an identity matrix of order $2\times 2$ and $\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$. Then the value of $\mathrm{n}\in \mathrm{N}$ for which $P^n=5I-8P$ is equal to

Q23.Let $f:R\to R$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in R$ and $f(x)\ne 0$ for any $x\in R$. If the function $f$ is differentiable at $x=0$ and $f^{\prime }(0)=3$, then ${\lim }_{h\to 0}\frac{1}{h}(f(h)-1)$ is equal to

Q24.Let the mirror image of the point $(1,3,a)$ with respect to the plane $\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})-b=0$ be $(-3,5,2)$. Then, the value of $|a+b|$ is equal to

Q25.Let $P$ be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1,\alpha )$ lies on the plane $P$, then the value of $|5\alpha |$ is equal to....

Q26.Let $P(x)$ be a real polynomial of degree 3 which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and ${\int }_{-1}^{1}P(x)dx=18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to......

Q27.The term independent of $x$ in the expansion of ${\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]}^{10},x\ne 1$, is equal to.....

Q28.If ${\sum }_{r=1}^{10}r!(r^3+6r^2+2r+5)=\alpha (11!)$, then the value of $\alpha$ is equal to.......

Q29.If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f(x^3)+xg(x^3)$ is divisible by $x^2+x+1$, then $P(1)$ is equal to ______

Q30.Let $y=y(x)$ be the solution of the differential equation $xdy-ydx=\sqrt{(x^2-y^2)}dx,x\ge 1$, with $y(1)=0$. If the area bounded by the line $x=1,x=e^{\pi },y=0$ and $y=y(x)$ is $\alpha e^{2\pi }+\beta$, then the value of $10(\alpha +\beta )$ is equal to......