JEE Main 2021 · 25 Feb 2021 · Shift 1

Q1.A solid sphere of radius R gravitationally attracts a particle placed at $3R$ from its centre with a force $F_1$. Now a spherical cavity of radius $\left(\frac{\mathrm{R}}{2}\right)$ is made in the sphere (as shown in figure) and the force becomes $F_2$. The value of …

Q2.An $\alpha$ particle and a proton are accelerated from rest by a potential difference of 200 V . After this, their de Broglie wavelengths are ${\lambda }_{\alpha }$ and ${\lambda }_{\mathrm{P}}$ respectively. The ratio $\frac{{\lambda }_{\mathrm{P}}}{{\lambda }_{\alpha }}$ is :

Q3.The angular frequency of alternating current in a L-C- R circuit is $100\mathrm{rad}/\mathrm{s}$. The components connected are shown in the figure. Find the value of inductance of the coil and capacity of condenser.

Q4.The current ( i ) at time $\mathrm{t}=0$ and $\mathrm{t}=\infty$ respectively for the given circuit is :

Q5.In an octagon $ABCDEFGH$ of equal side, what is the sum of $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{AE}}+\overrightarrow{\mathrm{AF}}+\overrightarrow{\mathrm{AG}}+\overrightarrow{\mathrm{AH}}$, if, …

Q6.Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $\frac{I_{\text{max }}-I_{\text{min }}}{I_{\text{max }}+I_{\text{min }}}$ will be :

Q7.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The escape velocities of planet $A$ and $B$ are same. But $A$ and $B$ are of unequal mass.
Reason R: The product of their mass and radius must be same. …

Q8.Given below are two statements :
Statement I:A speech signal of 2 kHz is used to modulate a carrier signal of 1 MHz . The bandwidth requirement for the signal is 4 kHz .
Statement II : The side band frequencies are 1002 kHz and 998 kHz .
In the light of the above statements, choo …

Q9.The pitch of the screw gauge is 1 mm and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is cl …

Q10.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: When a rod lying freely is heated, no thermal stress is developed in it.
Reason R: On heating, the length of the rod increases.
In the light of the above statements …

Q11.A 5 V battery is connected across the points X and Y . Assume ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$ to be normal silicon diodes. Find the current supplied by the battery if the +ve terminal of the battery is connected to point X .

Q12.A diatomic gas, having $C_P=\frac{7}{2}R$ and $C_V=\frac{5}{2}R$, is heated at constant pressure. The ratio dU : dQ : dW :

Q13.If the time period of a two meter long simple pendulum is $2s$, the acceleration due to gravity at the place where pendulum is executing S.H.M. is :

Q14.Two radioactive substances $X$ and $Y$ originally have $N_1$ and $N_2$ nuclei respectively. Half life of $X$ is half of the life of Y . After three half lives of Y , number of nuclei of both are equal.
The ratio $\frac{{\mathrm{N}}_1}{{\mathrm{N}}_2}$ will be equ …

Q15.Magnetic fields at two points on the axis of a circular coil at a distance of 0.05 m and 0.2 m from the centre are in the ratio $8:1$. The radius of coil is

Q16.A student is performing the experiment of resonance column. The diameter of the column tube is 6 cm . The Frequency of the tuning fork is 504 Hz . Speed of the sound at the given temperature is $336\mathrm{m}/\mathrm{s}$. The zero of the metre scale coincides with the top e …

Q17.Match List - I with List - II :

List - I
(a) h (Planck's constant)
(b) E (kinetic energy)
(c) V (electric potential)
(d) P (liner momentum )

List - II
(i) $\left[\mathrm{ML}{\mathrm{T}}^{-1}\right]$
(ii) $\left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-1}\right]$
(ii …

Q18.A proton, a deuteron and an $\alpha$ particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is ____ and their speed is ____ in the ratio.

Q19.Two satellites A and B of masses 200 kg and 400 kg are revolving round the earth at height of 600 km and 1600 km respectively.
If $T_A$ and $T_B$ are the time periods of $A$ and $B$ respectively then the value of $T_B-T_A$:
[ Given : radius of earth …

Q20.An engine of a train, moving with uniform acceleration, passes the signal - post with velocity $u$ and the last compartment with velocity $v$. The velocity with which middle point of the train passes the signal post is :

Q21.In a certain thermodynamical process, the pressure of a gas depends on its volume as ${\mathrm{kV}}^3$. The work done when the temperature changes from $100^{\circ }\mathrm{C}$ to $300^{\circ }\mathrm{C}$ will be ____ nR , where n denotes numbers of moles of a gas.

Q22.The same size images are formed by a convex lens when the object is placed at 20 cm or at 10 cm from the lens. The focal length of convex lens is ____ cm.

Q23.A transmitting station releases waves of wavelength 960 m . A capacitor of $2.56\mu \mathrm{F}$ is used in the resonant circuit. The self inductance of coil necessary for resonance is ____ $\times 10^{-8}\mathrm{H}$.

Q24.512 identical drops of mercury are charged to a potential of 2 V each. The drops are joined to form a single drop. The potential of this drop is ____ V.

Q25.A coil of inductance 2 H having negligible resistance is connected to a source of supply whose voltage is given by $\mathrm{V}=3t$ volt. (where $t$ is in second). If the voltage is applied when $t=0$, the energy stored in the coil after 4 s is ____ J.

Q26.The potential energy ( U ) of a diatomic molecule is a function dependent on r (interatomic distance) as $\mathrm{U}=\frac{\alpha }{{\mathrm{r}}^{10}}-\frac{\beta }{{\mathrm{r}}^5}-3$
Where, $\alpha$ and $\beta$ are positive constants. The equilibrium between two atoms will be …

Q27.A mono-atomic gas of mass $4.0u$ is kept in an insulated container. Container is moving with velocity $30\mathrm{m}/\mathrm{s}$. If container is suddenly stopped then change in temperature of the gas $(R=$ gas constant) is $\frac{x}{3R}$. Value of $x$ …

Q28.A small bob tied at one end of a thin string of length 1 m is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio $5:1$. The velocity of the bob at the highest position is ____ $\mathrm{m}/\mathrm{s}.($ Take …

Q29.The electric field in a region is given by $\vec{E}=\left(\frac{3}{5}{E}_0\hat{i}+\frac{4}{5}{E}_0\hat{j}\right)\frac{N}{C}$. The ratio of flux of reported field through the rectangular surface of area $0.2m^2$ (parallel to $y-z$ plane) to that of the surface of are …

Q1.Identify A and B in the chemical reaction.

Q2.The Plots of radial distribution functions for various orbitals of hydrogen atom against '$r$' are given below:

(a) (see plot)
(b) (see plot)
(c) (see plot)
(d) (see plot)

The correct plot for 3s orbital is:

Q3.Which statement is correct?

Q4.Given below are two statements:
Statement I: An allotrope of oxygen is an important intermediate in the formation of reducing smog.
Statement II: Gases such as oxides of nitrogen and sulphur present in troposphere contribute to the formation of photochemical smog.
In the light of …

Q5.Ellingham diagram is a graphical representation of:

Q6.Compound(s) which will liberate carbon dioxide with sodium bicarbonate solution is/are:

$A=$ (see structure)

$B=$ (see structure)

$C=$ (see structure)

Q7.Complete combustion of 1.80 g of an oxygen containing compound $({\mathrm{C}}_{\mathrm{x}}{\mathrm{H}}_{\mathrm{y}}{\mathrm{O}}_{\mathrm{z}})$ gave 2.64 g of ${\mathrm{CO}}_2$ and 1.08 g of ${\mathrm{H}}_2\mathrm{O}$. The percentage of oxygen in the organic compound is

Q8.Which of the following reaction/s will not give $p$-aminoazobenzene?

A - (see structure)
i) $\mathrm{Sn}/\mathrm{HCl}$
ii) ${\mathrm{HNO}}_2$
iii) Aniline

B - (see structure)
i) ${\mathrm{NaBH}}_4$
ii) NaOH
iii) Aniline

C - (see structure)
i) ${\mathrm{HNO}}_2$
ii) Aniline, …

Q9.Which of the glycosidic linkage between galactose and glucose is present in lactose?

Q10.The hybridization and magnetic nature of ${\left[\mathrm{Mn}(\mathrm{CN}{)}_6\right]}^{4-}$ and ${\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]}^{3-}$, respectively are:

Q11.Which one of the following reactions will not form acetaldehyde?

Q12.The major product of the following chemical reaction is:

Q13.The solubility of AgCN in a buffer solution of $\mathrm{pH}=3$ is x . The value of x is :
[Assume : No cyano complex is formed ; ${\mathrm{K}}_{\mathrm{sp}}(\mathrm{AgCN})=2.2\times 10^{-16}$ and ${\mathrm{K}}_{\mathrm{a}}(\mathrm{HCN})=6.2\times 10^{-10}$ ]

Q14.Identify $A$ in the given chemical reaction.
${\mathrm{H}}_2{\mathrm{C}}^{/{\mathrm{CH}}_3}$
$\underset{{\mathrm{CH}}_2}{\overset{\mathrm{HC}}{\to }}{}_{{\mathrm{CH}}_3}^{\mathrm{CH}}$
'A'
major product

Q15.The correct statement about $B_2{H}_6$ is :

Q16.According to molecular orbital theory, the species among the following that does not exist is :

Q17.Given below are two statements :
Statement I: ${\mathrm{CeO}}_2$ can be used for oxidation of aldehydes and ketones.
Statement II : Aqueous solution of ${\mathrm{EuSO}}_4$ is a strong reducing agent.
In the light of the above statements, choose the correct answer from the opt …

Q18.Which of the following equation depicts the oxidizing nature of ${\mathrm{H}}_2{\mathrm{O}}_2$ ?

Q19.In Freundlich adsorption isotherm at moderate pressure, the extent of adsorption $\left(\frac{x}{m}\right)$ is directly proportional to ${\mathrm{P}}^{\mathrm{x}}$. The value of x is :

Q20.In which of the following pairs, the outer most electronic configuration will be the same ?

Q21.For the reaction $\mathrm{aA}+\mathrm{bB}\to \mathrm{cC}+\mathrm{dD}$, the plot of $\log \mathrm{K}$ vs $\frac{1}{\mathrm{T}}$ is given below :
The temperature at which the rate constant of the reaction is $10^{-4}{\mathrm{s}}^{-1}$ is ____ K.
(Rounded - off …

Q22.In basic medium ${\mathrm{CrO}}_4{}^{2-}$ oxidises ${\mathrm{S}}_2{\mathrm{O}}_3{}^{2-}$ to form ${\mathrm{SO}}_4{}^{2-}$ and itself changes into $\mathrm{Cr}(\mathrm{OH}{)}_4{}^{-}$.
The volume of $0.154\mathrm{M}{\mathrm{CrO}}_4{}^{2-}$ required to react with …

Q23.Among the following, the number of halide(s) which is / are inert to hydrolysis is ____
(A) ${\mathrm{BF}}_3$
(B) ${\mathrm{SiCl}}_4$
(C) ${\mathrm{PCl}}_5$
(D) ${\mathrm{SF}}_6$

Q24.0.4 g mixture of $\mathrm{NaOH},{\mathrm{Na}}_2{\mathrm{CO}}_3$ and some inert impurities was first titrated with $\frac{\mathrm{N}}{10}\mathrm{HCl}$ using phenolphthalein as an indicator, 17.5 mL of HCl was required at the end point. After this methyl orange was added and …

Q25.The reaction of cyanamide, ${\mathrm{NH}}_2{\mathrm{CN}}_{(\mathrm{s})}$ with oxygen was run in a bomb calorimeter and $\Delta U$ was found to be $-742.24\mathrm{kJ}{\mathrm{mol}}^{-1}$. The magnitude of $\Delta {\mathrm{H}}_{298}$ for the reaction …

Q26.1 molal aqueous solution of an electrolyte $A_2{B}_3$ is $60\%$ ionised. The boiling point of the solution at 1 atm is _____ K. (Rounded-off to the nearest integer)
[Given ${\mathrm{K}}_{\mathrm{b}}$ for …

Q27.A car tyre is filled with nitrogen gas at 35 psi at $27^{\circ }\mathrm{C}$. It will burst if pressure exceeds 40 psi. The temperature in ${}^{\circ }\mathrm{C}$ at which the car tyre will burst is _____ (Rounded-off to the nearest integer)

Q28.The ionization enthalpy of ${\mathrm{Na}}^{+}$ formation from ${\mathrm{Na}}_{(\mathrm{g})}$ is $495.8\mathrm{kJ}{\mathrm{mol}}^{-1}$, while the electron gain enthalpy of Br is $-325.0\mathrm{kJ}{\mathrm{mol}}^{-1}$. Given the lattice enthalpy of NaBr is …

Q29.Using the provided information in the following paper chromatogram:
Fig: Paper chromatography for compounds A and B. The calculated ${\mathrm{R}}_f$ value of A is _____ $\times 10^{-1}$.

Q30.Consider the following chemical reaction.

The number of ${\mathrm{sp}}^2$ hybridized carbon atom(s) present in the product is _____

Q1.If a curve passes through the origin and the slope of the tangent to it at any point $(x,y)$ is $\frac{x^2-4x+y+8}{x-2}$, then this curve also passes through the point :

Q2.Let $f,g:\mathbb{N}\to \mathbb{N}$ such that $f(n+1)=f(n)+f(1)\forall n\in \mathbb{N}$ and $g$ be any arbitrary function. Which of the following statements is NOT true ?

Q3.The value of ${\int }_{-1}^1{x}^2{e}^{[x^3]}dx$, where $[t]$ denotes the greatest integer $\le t$, is :

Q4.If Rolle's theorem holds for the function $f(x)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{\prime }\left(\frac{4}{3}\right)=0$, then ordered pair $(a,b)$ is equal to :

Q5.If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of $90^{\circ }$, then which of the following relations is TRUE ?

Q6.The statement $A\to (B\to A)$ is equivalent to :

Q7.A tangent is drawn to the parabola $y^2=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does NOT lie on it ?

Q8.${\lim }_{n\to \infty }{\left(1+\frac{1+\frac{1}{2}+\dots +\frac{1}{n}}{n^2}\right)}^n$ is equal to :

Q9.When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}$. If three missiles are fired independently from the ship, then probability that al …

Q10.If $0<\theta ,\varphi <\frac{\pi }{2},x={\sum }_{n=0}^{\infty }{\cos }^{2n}\theta ,y={\sum }_{n=0}^{\infty }{\sin }^{2n}\varphi$ and $z={\sum }_{n=0}^{\infty }{\cos }^{2n}\theta \cdot {\sin }^{2n}\varphi$ then :

Q11.Let the lines $(2-i)z=(2+i)\bar{z}$ and $(2+i)z+(i-2)\bar{z}-4i=0$, (here $i^2=-1$) be normal to a circle $C$. If the line $iz+\bar{z}+1+i=0$ is tangent to this circle $C$, then its radius is :

Q12.All possible values of $\theta \in [0,2\pi ]$ for which $\sin 2\theta +\tan 2\theta >0$ lie in :

Q13.A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is $30^{\circ }$ (Ignore man's height). After sailing for 20 seconds, towards the base of th …

Q14.The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

Q15.Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^2+m^2-n^2=0$. Then the value of ${\sin }^4\alpha +{\cos }^4\alpha$ is :

Q16.The value of the integral
$\int \frac{\sin \theta \sin 2\theta \left({\sin }^6\theta +{\sin }^4\theta +{\sin }^2\theta \right)\sqrt{2{\sin }^4\theta +3{\sin }^2\theta +6}}{1-\cos 2\theta }d\theta$
(where c is a constant of integration)

Q17.The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^2+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :

Q18.The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :

Q19.The integer '$k$', for which the inequality $x^2-2(3k-1)x+8k^2-7>0$ is valid for every $x$ in $R$, is :

Q20.The image of the point $(3,5)$ in the line $x-y+1=0$, lies on :

Q21.The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1,2, 3,4,5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is

Q22.The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area $A$. Then $A^4$ is equal to

Q23.The number of points, at which the function $f(x)=|2x+1|-3|x+2|+\left|x^2+x-2\right|,x\in R$ is not differentiable, is

Q24.If $\mathrm{A}=\left[\begin{array}{cc}0& -\tan \left(\frac{\theta }{2}\right)\\ \tan \left(\frac{\theta }{2}\right)& 0\end{array}\right]$ and …

Q25.Let $\vec{a}=\hat{i}+2\hat{j}-\hat{k},\vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $\vec{r}\cdot \vec{a}$ is equal t …

Q26.Let $A=\left[\begin{array}{lll}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$, where $x,y$ and $z$ are real numbers such that $x+y+z>0$ and $xyz=2$. If $A^2=I_3$, then the value of $x^3+y^3+z^3$ is ____

Q27.Let $A_1,A_2,A_3,\dots$ be squares such that for each $n\ge 1$, the length of the side of $A_n$ equals the length of diagonal of $A_{n+1}$. If the length of $A_1$ is 12 cm then smallest value of $n$ for which area of $A_n$ is less than one, is ____

Q28.The locus of the point of intersection of the lines $(\sqrt{3})kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4(\sqrt{3})k=0$ is a conic, whose eccentricity is ____

Q29.Let $f(x)$ be a polynomial of degree 6 in $x$, in which the coefficient of $x^6$ is unity and it has extrema at $x=-1$ and $x=1$. If ${\lim }_{x\to 0}\frac{f(x)}{x^3}=1$, then $5f(2)$ is equal to ____

Q30.If the system of equations

has infinitely many solutions, then $k$ is equal to ____