JEE Main 2021 · 16 Mar 2021 · Shift 2

Q1.A mosquito is moving with a velocity $\vec{v}=0.5t^2\hat{i}+3t\hat{j}+9\hat{k}\mathrm{m}/\mathrm{s}$ and accelerating in uniform conditions. What will be the direction of mosquito after 2 s ?

Q2.In order to determine the young's modulus of a wire of radius 0.2 cm (measured using a scale of least count $=0.001\mathrm{cm}$) and length 1 m (measured using a scale of least count $=1\mathrm{mm}$), a weight of mass 1 kg (measured using a scale of least count …

Q3.The magnetic field in a region is given by $\vec{B}=B_0\left(\frac{x}{a}\right)\hat{k}$. A square loop of side d is placed with its edges along the x and y axes. The loop is moved with a constant velocity $\vec{v}=v_0\hat{i}$. The emf induced in the loop is:

Q4.Calculate the value of mean free path ($\lambda$) for oxygen molecules at temperature $27^{\circ }\mathrm{C}$ and pressure $1.01\times 10^5\mathrm{Pa}$. Assume the molecular diameter 0.3 nm and the gas is ideal. $(\mathrm{k}=1.38\times 10^{-23}{\mathrm{JK}}^{-1})$

Q5.Calculate the time interval between $33\%$ decay and $67\%$ decay if half-life of a substance is 20 minutes.

Q6.The following logic gate is equivalent to:

Q7.Red light differs from blue light as they have:

Q8.What will be the nature of flow of water form a circular tap, when its flow rate increased from $0.18\mathrm{L}/\min$ to $0.48\mathrm{L}/\min$ ? The radius of the tap and viscosity of water are 0.5 cm and $10^{-3}\mathrm{Pa}\mathrm{s}$, respectively.
( …

Q9.A large block of wood of mass $\mathrm{M}=5.99\mathrm{kg}$ is hanging form two long massless cords. A bullet of mass $\mathrm{m}=$ 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance …

Q10.A charge $Q$ is moving $d\vec{\ell }$ distance in the magnetic field $\vec{B}$. Find the value of work done by $\vec{B}$.

Q11.Two identical antennas on identical towers are separated from each other by a distance of 45 km . What should be the minimum height of receiving antenna to receive the signals in line of sight? (Assume radius of earth is 6400 km )

Q12.Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass $=500\mathrm{g}$, Decay constant $=20\mathrm{g}/\mathrm{s}$ then how much time is required for the amplitude of the system to drop to half of its initial value?
( …

Q13.The half- life of ${\mathrm{Au}}^{198}$ is 2.7 days. The activity of $1.50{\mathrm{mg}}^{198}$ of ${\mathrm{Au}}^{198}$ if atomic weight is $198\mathrm{g}{\mathrm{mol}}^{-1}$ is, $\left({\mathrm{N}}_{\mathrm{A}}=6\times 10^{23}/\mathrm{mol}\right)$

Q14.The de - Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of 100 V . What should be the ratio of their wavelengths? …

Q15.A bimetallic strip consists of metals A and B . It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that metal B . When bimetallic strip is placed in a cold bath, it will

Q16.The refractive index of converging lens is 1.4 . What will be the focal length of this if it is placed in a medium of same refractive index? Assume the radii of curvature of the faces of lens are $R_1$ and $R_2$ respectively.

Q17.Find out the surface charge density at the intersection of point $x=3m$ plane and $x$-axis, in the region of uniform line charge of $8\mathrm{nC}/\mathrm{m}$ lying along the $z$-axis in free space.

Q18.Statement I : A cyclist is moving on an un-banked road with a speed of $7{\mathrm{kmh}}^{-1}$ and takes a sharp circular turn along a path of radius of 2 m without reducing the speed. The static friction coefficient is 0.2 The cyclist will not slip and pass the curve. ( …

Q19.For the given circuit, comment on the type of transformer used.

Q20.A resistor develops 500 J of thermal energy in 20 s when a current of 1.5 A is passed through it. If the current is increased from 1.5 A to 3 A , what will be the energy developed in 20 s .

Q21.A closed-organ pipe of length $L$ and an open pipe contain gases of densities ${\rho }_1$ and ${\rho }_2$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is …

Q22.In a parallel plate capacitor set up, plate area of capacitor is $2{\mathrm{m}}^2$ and the plates are separated by 1 m . If the space between the plates are filled with a dielectric material of thickness 0.5 m and area $2{\mathrm{m}}^2$ (See fig) the capacitance of the set- u …

Q23.A swimmer can swim with velocity of $12\mathrm{km}/\mathrm{h}$ in still water. Water flowing in a river has velocity 6 $\mathrm{km}/\mathrm{h}$. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank j …

Q24.A body of mass 2 kg moves under a force of $(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+5\hat{\mathrm{k}})\mathrm{N}$. It starts from rest and was at the origin initially. After 4 s , its new coordinates are ( $8,b,20$ ). The value of $b$ is ____ (Round off the Nearest Integer …

Q25.The energy dissipated by a resistor is 10 mJ in 1 s when an electric current of 2 mA flows through it. The resistance is ____ $\Omega$. (Round off to the Nearest Integer)

Q26.A solid disc of radius 'a' and mass 'm' rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the will be $\frac{2}{b}g\sin \theta$ where $b$ is ____ (Round off to the Nearest Integer)
( $\mathrm{g}=$ acceleration …

Q27.A deviation of $2^0$ is produced in the yellow ray when prism of crown and flint glass are achromatically combined. Taking dispersive powers of crown and flint glass as 0.02 and 0.03 respectively and refractive index for yellow light for these glasses are 1.5 and 1.6 respective …

Q28.A force $\vec{F}=4\hat{i}+3\hat{j}+4\hat{k}$ is applied on an intersection point of $x=2$ plane and $x$-axis. The magnitude of torque of this force about a point $(2,3,4)$ is ⋯---------. (Round off to the Nearest Integer)

Q29.If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs so be supplied will be $\frac{x}{5}\frac{G{M}^2}{R}$ where $x$ is ⋯----. (Round off to the Nearest Integer)
( $M$ is the mass of earth, $R$ is the ra …

Q30.For an ideal heat engine, the temperature of the source is $127^{\circ }\mathrm{C}$. In order to have $60\%$ efficiency the temperature of the sink should be ⋯⋯----- ${}^{\circ }\mathrm{C}$. (Round off to the Nearest Integer)

Q1.Ammonolysis of Alkyl halides followed by the treatment with NaOH solution can be used to prepare primary, secondary and tertiary amines. The purpose of NaOH in the reaction is:

Q2.The characteristics of elements $\mathrm{X}$, $\mathrm{Y}$ and Z with atomic numbers, respectively, 33, 53, and 83 are:

Q3.The INCORRECT statements below regarding colloidal solutions is:

Q4.Statement I : Sodium hydride can be used as an oxidizing agent.
Statement II : The lone pair of electrons on nitrogen in pyridine makes it basic.
Choose the CORRECT answer from the options given below:

Q5.The green house gas/es is (are) :
(A) Carbon dioxide
(B) Oxygen
(C) Water vapour
(D) Methane

Choose the most appropriate answer from the options given below:

Q6.${\mathrm{Fex}}_2$ and ${\mathrm{Fey}}_3$ are known when x and y are:

Q7.Identify the elements X and Y using the ionization energy values given below:
Ionization energy ( $\mathrm{kJ}/\mathrm{mol}$ )

Q8.Identify the reagent(s) 'A' and condition(s) for the reaction

Q9.Which of the following is least basic?

Q10.The INCORRECT statement regarding the structure of ${\mathrm{C}}_{60}$ is:

Q11.Which of the following reduction reaction CANNOT be carried out with coke?

Q12.(i) ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{MgBr}$
$\underset{\text{ (ii) }{\mathrm{H}}_3{\mathrm{O}}^{+}}{\overset{\text{(1.0 equivalent), dry Ether }}{\to }}\underset{\text{ Major Product }}{\mathrm{X}}$

The structure of $X$ is:

Q13.An unsaturated hydrocarbon $X$ on ozonolysis gives $A$. Compound $A$ when warmed with ammonical silver nitrate forms a bright silver mirror along the sides of the test tube. The unsaturated hydrocarbon $X$ is:

Q14.In the above reaction, the reagent "A" is:

Q15.The correct statement about ${\mathrm{H}}_2{\mathrm{O}}_2$ are:
(A) Used in the treatment of effluents.
(B) Used as both oxidizing and reducing agents
(C) The two hydroxyl groups lie in the same plane.
(D) Miscible with water.

Choose the correct answer from the options given …

Q16.Match List - I with List - II

List - I: Test / Reagents / Observation(s)
List - II: Species detected

The correct match is:

Q17.Arrange the following metal complex / compounds in the increasing order of spin only magnetic moment. Presume all the three, high spin system.
(Atomic numbers $\mathrm{Ce}=58,\mathrm{Gd}=64$ and $\mathrm{Eu}=63$.)
(a) …

Q18.The secondary structure of protein is stabilised by:

Q19.The exact volumes of 1 M NaOH solution required to neutralize 50 mL of $1\mathrm{M}{\mathrm{H}}_3{\mathrm{PO}}_3$ solution and 100 mL of $2\mathrm{M}{\mathrm{H}}_3{\mathrm{PO}}_2$ solution respectively, are:

Q20.Which of the following polymer is used in the manufacture of wood laminates?

Q21.Sulphurous acid $\left({\mathrm{H}}_2{\mathrm{SO}}_3\right)$ has ${\mathrm{Ka}}_1=1.7\times 10^{-2}$ and ${\mathrm{Ka}}_2=6.4\times 10^{-8}$. The pH of $0.588\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_3$ is ______. (Round off to the nearest integer).

Q22.The number of orbitals with $\mathrm{n}=5,{\mathrm{m}}_{\mathrm{l}}=+2$ is ______. (Round off to the nearest integer).

Q23.A and B decompose via first order kinetics with half-lives 54.0 min and 18.0 min respectively. Starting from an equimolar non reactive mixture of A and B, the time taken for the concentration of $A$ to become 16 times that of $B$ is ______ min.
(Round off to the nearest integ …

Q24.Ga (atomic mass 70u) crystallizes in a hexagonal close packed structure. The total number of voids in 0.581 g of Ga is ______ $\times 10^{21}$. (Round off to the nearest integer).
[Given : ${\mathrm{N}}_{\mathrm{A}}=6.023\times 10^{23}$]

Q25.When 35 mL of 0.15 M lead nitrate solution is mixed with 20 mL of 0.12 M chromic sulphate solution, _____ $\times 10^{-5}$ moles of lead sulphate precipitate out.
(Round off to the nearest integer).

Q26.In Duma's method of estimation of nitrogen, 0.1840 g of an organic compound gave 30 mL of nitrogen collected at 287 K and 758 mm of Hg pressure. The percentage composition of nitrogen in the compound is _____. (Round off to the nearest integer).
[Given: Aqueous tension at …

Q27.At $25^{\circ }\mathrm{C},50\mathrm{g}$ of iron reacts with HCl to form ${\mathrm{FeCl}}_2$. The evolved hydrogen gas expands against a constant pressure of 1 bar. The work done by the gas during this expansion is _____ J. (Round off to the nearest integer).
[Given: …

Q28.At 363 K, the vapour pressure of $A$ is 21 kPa and that of $B$ is 18 kPa. One mole of $A$ and 2 moles of $B$ are mixed. Assuming that this solution is ideal, the vapour pressure of the mixture is _____ kPa. (Round off to the nearest integer).

Q29.${\left[\mathrm{Ti}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6\right]}^{3+}$ absorbs light of wavelength 498 nm during a d-d transition. The octahedral splitting energy for the above complex is _____ $\times 10^{-19}\mathrm{J}$. (Round off to the nearest integer). …

Q30.A $5.0\mathrm{mmol}{\mathrm{dm}}^{-3}$ aqueous solution of KCl has a conductance of 0.55 mS when measured in a cell of cell constant $1.3{\mathrm{cm}}^{-1}$. The molar conductivity of this solution is _____ ${\mathrm{mSm}}^2{\mathrm{mol}}^{-1}$. (Round off to the nearest integer) …

Q1.Consider the integral
$\mathrm{I}={\int }_0^{10}\frac{[\mathrm{x}]{\mathrm{e}}^{[\mathrm{x}]}}{{\mathrm{e}}^{\mathrm{x}-1}}\mathrm{dx}$, where $[\mathrm{x}]$ denotes the greatest integer less than or equal to x. Then the value of I is equal to :

Q2.The maximum value of $f(x)=\left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|,x\in R$ is :

Q3.If $(x,y,z)$ be an arbitrary point lying on a plane $P$ which passes through the points $(42,0,0),(0,42,0)$ and $(0,0,42)$, then the value of the expression …

Q4.If the foot of the perpendicular from point $(4,3,8)$ on the line $L_1:\frac{x-a}{\ell }=\frac{y-2}{3}=\frac{z-b}{4},\ell \ne 0$ is (3, 5, 7) then the shortest distance between the line $L_1$ and line $L_2:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is equal to :

Q5.Let $f$ be a real valued function, defined on $R-\{-1,1\}$ and given by $f(x)=3{\log }_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$. Then in which of the following intervals, function $f(x)$ is increasing?

Q6.Consider a rectangle $ABCD$ having $5,7,6,9$ points in the interior of the segments $AB,CD,BC,DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these point …

Q7.Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy ${\sin }^{-1}\left(\frac{3x}{5}\right)+{\sin }^{-1}\left(\frac{4x}{5}\right)={\sin }^{-1}x$ is equal to :

Q8.Let $\alpha \in R$ be such that the function $f(x)=\{\begin{array}{cl}\frac{{\cos }^{-1}\left(1-\{x{\}}^2\right){\sin }^{-1}(1-\{x\})}{\{x\}-\{x{\}}^3}& ,x\ne 0\\ \alpha & ,x=0\end{array}$ is continuous at $x=0$, where $\{x\}=x-[x],[x]$ is the greatest inte …

Q9.Let $A=\{2,3,4,5,\dots ,30\}$ and ' $\sim$ ' be an equivalence relation on $A\times A$, defined by $(a,b)\sim (c,d)$, if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

Q10.Let the lengths of intercepts on $x$-axis and $y$-axis made by the circle $x^2+y^2+ax+2ay+c=0,(a<0)$ be $2\sqrt{2}$ and $2\sqrt{5}$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x+2y=0$, is …

Q11.Let $C$ be the locus of the mirror image of a point on the parabola $y^2=4x$ with respect to the line $y=x$. Then the equation of tangent to $C$ at $P(2,1)$ is :

Q12.The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\log }_{e}2\right)\ge {\log }_{\sqrt{2}}|5\sqrt{7}+9i|,i=\sqrt{-1}$, is equal to :

Q13.Let A denote the event that a 6-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, by divisible by 3. Then probability of event A is equal to :

Q14.If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\tan x)y=\sin x,0\le x\le \frac{\pi }{3}$, with $y(0)=0$, then $y\left(\frac{\pi }{4}\right)$ equal to:

Q15.Let $P(x)=x^2+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_0^{1}P(x)dx=1$ and $P(x)$ leaves remainder 5 when it is divided by $(x-2)$. Then the value of $9(b+c)$ is equal to :

Q16.Let $f:S\to S$ where $S=(0,\infty )$ be a twice differentiable function such that $f(x+1)=xf(x)$. If $g:S\to R$ be defined as $g(x)={\log }_{e}f(x)$, then the value of $\left|g^{\prime \prime }(5)-g^{\prime \prime }(1)\right|$ is equal to :

Q17.Let ${\mathrm{C}}_1$ be the curve obtained by the solution of differential equation $2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{y}}^2-{\mathrm{x}}^2,\mathrm{x}>0$. Let the curve $C_2$ be the solution of $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}$. If both t …

Q18.Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}-3\hat{j}+5\hat{k}$.
If $\vec{r}\times \vec{a}=\vec{b}\times \vec{r},\vec{r}.(a\hat{i}+2\hat{j}+\hat{k})=3$ and $\vec{r}.(2\hat{i}+5\hat{j}-\alpha \hat{k})=-1,\alpha \in R$, then the value of …

Q19.Let $A(-1,1),B(3,4)$ and $C(2,0)$ be given three points. A line $y=mx,m>0$, intersects lines $AC$ and BC at point P and Q respectively. Let ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ be the area of $△\mathrm{ABC}$ and $△PQC$ respectively, such tha …

Q20.If the points intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4b,b>4$ lie on the curve $y^2=3x^2$, then $b$ is equal to :

Q21.Let $S_n(x)={\log }_{a^{1/2}}x+{\log }_{a^{1/3}}x+{\log }_{a^{1/6}}x+{\log }_{a^{1/11}}x+{\log }_{a^{1/18}}x+{\log }_{a^{1/27}}x+\dots$. up to $n$-terms. Where $a>$ 1. If $S_{24}(x)=1093$ and $S_{12}(2x)=265$, then the value of $a$ is equal to.....

Q22.For real numbers $\alpha ,\beta ,\gamma$ and $\delta$, if $\int \frac{\left(x^2-1\right)+{\tan }^{-1}\left(\frac{x^2+1}{x}\right)}{\left(x^4+3x^2+1\right){\tan }^{-1}\left(\frac{x^2+1}{x}\right)}dx$ …

Q23.Let $A=\left[\begin{array}{l}{a}_1\\ a_2\end{array}\right]$ and $B=\left[\begin{array}{l}{b}_1\\ b_2\end{array}\right]$ be two $2\times 1$ matrices with real entries such that $A=XB$, where …

Q24.If the distance of the point $(1,-2,3)$ from the plane $x+2y-3z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac{7}{2}}$, then the value of $|m|$ is equal to......

Q25.Let $n$ be a positive integer. Let $A={\sum }_{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})\left[{\left(\frac{1}{2}\right)}^k+{\left(\frac{3}{4}\right)}^k+{\left(\frac{7}{8}\right)}^k+{\left(\frac{15}{16}\right)}^k+{\left(\frac{31}{32}\right)}^k\right]$. If $63A=1-\frac{1}{2^{30}}$, then …

Q26.Let $\frac{1}{16}$, $a$ and $b$ be in G.P. and $\frac{1}{a},\frac{1}{b},6$ be in A.P., where $a,b>0$. Then $72(a+b)$ is equal to ______

Q27.In $△ABC$, the lengths of sides $AC$ and $AB$ are 12 cm and 5 cm, respectively. If the area of $△ABC$ is $30{\mathrm{cm}}^2$ and $R$ and $r$ are respectively the radii of circumcircle and incircle of $△ABC$, then the value of $2R+r$ …

Q28.Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be defined as
$f(x)=\{\begin{array}{ll}x+a,& x<0\\ |x-1|,& x\ge 0\end{array}$ and $g(x)=\{\begin{array}{ll}x+1,& x<0\\ (x-1{)}^2+b,& x\ge 0\end{array}$, where $a,b$ are non-negative rea …

Q29.Consider the statistics of two sets of observations as follows:

If the variance of the combined set of these two observations is $\frac{17}{9}$, then the value of …

Q30.Let $\vec{c}$ be a vector perpendicular to the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+3\hat{k})=8$ then the value of $\vec{c}\cdot (\vec{a}\times \vec{b})$ is equal to ______