JEE Main 2021 · 27 Jul 2021 · Shift 2

Q1.The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of $9.0\times 10^3\mathrm{km}$. Find the mass of Mars.
$\{$ Given $\frac{4{\pi }^2}{\mathrm{G}}=6\times 10^{11}{\mathrm{N}}^{-2}{\mathrm{kg}}^2\}$

Q2.A raindrop with radius $\mathrm{R}=0.2\mathrm{mm}$ falls from a cloud at a heighth $=2000\mathrm{m}$ above the ground. Assume that the drop is spherical through its fall and the force of buoyance may be neglected, then the terminal speed attained by the raindrop is :
[Density …

Q3.Figure A and B show long straight wires of circular cross - section ( a and b with $\mathrm{a}<\mathrm{b}$ ), carrying current I which is uniformly distributed across the cross - section. The magnitude of magnetic field $B$ varies with radius $r$ and can be represented as :

Q4.The resistance of a conductor at $15^{\circ }\mathrm{C}$ is $16\Omega$ and at $100^{\circ }\mathrm{C}$ is $20\Omega$. What will be the temperature coefficient of resistance of the conductor ?

Q5.Two identical particles of mass 1 kg each go round a circle of radius R , under the action of their mutual gravitational attraction. The angular speed of each particle is :

Q6.Consider the following statements:
(A) Atoms of each element emit characteristics spectrum.
(B) According to Bohr's Postulate, an electron in a hydrogen atom, revolves in a certain stationary orbit.
(C) The density of nuclear matter depends on the size of the nucleus.
(D) A free …

Q7.Match List I with List II.

List I:
(a) Capacitance, C
(b) Permittivity of free space, ${\epsilon }_0$
(c) Permeability of free space, ${\mu }_0$
(d) Electric field, E

List II:
(i) ${\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}$
(ii) …

Q8.One mole of an ideal gas is taken through an adiabatic process where the temperature rises from $27^{\circ }\mathrm{C}$ to $37^{\circ }\mathrm{C}$. If the ideal gas is composed of polyatomic molecule that has 4 vibrational modes, which of the following of the following is true? …

Q9.Two Carnot engines $A$ and operate in series such that engine $A$ absorbs heat at $T_1$ and rejects heat to a sink at temperature T. Engine B absorbs half of the heat rejected by Engine A and rejects heat to the sink at $T_3$. When work done in both the cases is equal, the va …

Q10.An electron and proton are separated by a large distance. The electron starts approaching the proton with energy 3 eV. The proton captures the electron and forms a hydrogen atom in second excited state. The resulting photon is incident on a photosensitive metal of threshold wavel …

Q11.An object of mass 0.5 kg executing simple harmonic motion. Its amplitude is 5 cm and time period $(T)$ is 0.2 s. What will be the potential energy of the object at an instant $t=\frac{T}{4}$ s starting from mean position. Assume that the initial phase of the oscillation is zero.

Q12.What will be the magnitude of electric field at point O as shown in figure? Each side of the figure is $\ell$ and perpendicular to each other?

Q13.A $100\Omega$ resistance, a $0.1\mu \mathrm{F}$ capacitor and an inductor are connected in series across a 250 V supply at variable frequency. Calculate the value if inductance of inductor at which resonance will occur. Given that the resonant frequency is 60 Hz.

Q14.An automobile of mass '$m$' accelerates starting from origin and initially at rest, while the engine supplies constant power P. The position is given as a function of time by:

Q15.Find the truth table for the function $Y$ of $A$ and $B$ represented in the following figure.

Q16.A simple pendulum of mass '$m$', length '$\ell$' and charge '$+q$' suspended in the electric field produced by two conducting parallel plates as shown. The value of deflection of pendulum in equilibrium position will be:

Q17.Given below is the plot of a potential energy function $\mathrm{U}(\mathrm{x})$ for a system, in which a particle is in one dimensional motion, while a conservation force $F(x)$ acts on it. Suppose that $E_{\text{mech }}=8\mathrm{J}$, the incorrect statement for this system is …

Q18.The expected graphical representation of the variation of angle of deviation '$\delta$' with angle of incidence '$i$' in a prism is :

Q19.A physical '$y$' is represented by the formula $y=m^2{r}^{-4}{g}^x{\ell }^{-\frac{3}{2}}$ if the percentage errors found in $y,m$, $\mathrm{r},\ell$ and g are 18, $1,0.5,4$ and p respectively, then find the value of x and p .

Q20.A particle of mass M originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation


Where $F_0$ and $T$ are constants. The force acts only for the ti …

Q21.A swimmer wants to cross a river from point A to point B . Line $AB$ makes an angle of $30^{\circ }$ with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle $\theta$ with the line $AB$ should be
____ ${}^{\circ }$, so th …

Q22.For the circuit shown, the value of current at time $t=3.2\mathrm{s}$ will be
____ A.

[Voltage distribution $\mathrm{V}(\mathrm{t})$ is shown by Fig.(1) and the circuit is shown in Fig.(2)]

Q23.The difference in the number of waves when yellow light propagates through air and vacuum columns of the same thickness is one. The thickness of the air column is ____ mm .
[Refractive index of air $=1.0003$, wavelength of yellow light in vacuum $=6000\mathring{A}$ ]

Q24.In the given figure the magnetic flux through the loop increases according to the relation ${\varphi }_{\beta }(t)10t^2+20t$, where ${\varphi }_{\beta }$ is in milliwebers and t is in seconds. The magnitude of current through $\mathrm{R}=2\Omega$ resistor at …

Q25.The water is filled upto height of 12 m in a tank having vertical sidewalls. A hole is made in one of the walls at a depth ' $h$ ' below the water level. The value of ' $h$ ' for which the emerging stream of water strikes the ground at the maximum range is ____ m.

Q26.The ${\mathrm{K}}_{\alpha }$-X ray of molybdenum has wavelength 0.071 nm . If the energy of a molybdenum atom with a K electron knocked out is 27.5 keV , the energy of this atom when an L electron is knocked out will be ____ ke V. (Round off to the nearest integer) …

Q27.The maximum amplitude wave is found to be 12 V while the minimum amplitude is found to be 3 V . The modulation index is 0.6 x where x is ____ .

Q28.A small block slides down from the top of hemisphere of radius $R=3\mathrm{m}$ as shown in the figure. The height ' $h$ ' at which the block will lose contact with the surface of the sphere is ____ m.
(Assume there is no friction between the block and the hemisphere)

Q29.In the given figure, two wheels P and Q are connected by a belt B . The radius of P is three times as that of Q . In case of same rotational kinetic energy, the ratio of rotational inertias $\left(\frac{I_1}{I_2}\right)$ will be $x:1$. The value of $x$ will
be ____ …

Q30.A particle executes simple harmonic motion represented by displacement function as $x(t)=A\sin (\omega t+\varphi )$
If the position and velocity of the particle at $\mathrm{t}=0\mathrm{s}$ are 2 cm and $2\omega \mathrm{cm}{\mathrm{s}}^{-1}$ respectively, then its amplitude is …

Q1.Select the correct statements.
(A) Crystalline solids have long range order.
(B) Crystalline solids are isotropic.
(C) Amorphous solids are sometimes called pseudo solids
(D) Amorphous solids soften over a range of temperatures.
(E) Amorphous solids have a definite heat of fusion …

Q2.If the Thompson model of the atom was correct, then the result of Rutherford's gold foil experiment would have been:

Q3.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: ${\mathrm{SO}}_2(\mathrm{g})$ is adsorbed to a larger extent than ${\mathrm{H}}_2(\mathrm{g})$ on activated charcoal.
Reason R: ${\mathrm{SO}}_2(\mathrm{g})$ has …

Q4.Compound A gives D-Galactose and D-Glucose on hydrolysis. The compound A is:

Q5.Given below are two statements:
Statement I: Hyper conjugation is permanent effect.
Statement II : Hyper conjugation in ethyl cation $({\mathrm{CH}}_3-\stackrel{+}{\mathrm{C}}{\mathrm{H}}_2)$ involves the overlapping of ${\mathrm{Csp}}^2-{\mathrm{H}}_{1\mathrm{s}}$ bond with emp …

Q6.What is A in the following reaction?

Q7.Match List - I with List - II:

List - I
(a) Li
(b) Na
(c) K
(d) Cs

List - II
(i) photoelectric cell
(ii) absorbent of ${\mathrm{CO}}_2$
(iii) coolant in fast breeder nuclear reactor
(iv) treatment of cancer
(v) bearings for motor engines

Choose the correct answer from the opti …

Q8.The correct sequence of correct reagent for the following transformation is:

Q9.$\mathrm{R}-\mathrm{CN}\underset{\text{ (ii) }{\mathrm{H}}_2\mathrm{O}}{\overset{\text{ (i) }\text{ DIBAL-H }}{\to }}\mathrm{R}-\mathrm{Y}$
Consider the above reaction and identify " Y "

Q10.Given below are two statements :
Statement I: Penicillin is a bacteriostatic type antibiotic.
Statement II: The general structure of penicillin is:

Q11.Which one of the following set of elements can be detected using sodium fusion extract?

Q12.The addition of silica during the extraction of copper from its sulphide ore

Q13.Consider the above reaction, the major product "P" formed is

Q14.The number of neutrons and electrons respectively, present in the radioactive isotope of hydrogen is:

Q15.The CORRECT order of first ionisation enthalpy is:

Q16.Given below are two statements:
Statement I : ${\left[\mathrm{Mn}(\mathrm{CN}{)}_6\right]}^{3-},{\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]}^{3-}$ and ${\left[\mathrm{Co}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_3\right]}^{3-}$ ared ${}^2{\mathrm{sp}}^3$ hybridised
Statement II : …

Q17.To an aqueous solution containing ions such as ${\mathrm{Al}}^{3+},{\mathrm{Zn}}^{2+},{\mathrm{Ca}}^{2+},{\mathrm{Fe}}^{3+},{\mathrm{Ni}}^{2+},{\mathrm{Ba}}^{2+}$ and ${\mathrm{Cu}}^{2+}$ was added conc. HCl , followed by ${\mathrm{H}}_2\mathrm{S}$.
The total number of cations precipita …

Q18.Match List - I with List - II:

List - I (compound)
(a) Carbon monoxide
(b) Sulphur dioxide
(c) Polychlorinated biphenyls
(d) Oxides of nitrogen

List - II (effect / affected species)
(i) Carcinogenic
(ii) Metabolized by pyrus plants
(iii) Haemoglobin
(iv) Stiffness of flower bud …

Q19.Consider the above reaction, and choose the correct statement:

Q20.Number of $\mathrm{Cl}=\mathrm{O}$ bonds in chlorous acid, chloric acid and perchloric acid respectively are:

Q21.In a solvent 50% of an acid HA dimerizes and the rest dissociates. The van't Hoff factor of the acid is $\_\_\_\_$ $\times 10^{-2}$. (Round off to the Nearest Integer).

Q22.For the first order reaction $\mathrm{A}\to 2\mathrm{B}$, 1 mole of reactant A gives 0.2 moles of B after 100 minutes. The half life of the reaction is $\_\_\_\_$ min. (Round off to the Nearest Integer).
[Use : $\ln 2=0.69,\ln 10=2.3$ Properties of logarithms : …

Q23.$2{\mathrm{SO}}_2(\mathrm{g})+{\mathrm{O}}_2(\mathrm{g})\to 2{\mathrm{SO}}_3(\mathrm{g})$
The above reaction is carried out in a vessel starting with partial pressure ${\mathrm{P}}_{{\mathrm{SO}}_2}=250\mathrm{m}$ bar, ${\mathrm{P}}_{{\mathrm{O}}_2}=750\mathrm{m}$ b …

Q24.The equilibrium constant for the reaction
$\mathrm{A}(\mathrm{s})\rightleftharpoons \mathrm{M}(\mathrm{s})+\frac{1}{2}{\mathrm{O}}_2(\mathrm{g})$
In ${\mathrm{K}}_{\mathrm{p}}=4$. At equilibrium, the partial pressure of ${\mathrm{O}}_2$ is $\_\_\_\_$ atm.
(Round off to the Neare …

Q25.For the cell $\mathrm{Cu}(\mathrm{s})\left|{\mathrm{Cu}}^{2+}(\mathrm{aq})(0.1\mathrm{M})\right|\left|{\mathrm{Ag}}^{+}(\mathrm{aq})(0.01\mathrm{M})\right|\mathrm{Ag}(\mathrm{s})$
The cell potential ${\mathrm{E}}_1=0.3095\mathrm{V}$
For the cell …

Q26.The dihedral angle in staggered form the Newman projection of $1,1,1,-$ Trichloro ethane is
____ degree. (Round off to the Nearest Integer).

Q27.The total number of electrons in all bonding molecular orbitals of ${\mathrm{O}}_2^{2-}$ is ____.
(Round off to the Nearest Integer).

Q28.3 moles of metal complex with formula $\mathrm{Co}(\mathrm{en}{)}_2{\mathrm{Cl}}_3$, gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of Co in the complex is ____. (Round off to the Nearest Integer).

Q29.When 400 mL of $0.2\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ solution is mixed with 600 mL of of 0.1 M NaOH solution the increase in temperature of the final solution is ____ $\times 10^{-2}\mathrm{K}$.
(Round off to the Nearest Integer).
[Use: …

Q30.10.0 ml of $0.05\mathrm{M}{\mathrm{KMnO}}_4$ solution was consumed in a titration with 10.0 mL of given oxalic acid dehydrate solution. The strength of given oxalic acid solution is ____ $\times 10^{-2}\mathrm{g}/\mathrm{L}$ (Round off to the Nearest Integer).

Q1.Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b}\times (\vec{b}\times \vec{c})$. If magnitudes of the vectors $\vec{a},\vec{b}$ and $\vec{c}$ are $\sqrt{2},1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is …

Q2.Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x,y)\in N\times N:x^3-3x^{2}y-x{y}^2+3y^3=0\right\}$. Then the relation $R$ is :

Q3.Let $y=y(x)$ be the solution of the differential equation
$(x-x^3)dy=(y+y{x}^2-3x^4)dx,x>2$. If $y(3)=3$ then $y(4)$ is equal to

Q4.Let $f:[0,\infty )\to [0,3]$ be a function defined by $f(x)=\{\begin{array}{cc}\max \{\sin t:0\le t\le x\},& 0\le x\le \pi \\ 2+\cos x,& x>\pi \end{array}$
Then which of the following is true?

Q5.If $\tan \left(\frac{\pi }{9}\right),\mathrm{x},\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),\mathrm{y},\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to :

Q6.The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :

Q7.Let $f:R\to R$ be defined as $f(x+y)+f(x-y)=2f(x)f(y),f\left(\frac{1}{2}\right)=-1$. Then, the value of ${\sum }_{k=1}^{20}\frac{1}{\sin (k)\sin (k+f(k))}$ is equal to :

Q8.Let the mean and variance of the frequency distribution

x: ${\mathrm{x}}_1=2$, ${\mathrm{x}}_2=6$, ${\mathrm{x}}_3=8$, ${\mathrm{x}}_4=9$

f: 4, 4, $\alpha$, $\beta$

be 6 and 6.8 respectively. If $x_3$ is changed from 8 to 7, then the mean for the new data will be :

Q9.The point $\mathrm{P}(\mathrm{a},\mathrm{b})$ undergoes the following three transformations successively :
(a) reflection about the line $\mathrm{y}=\mathrm{x}$
(b) translation through 2 units along the positive direction of $x$-axis.
(c) rotation through angle $\frac{\pi }{4}$ a …

Q10.Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6\sqrt{5}$ on the $x$-axis. Then the radius of the circle $C$ is equal to :

Q11.Let $A$ and $B$ be two $3\times 3$ real matrices such that $(A^2-B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3{B}^2=A^2{B}^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to :

Q12.Let $\alpha ={\max }_{x\in R}\left\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\right\}$ and $\beta ={\min }_{x\in R}\left\{8^{2\sin 3x}\cdot 4^{4\cos 3x}\right\}$. If $8x^2+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5}$, then the value of $c-b$ …

Q13.Let c be the set of all complex numbers. Let
$S_1=\{z\in C:|z-2|\le 1\}$ and
$S_2=\{z\in C:z(1+i)+\bar{z}(1-i)\ge 4\}$.
Then, the maximum value of ${\left|z-\frac{5}{2}\right|}^2$ for $z\in S_1\cap S_2$ is equal to :

Q14.A possible value of '$x$', for which the ninth term in the expansion of ${\left\{3^{{\log }_3\sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right){\log }_3\left(5^{x-1}+1\right)}\right\}}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right){\log }_3\left(5^{x-1}+1\right)}$ …

Q15.The value of ${\lim }_{x\to 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to :

Q16.For real numbers $\alpha$ and $\beta \ne 0$, if the point of intersection of the straight lines $\frac{x-\alpha }{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta }=\frac{y-6}{3}=\frac{z-7}{3}$, lies on the plane $x+2y-z=8$, then $\alpha -\beta$ is equal to :

Q17.Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point :

Q18.Which of the following is the negation of the statement "for all $M>0$, there exists $x\in S$ such that $x\ge M$"?

Q19.A student appeared in an examination consisting of 8 true - false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least '$n$' correct answers is less than $\frac{1}{2}$, is :

Q20.Let $f:(a,b)\to R$ be twice differentiable such that $f(x)={\int }_a^{x}g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g^{\prime }(x)=0$ has at least :

Q21.The number of real roots of the equation
$e^{4x}-e^{3x}-4e^{2x}-e^x+1=0$ is equal to ____

Q22.If the real part of the complex number $z=\frac{3+2i\cos \theta }{1-3i\cos \theta },\theta \in \left(0,\frac{\pi }{2}\right)$ is zero, then the value of ${\sin }^{2}3\theta +{\cos }^2\theta$ is equal to ____

Q23.Let n be a non-negative integer. Then the number of divisors of the form "$4\mathrm{n}+1$" of the number $(10{)}^{10}\cdot (11{)}^{10}\cdot (13{)}^{13}$ is equal to ____

Q24.The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points $\mathrm{Q}(3,-4,-5)$ and $\mathrm{R}(2,-3,1)$ and the plane $2\mathrm{x}+\mathrm{y}+\mathrm{z}=7$, is equal to ____

Q25.Let $A=\left\{n\in N\mid n^2\le n+10,000\right\},B=\{3k+1\mid k\in N\}$ and $C=\{2k\mid k\in N\}$, then the sum of all the elements of the set $A\cap (B-C)$ is equal to ______

Q26.If ${\int }_0^{\pi }\left({\sin }^3\mathrm{x}\right){\mathrm{e}}^{-{\sin }^2\mathrm{x}}\mathrm{dx}=\alpha -\frac{\beta }{\mathrm{e}}{\int }_0^1\sqrt{\mathrm{t}}{\mathrm{e}}^{\mathrm{t}}\mathrm{dt}$, then $\alpha +\beta$ is equal to ______

Q27.If $A=\left[\begin{array}{lll}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$ and $M=A+A^2+A^3+\dots .+A^{20}$, then the sum of all the elements of the matrix $M$ is equal to ______

Q28.Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus at $(4,-4)$ and one vertex at $(5,-4)$. If $mx-y=4,m>0$ is a tangent to the ellipse $E$, then the value of $5{\mathrm{m}}^2$ is equal to ______

Q29.Let $y=y(x)$ be the solution of the differential equation $dy=e^{\alpha x+y}dx;\alpha \in N$. If $y\left({\log }_{e}2\right)={\log }_{e}2$ and $y(0)={\log }_e\left(\frac{1}{2}\right)$, then the value of $\alpha$ is equal to ______

Q30.Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k},\vec{b}=3\hat{i}+\beta \hat{j}-\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a}\cdot \vec{b}=-1$ and $\vec{b}\cdot \vec{c}=10$, then …