JEE Main 2021 · 17 Mar 2021 · Shift 1

Q1.A current of 10 A exists in a wire of cross-sectional area of $5{\mathrm{mm}}^2$ with a drift velocity of $2\times 10^{-3}{\mathrm{ms}}^{-1}$. The number of free electrons in each cubic meter of the wire is:

Q2.An electron of mass $m$ and a photon have same energy $E$. The ratio of wavelength of electron to that of photon is: (c being the velocity of light)

Q3.The thickness at the centre of a plano-convex lens is 3 mm and the diameter is 6 cm. If the speed of light in the material of the lens is $2\times 10^8{\mathrm{ms}}^{-1}$. The focal length of the lens is:

Q4.The vernier-scale used for measurement has a positive zero error of 0.2 mm. If while taking a measurement it was noted that '0' on the vernier-scale lies between 8.5 cm and 8.6 cm, vernier coincidence is 6, then the correct value of measurement is (least count $=0.01$ cm):

Q5.A triangular plate is shown. A force $\vec{F}=4\hat{i}-3\hat{j}$ is applied at point P. The torque at point P with respect to point 'O' and 'Q' are:

Q6.A car accelerates from rest at a constant rate $\alpha$ for some time after which it decelerates at a constant rate $\beta$ to come to rest. If the total time elapsed is t seconds, the total distance traveled is :

Q7.A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of $20{\mathrm{ms}}^{-1}$. The ball gets deflected by an obstacle on the way. After deflection it moves with $5\%$ of its initial kinetic energy. What is the speed of the ball now?

Q8.A mass M hangs on a massless rod of length $\ell$ which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular …

Q9.Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

Q10.A modern grand - prix racing car of mass $m$ is traveling on a flat track in a circular arc of radius R with a speed u . If the coefficient of static between the tyres and the track is ${\mu }_s$, then the magnitude of negative lift $F_L$ acting downwards on the car is :
( …

Q11.The output of the given combination gates represents:

Q12.A polyatomic ideal gas has 24 vibrational modes. What is the value of $\gamma$ ?

Q13.A solenoid of 1000 turns per meter has a core with relative permeability 500 . Insulated windings of the solenoid carry an electric current of 5 A . The magnetic flux density produced by the solenoid is: (permeability of free space …

Q14.A Carnot's engine working between 400 K and 800 K has a work output of 1200 J per cycle. The amount of heat energy supplied to the engine from the source in each cycle is:

Q15.When two soap bubbles of radii a and $\mathrm{b}(\mathrm{b}>\mathrm{a})$ coalesce, the radius of curvature of common surface is:

Q16.For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

Q17.An AC current is given by $I=I_1\sin \omega t+I_2\cos \omega t$. A hot wire ammeter will give a reading:

Q18.Two ideal polyatomic gages at temperatures $T_1$ and $T_2$ are mixed so that there is no loss of energy. If $F_1$ and $F_2,m_1$ and $m_2,n_1$ and $n_2$ be the degrees of freedom, masses, number of molecules of the first and second gas respectively, t …

Q19.Two identical metal of thermal conductivities ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ respectively are connected in series. The effective thermal conductivity of the combination is:

Q20.If an electron is moving in the ${\mathrm{n}}^{\text{th }}$ orbit of the hydrogen atom, then its velocity $\left(v_{\mathrm{n}}\right)$ for the ${\mathrm{n}}^{\text{th }}$ orbit is given as :

Q21.A parallel plate capacitor whose capacitance C is 14 pF is charged by a battery to a potential difference $\mathrm{V}=12\mathrm{V}$ between its plates. The charging battery is now disconnected and a porcelain plate with $\mathrm{k}=7$ is inserted between the plates, then th …

Q22.The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere
of same mass ' $m$ ' and radius ' $R$ ' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclin …

Q23.The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is ______ (Assuming the acceleration to be uniform) .

Q24.For VHF signal broadcasting ______ ${\mathrm{km}}^2$ of maximum service area will be covered by an antenna tower of height 30 m , if the receiving antenna is placed at ground. Let radius of the earth be 6400 km . (Round off to the Nearest Integer) (Take $\pi$ as 3.14)

Q25.The radius in kilometre to which the radius of earth $(\mathrm{R}=6400\mathrm{km})$ to be compressed so that the escape velocity is increased 10 times is ______ .

Q26.If $2.5\times 10^{-6}\mathrm{N}$ average force is exerted by a light wave on a non- reflecting surface of $30{\mathrm{cm}}^2$ area during 40 minutes of time span, the energy flux of light just before it falls on the surface is ______ $\mathrm{W}/{\mathrm{cm}}^2$. (Rou …

Q27.Two blocks ( $\mathrm{m}=0.5\mathrm{kg}$ and $\mathrm{M}=4.5\mathrm{kg}$ ) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is $\frac{3}{7}$. Then the maximum horizontal force that can be appli …

Q28.The equivalent resistance of series combination of two resistors is ' $s$ '. When they are connected in parallel, the equivalent resistance is ' $p$ '. If $s=np$, then the minimum value for $n$ is ______ (Round off to the Nearest Integer)

Q29.Four identical rectangular plates with length, $\ell =2\mathrm{cm}$ and breadth, $\mathrm{b}=\frac{3}{2}\mathrm{cm}$ are arranged as shown in figure. The equivalent capacitance between $A$ and $C$ is $\frac{X{\epsilon }_0}{d}$. The value of $x$ is ⋯------. (Roun …

Q30.Consider two identical springs each of spring constant k and negligible mass compared to the mass $M$ as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac{T_b}{T_a}=\sqrt{x}$, where …

Q1.Reducing smog is a mixture of :

Q2.Which of the following compound CANNOT act as a Lewis base?

Q3.Given below are two statements:
Statements I :Potassium permanganate on heating at 573 K forms potassium manganate.
Statement II: Both potassium permanganate and potassium manganate are tetrahedral and paramagnetic in nature.
In the light of the above statements, choose the most …

Q4.The above reaction requires which of the following reaction conditions?

Q5.A central atom in a molecule has two lone pairs of electrons and forms three single bonds. The shape of this molecule is:

Q6.Given below are two statements:
Statement I : Retardation factor $({\mathrm{R}}_{\mathrm{f}})$ can be measured in meter / centimeter.
Statement II: ${\mathrm{R}}_{\mathrm{f}}$ value of a compound remains constant in all solvents.
Chose the most appropriate answer from the options giv …

Q7.Mesityl oxide is a common name of:

Q8.The absolute value of the electron gain enthalpy of halogens satisfies :

Q9.A colloidal system consisting of a gas dispersed in a solid is called a/an:

Q10.The INCORRECT statement(s) about heavy water is (are)
(A) used as a moderator in nuclear reactor
(B) obtained as by-product in fertilizer industry
(C) used for the study of reaction mechanism
(D) has a higher dielectric constant than water
Choose the correct answer from the optio …

Q11.Product "A" in the above chemical reaction is:

Q12.Which of the following is correct structure of tyrosine?

Q13.Hoffmann bromomide degradation of benzamide gives product A, which upon heating with ${\mathrm{CHCl}}_3$ and NaOH gives product $B$.
The structures of $A$ and $B$ are:

Q14.The correct order of conductivity of ions in water is:

Q15.What is the spin- only magnetic moment value (BM) of divalent metal ion with atomic number 25, in it's aqueous solution?

Q16.With respect to drug-enzyme interaction, identify the wrong statement.

Q17.The product " A " in the above reaction is:

Q18.Which of the following reaction is an example of ammonolysis?

Q19.The point of intersection and sudden increase in the slope, in the diagram given below, respectively, indicates:

Q20.Which of the following is an aromatic compound ?

Q21.The oxygen dissolved in water exerts a partial pressure of 20 kPa in the vapour above water.
The molar solubility of oxygen in water is ____ $\times 10^{-5}\mathrm{mol}{\mathrm{dm}}^{-3}$. (Round off to the nearest integer).
[ Given : Henry's law constant …

Q22.For a certain first order reaction $32\%$ of the reactant is left after 570 s . The rate constant of this reaction is ____ $\times 10^{-3}{\mathrm{s}}^{-1}$. (Round off to the nearest integer).
[Given : ${\log }_{10}2=0.301,\ln 10=2.303$ ]

Q23.The pressure exerted by a non- reactive gaseous mixture of 6.4 g of methane and 8.8 g of carbon dioxide in a 10 L vessel at $27^{\circ }\mathrm{C}$ is ____ kPa .
(Round off to the nearest integer).
[ Assume gases are ideal, …

Q24.0.01 moles of a weak acid $\mathrm{HA}\left(\mathrm{Ka}=2.0\times 10^{-6}\right)$ is dissolved in 1.0 L of 0.1 M HCl solution. The degree of dissociation of HA is ____ $\times 10^{-5}$.
(Round off to the nearest integer).
[ Neglect volume change on adding HA.
Assume degr …

Q25.The reaction of white phosphorous on boiling with alkali in inert atmosphere resulted in the formation of product 'A'. The reaction of 1 mol of 'A' with excess of ${\mathrm{AgNO}}_3$ in aqueous medium gives ______ $\mathrm{mol}(\mathrm{s})$ of Ag.
(Round off to the nearest intege …

Q26.15 mL of aqueous solution of ${\mathrm{Fe}}^{2+}$ in acidic medium completely reacted with 20 mL of 0.03 M aqueous ${\mathrm{Cr}}_2{\mathrm{O}}_7^{2-}$. The molarity of the ${\mathrm{Fe}}^{2+}$ solution is ______ $\times 10^{-2}$ M
(Round off to the nearest integer).

Q27.A certain orbital has $n=4$ and $m_{\ell }=-3$. The number of radial nodes in this orbital is ______. (Round off to the nearest integer).

Q28.The mole fraction of a solute in a 100 molal aqueous solution is ______ $\times 10^{-2}$.
(Round off to the nearest integer).
[Given : Atomic masses : H : 1.0u, O : 16.0u]

Q29.In the above reaction, 3.9 g of benzene on nitration gives 4.92 g of nitrobenzene. The percentage yield of nitrobenzene in the above reaction is ______ %
(Round off to the nearest integer).
(Given atomic mass : C: 12.0u, H:1.0u, O:16.0u, N:14.0u)

Q30.The standard enthalpies of formation of ${\mathrm{Al}}_2{\mathrm{O}}_3$ and CaO are $-1675{\mathrm{kJ}\mathrm{mol}}^{-1}$ and $-635{\mathrm{kJ}\mathrm{mol}}^{-1}$ respectively.
For the reaction
$3\mathrm{CaO}+2\mathrm{Al}\to 3\mathrm{Ca}+{\mathrm{Al}}_2{\mathrm{O}}_3$ the standard reactio …

Q1.Two dices are rolled. If both dices have six faces numbered $1,2,3,5,7$ and $11$, then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is:

Q2.Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y$; $y(0)=0$:

Q3.The system of equation $kx+y+z=1$, $x+ky+z=k$ and $x+y+zk=k^2$ has no solution if $k$ is equal to:

Q4.Team 'A' consists of $7$ boys and $n$ girls and Team 'B' has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:

Q5.The inverse of $y=5^{\log x}$ is:

Q6.If ${\cot }^{-1}(\alpha )={\cot }^{-1}2+{\cot }^{-1}8+{\cot }^{-1}18+{\cot }^{-1}32+\dots$ upto $100$ terms, then $\alpha$ is:

Q7.The equation of the plane which contains the $y$-axis and passes through the point $(1,2,3)$ is:

Q8.If the Boolean expression $(p\Rightarrow q)\iff (q\ast (\sim p))$ is a tautology, then the Boolean expression $p\ast (\sim q)$ is equivalent to:

Q9.In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Q10.In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x-y+2=0$, then the centre of the circumcircle of the $△PQR$ is:

Q11.Choose the incorrect statement about the two circles whose equations are given below:
$x^2+y^2-10x-10y+41=0$ and $x^2+y^2-16x-10y+80=0$

Q12.If the fourth term in the expansion of ${\left(x+x^{{\log }_{2}x}\right)}^7$ is $4480$, then the value of $x$ where $x\in \mathbb{N}$ is equal to:

Q13.The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\dots \infty }}}}$ is :

Q14.The value of ${\lim }_{x\to 0^{+}}\frac{{\cos }^{-1}\left(x-[x{]}^2\right)\cdot {\sin }^{-1}\left(x-[x{]}^2\right)}{x-x^3}$, where $[x]$ denotes the greatest integer $\le x$ is :

Q15.The line $2x-y+1=0$ is a tangent to the circle at the point $(2,5)$ and the centre of the circle lies on $x-2y=4$. Then, the radius of the circle is :

Q16.Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=7\hat{i}+\hat{j}-6\hat{k}$.
If $\vec{r}\times \vec{a}=\vec{r}\times \vec{b},\vec{r}\cdot (\hat{i}+2\hat{j}+\hat{k})=-3$, then $\vec{r}\cdot (2\hat{i}-3\hat{j}+\hat{k})$ is equal to :

Q17.Which of the following statements is incorrect for the function $\mathrm{g}(\alpha )$ for $\alpha \in \mathrm{R}$ such that $g(\alpha )={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{{\sin }^{\alpha }x}{{\cos }^{\alpha }x+{\sin }^{\alpha }x}dx$

Q18.The sum of possible values of $x$ for ${\tan }^{-1}(x+1)+{\cot }^{-1}\left(\frac{1}{x-1}\right)={\tan }^{-1}\left(\frac{8}{31}\right)$ is :

Q19.The area of the triangle vertices $\mathrm{A}(\mathrm{z}),\mathrm{B}(\mathrm{iz})$ and $\mathrm{C}(\mathrm{z}+\mathrm{iz})$ is :

Q20.If $\mathrm{A}=\left(\begin{array}{cc}0& \sin \alpha \\ \sin \alpha & 0\end{array}\right)$ and $\det \left({\mathrm{A}}^2-\frac{1}{2}\mathrm{I}\right)=0$, then a possible value of $\alpha$ is :

Q21.If $(2021{)}^{3762}$ is divided by 17, then the remainder is ____

Q22.If $f(x)=\sin \left({\cos }^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{b}{a}{\log }_{e}2$ when $x=1$, where $a$ and $b$ are integers, then the minimum value of $\left|a^2-b^2\right|$ is. ____

Q23.Let there be three independent events ${\mathrm{E}}_1,{\mathrm{E}}_2$ and ${\mathrm{E}}_3$. The probability that only ${\mathrm{E}}_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let ' $p$ ' denote the probability of none of events occ …

Q24.If $\mathrm{A}=\left[\begin{array}{cc}2& 3\\ 0& -1\end{array}\right]$, then the value of $\det \left({\mathrm{A}}^4\right)+\det \left({\mathrm{A}}^{10}-(\mathrm{Adj}(2\mathrm{A}){)}^{10}\right)$ is equal to.....

Q25.The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circles' equations
$x^2+y^2-10x-10y+41=0$
$x^2+y^2-24x-10y+160=0$ is ____

Q26.If $\vec{a}=\alpha \hat{i}+\beta \hat{j}+3\hat{k}$,
$\overrightarrow{\mathrm{b}}=-\beta \hat{\mathrm{i}}-\alpha \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and
$\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}-\hat{\mathrm{k}}$
such that $\vec{a}\cdot \vec{b}=1$ a …

Q27.The maximum value of $z$ in the following equation $z=6xy+y^2$, where $3x+4y\le 100$ and $4x+3y\le 75$ for $x\ge 0$ and $y\ge 0$ is ____

Q28.If the equation of the plane passing through the line of intersection of the planes $2x-7y+4z-3=0,3x-5y+4z+11=0$ and the point $(-2,1,3)$ is $ax+by+cz-7=0$, then the value of $2a+b+c-7$ is ____

Q29.If [.] represents the greatest integer function, then the value of $\left|{\int }_0^{\sqrt{\frac{\pi }{2}}}\left[\left[x^2\right]-\cos x\right]dx\right|$ is ____

Q30.If the function $f(x)=\frac{\cos (\sin x)-\cos x}{x^4}$ is continuous at each point in its domain and $f(0)=\frac{1}{k}$, then k is ____