JEE Main 2021 · 16 Mar 2021 · Shift 1

Q1.A conducting wire of length '$\ell$', area of cross-section A and electric resistivity $\rho$ is connected between the terminals of a battery. A potential difference V is developed between its ends, causing an electric current.
If the length of the wire of the same material i …

Q2.An RC circuit as shown in the figure is driven by a AC source generating a square wave. The output wave pattern monitored by CRO would look close to :

Q3.A block of mass $m$ slides along a floor while a force of magnitude F is applied to it at an angle $\theta$ as shown in figure. The coefficient of kinetic is ${\mu }_{\mathrm{K}}$. Then, the block's acceleration '$a$' is given by :
( g is acceleration due to gravity)

Q4.Four equal masses, $m$ each are placed at the corners of a square of length $(\ell )$ as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be :

Q5.The velocity- displacement graph describing the motion of a bicycle is shown in the figure. The acceleration-displacement graph of the bicycle's motion is best described by:

Q6.For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant K is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is $\frac{3}{4}d$, where ' $d$ ' is the separation between t …

Q7.A plane electromagnetic wave of frequency 500 MHz is traveling in vacuum along y-direction. At a particular point in space and time, $\vec{B}=8.0\times 10^{-8}\hat{z}T$. The value of electric field at this point is: $($ speed of light …

Q8.Time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration $\mathrm{g}/2$, the time period of pendulum will be:

Q9.The stopping potential in the context of photoelectric depends on the following property of incident electromagnetic radiation:

Q10.A bar magnet of length 14 cm is placed in the magnetic meridian with its north pole pointing towards the geographic north pole. A neutral point is obtained at a distance of 18 cm from the centre of the magnet. If ${\mathrm{B}}_{\mathrm{H}}=0.4\mathrm{G}$, the magnetic moment of …

Q11.A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm . If the block takes 40 s to complete one round, the normal force by the side walls of the groove is:

Q12.The volume V of an enclosure contains a mixture of three gases, 16 g of oxygen, 28 g of nitrogen and 44 g of carbon dioxide at absolute temperature T . Consider R as universal gas constant. The pressure of the mixture of gases is :

Q13.The pressure acting on a submarine is $3\times 10^5\mathrm{Pa}$ at a certain depth. If the depth is doubled, the percentage in the pressure acting on the submarine would be: (Assume that atmospheric pressure is $1\times 10^5\mathrm{Pa}$ density of water is …

Q14.The maximum and minimum distances of a comet from the Sun are $1.6\times 10^{12}\mathrm{m}$ and $8.0\times 10^{10}\mathrm{m}$ respectively. If the speed of the comet at the nearest point is $6\times 10^4{\mathrm{ms}}^{-1}$, the speed at the farthest point is:

Q15.A conducting bar of length $L$ is free to slide on two parallel conducting rails as shown in the figure. Two resistors ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are connected across the ends of the rails. There is a uniform magnetic field $\vec{B}$ pointing into the page. An …

Q16.One main scale division of a vernier callipers is ' $a$ ' $cm$ and $n^{\text{th }}$ division of the vernier scale coincide with $(n-1{)}^{\text{th }}$ division of the main scale. The least count of the callipers in mm is :

Q17.A 25 m long antenna is mounted on an antenna tower. The height of the antenna tower is 75 m . The wavelength (in meter) of the signal transmitted by this antenna would be:

Q18.The angle of deviation through a prism is minimum when
(A) Incident ray and emergent ray are symmetric to the prism
(B) The refracted ray inside the prism becomes parallel to its base
(C) Angle if incidence is equal to that of the angle of emergence
(D) When angle of emergence is …

Q19.For an electromagnetic wave traveling in free space, the relation between average energy densities due to electric ( ${\mathrm{U}}_{\mathrm{e}}$ ) and magnetic ( ${\mathrm{U}}_{\mathrm{m}}$ ) fields is :

Q20.In thermodynamics, heat and work are :

Q21.Consider a frame that is made up of two thin massless rods $AB$ and $AC$ as shown in the figure. A vertical force $\vec{P}$ of magnitude 100 N is applied at point A of the frame. Suppose the force is $\overrightarrow{\mathrm{P}}$ resolved parallel to the arms AB and AC of …

Q22.The value of power dissipated across the zener diode ( ${\mathrm{V}}_{\mathrm{z}}=15\mathrm{V}$ ) connected in the circuit as shown in the figure is $\mathrm{x}\times 10^{-1}$ watt.
The value of x , to the nearest integer, is ____

Q23.In the figure given, the electric current flowing through the 5 $\mathrm{k}\Omega$ resistor is ' x ' mA . The value of x to the nearest integer is ____

Q24.In the logic circuit shown in the figure, if input A and B are 0 to 1 respectively, then output at Y would be ' x '.
The value of $x$ is ____

Q25.The resistance $R=\frac{V}{I}$, where $V=(50\pm 2)V$ and $I=(20\pm 0.2)A$. The percentage error in $R$ is ' $x$ ' $\%$. The value of ' $x$ ' to the nearest integer is ____

Q26.A sinusoidal voltage of peak value 250 V is applied to a series LCR circuit, in which $\mathrm{R}=8\Omega ,\mathrm{L}=24\mathrm{mH}$ and $\mathrm{C}=60\mu \mathrm{F}$. The value of power dissipated at resonant condition is ' x ' kW . The value of x to the nearest integer …

Q27.Consider a 20 kg uniform circular disk of radius 0.2 m . It is pin supported at its centre and is at rest initially. The disk is acted upon by a constant force $\mathrm{F}=20\mathrm{N}$ through a massless string wrapped around its periphery as shown in the figure. Suppose the …

Q28.The first three spectral of H -atom in the Balmer series are given ${\lambda }_1,{\lambda }_2,{\lambda }_3$ considering the Bohr atomic model, the wave lengths of first and third spectral lines $\left(\frac{{\lambda }_1}{{\lambda }_3}\right)$ are related by a factor of approxima …

Q29.A fringe width of 6 mm was produced for two slits separated by 1 mm apart. The screen is placed 10 m away. The wavelength of light used is ' x ' nm .
The value of ' $x$ ' to the nearest integer is ____

Q30.A ball of mass 10 kg moving with a velocity $10\sqrt{3}{\mathrm{ms}}^{-1}$ along $\mathrm{X}-$ axis, hits another ball of mass 20 kg which is at rest. After collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces star …

Q1.Among the following, the aromatic compounds are:
(A) (see structure)
(B) (see structure)
(C) (see structure)
(D) (see structure)
Choose the correct answer from the following options:

Q2.The products "A" and "B" formed in above reactions are:

Q3.The product "P" in the above reaction is:

Q4.Assertion A : Enol form of acetone $[\mathrm{CH}{}_3\mathrm{COCH}{}_3]$ exists in $<0.1\%$ quantity. However, the enol form of acetyl acetone $[\mathrm{CH}{}_3\mathrm{COCH}{}_2\mathrm{OCCH}{}_3]$ exists in approximately $15\%$ quantity.
Reason R : Enol form of acetylacetone is stabilized by intramolecular hydrogen bonding, which …

Q5.Match List - I with List - II:

List - I (Industrial process)
(a) Haber's process
(b) Ostwald's process
(c) Contact process
(d) Hall-Heroult process

List - II (Application)
(i) $\mathrm{HNO}{}_3$ synthesis
(ii) Aluminium extraction
(iii) $\mathrm{NH}{}_3$ synthesis
(iv) $\mathrm{H}{}_2\mathrm{SO}{}_4$ synthesis …

Q6.Given below are two statement: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason R:
Assertion A: Size of $\mathrm{Bk}{}^{3+}$ ion is less than $\mathrm{Np}{}^{3+}$ ion.
Reason R: The above is a consequence of the lanthanoid contraction.
In the light of the above s …

Q7.Given below are two statements:
Statement I: The $E^{\circ }$ value for $\mathrm{Ce}{}^{4+}/\mathrm{Ce}{}^{3+}$ is +1.74 V
Statement II: Ce is more stable in $\mathrm{Ce}{}^{4+}$ state than $\mathrm{Ce}{}^{3+}$ state
In the light of the above statements, choose the most appropriate answer from the options giv …

Q8.Which among the following pairs of Vitamins is stored in our body relatively for longer duration?

Q9.Which of the following is Lindlar catalyst?

Q10.Given below are two statements: one is labelled as Assertion A and the other is labelled as Reasons R :
Assertion A : The H-O-H bond angle in water molecule is $104.5^{\circ }$.
Reason R : The lone pair- lone pair repulsion of electrons is higher than the bond pair- bond pair repu …

Q11.In chromatography technique, the purification of compound is independent of:

Q12.The type of pollution that gets increased during the day time and in the presence of $\mathrm{O}{}_3$ is:

Q13.In the above chemical reaction, intermediate "X" and reagent / condition "A" are:

Q14.Match List - I with List - II

List - I
Name of oxo acid
(a) Hypophosphorous acid
(b) Orthophosphoric acid
(c) Hypophosphoric acid
(d) Orthophosphorous acid

List - II
Oxidation state of "P"
(i) +5
(ii) +4
(iii) +3
(iv) +2
(v) +1

Choose the correct answer from the options given …

Q15.Which of the following reaction DOES NOT involve Hoffmann bromamide degradation?

Q16.Given below are two statements:
Statement I: Both CaCl₂·6H₂O and MgCl₂·8H₂O undergo dehydration on heating.
Statement II: BeO is amphoteric whereas the oxides of other elements in the same group are acidic.
In the light of the above statements, choose the correct answer from the …

Q17.A group 15 element, which is a metal and forms a hydride with strongest reducing power among group 15 hydrides. The element is:

Q18.The process that involves the removal of sulphur from the ores is:

Q19.Given below are two statements:
Statement I: H₂O₂ can act as both oxidizing and reducing agent in basic medium.
Statement II: In the hydrogen economy, the energy is transmitted in the form of dihydrogen.
In the light of the above statements, choose the correct answer from the opt …

Q20.The functions of antihistamine are:

Q21.A certain element crystallises in a bcc lattice of unit cell edge length 27 Å. If the same element under the same conditions crystallises in the fcc lattice, the edge length of the unit cell in Å will be ____. (Round off to the nearest Integer).
[Assume each lattice point has a s …

Q22.2MnO₄⁻ + bC₂O₄²⁻ + cH⁺ → xMn²⁺ + yCO₂ + zH₂O
If the above equation is balanced with integer coefficients, the value of c is ____.
(Round off to the nearest Integer).

Q23.Two salts A₂X and MX have the same value of solubility product of 4.0 × 10⁻¹². The ratio of their molar solubilities i.e S(A₂X)/S(MX) = ____.
(Round off to the Nearest Integer).

Q24.Complete combustion of 750 g of an organic compound provides 420 g of CO₂ and 210 g of H₂O. The percentage composition of carbon and hydrogen in organic compound is 15.3 and ____ respectively. (Round off to the nearest Integer).

Q25.A 6.50 molal solution of $\mathrm{KOH}(\mathrm{aq})$ has a density of $1.89\mathrm{g}{\mathrm{cm}}^{-3}$. The molarity of the solution is ____ $\mathrm{mol}{\mathrm{dm}}^{-3}$. (Round off to the nearest Integer).
[Atomic masses : …

Q26.$A{B}_2$ is $10\%$ dissociated in water to $A^{2+}$ and $B^{-}$. The boiling point of a 10.0 molal aqueous solution of ${\mathrm{AB}}_2$ is ____ ${}^{\circ }\mathrm{C}$. (Round off to the nearest Integer).
[Given : Molal elevation constant of water …

Q27.For the reaction $\mathrm{A}(\mathrm{g})\rightleftharpoons \mathrm{B}(\mathrm{g})$ at $495\mathrm{K},{\Delta }_{\mathrm{r}}{\mathrm{G}}^{\circ }=-9.478\mathrm{kJ}{\mathrm{mol}}^{-1}$. If we start the reaction in a closed container at 495 K with 22 millimoles of A , the amoun …

Q28.The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the trans-complex of ${\mathrm{CoCl}}_3$. $4{\mathrm{NH}}_3$ is ____ .
(Round off to the Nearest Integer).

Q29.The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is $1.0\times 10^{-3}{\mathrm{s}}^{-1}$ and the activation energy ${\mathrm{E}}_a=11.488\mathrm{kJ}{\mathrm{mol}}^{-1}$, the rate constant at 200 K is ____ …

Q30.When light of wavelength 248 nm falls on a metal of threshold energy 3.0 eV , the de-Broglie wavelength of emitted electrons is ____ $\stackrel{\circ }{A}$.
(Round off to the nearest Integer). [Use : …

Q1.Let a complex number $\mathrm{z},|\mathrm{z}|\ne 1$, satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|\mathrm{z}|+11}{(|\mathrm{z}|-1{)}^2}\right)\le 2$. Then, the largest value of $|\mathrm{z}|$ is equal to.........

Q2.Consider three observations $\mathrm{a},\mathrm{b}$ and c such that $\mathrm{b}=\mathrm{a}+\mathrm{c}$. If the standard deviation of $a+2,b+2,c+2$ is $d$, then which of the following is true?

Q3.The number of elements in the set $\{x\in R:(|x|-3)|x+4|=6\}$ is equal to :

Q4.The range of $a\in R$ for which the function $f(x)=(4a-3)\left(x+{\log }_{e}5\right)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),x\ne 2n\pi ,n\in N$ has critical points is:

Q5.If for $x\in \left(0,\frac{\pi }{2}\right),{\log }_{10}\sin x+{\log }_{10}\cos x=-1$ and ${\log }_{10}(\sin x+\cos x)=\frac{1}{2}\left({\log }_{10}{}^n-1\right),n>0$, then the value of $n$ is equal to :

Q6.Let the functions $f:R\to R$ and $g:R\to R$ be defined as :
$f(x)=\{\begin{array}{cc}x+2,& x<0\\ x^2,& x\ge 0\end{array}$ and $g(x)=\{\begin{array}{cc}{x}^3,& x<1\\ 3x-2,& x\ge 1\end{array}$
Then, the number of poin …

Q7.Let the position vectors of two points $P$ and $Q$ be $3\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}-4\hat{k}$ respectively. Let $R$ and S be two points such that the direction rations of lines PR and QS are $(4,-1,2)$ and $(-2,1,-2)$, respectively. Let lines …

Q8.If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx}+2y\tan x=\sin x,y\left(\frac{\pi }{3}\right)=0$, then the maximum value of the function $y(x)$ over $R$ is equal to :

Q9.Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha ,\beta ),(0,\beta )$ …

Q10.If $n$ is the number or irrational terms in the expansion of ${\left(3^{1/4}+5^{1/8}\right)}^{60}$, then ( $n-1$ ) is divisible by:

Q11.If the three normals drawn to the parabola, $y^2=2x$ pass through the point $(a,0)a\ne 0$, then ' $a$ ' must be greater than :

Q12.If for $\mathrm{a}>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2\mathrm{a},3)$ and $\mathrm{B}(0,4,5)$ on the plane $Ix+my+nz=0$ are points $\mathrm{C}(0,-a,-1)$ and $D$ respectively, then the length of line segment $CD$ is equal to :

Q13.The number of roots of the equation,
$(81{)}^{{\sin }^{2}x}+(81{)}^{{\cos }^{2}x}=30$ in the interval $[0,\pi ]$ is equal to :

Q14.Which of the following Boolean expression is a tautology?

Q15.The locus of the midpoints of the chord of the circle, $x^2+y^2=25$ which is tangent to the hyperbola, $\frac{x^2}{9}-\frac{y^2}{16}=1$ is :

Q16.Let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{i}& -\mathrm{i}\\ -\mathrm{i}& \mathrm{i}\end{array}\right],\mathrm{i}=\sqrt{-1}$. Then, the system of linear equation …

Q17.Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in N,{\left(1-x+x^3\right)}^n={\sum }_{j=0}^{3n}{a}_j{x}^j$, then …

Q18.Let ${\mathrm{S}}_{\mathrm{k}}={\sum }_{\mathrm{r}=1}^{\mathrm{k}}{\tan }^{-1}\left(\frac{6^{\mathrm{r}}}{2^{2\mathrm{r}+1}+3^{2\mathrm{r}+1}}\right)$. Then ${\lim }_{\mathrm{k}\to \infty }{\mathrm{S}}_{\mathrm{k}}$ is equal to :

Q19.Let $P$ be a plane $Ix+my+nz=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. If plane $P$ divides the line segment $AB$ joining points $A(-3,-6,1)$ and $B(2,4,-3)$ in ratio $k:1$ then the value of $k$ is equal to :

Q20.A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

Q21.Consider an arithmetic series and a geometric series having four initial terms from the set $\{11,8,21,16,26,32,4\}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ____

Q22.If the normal to the curve $y(x)={\int }_0^x\left(2t^2-15t+10\right)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5,a>1$, then the value of $|a+6b|$ is equal to ____

Q23.Let $f:R\to R$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in R$. If $I_1={\int }_0^{8}f(x)dx$ and $I_2={\int }_{-1}^{3}f(x)dx$, then the value of $I_1+2I_2$ is equal to ____

Q24.Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx}=2(x+1)$. If the numerical value of area bounded by the curve $y=y(x)$ and $x$-axis is $\frac{4\sqrt{8}}{3}$, then the value of $y(1)$ is equal to ____

Q25.Let $z$ and $\omega$ be two complex numbers such that $\omega =z\bar{z}-2z+2,\left|\frac{z+i}{z-3i}\right|=1$ and $\mathrm{Re}(\omega )$ has minimum value. Then, the minimum value of $\mathrm{n}\in \mathrm{N}$ for which ${\omega }^{\mathrm{n}}$ is real, is equal to

Q26.If ${\lim }_{x\to 0}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}=2$, then $a+b+c$ is equal to

Q27.Let $f:(0,2)\to R$ be defined as $f(x)={\log }_2\left(1+\tan \left(\frac{\pi x}{4}\right)\right)$. Then, ${\lim }_{n\to \infty }\frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\dots .+f(1)\right)$ is equal to

Q28.The total number of $3\times 3$ matrices A having entries from the set $\{0,1,2,3\}$ such that the sum of all the diagonal entries of $A{A}^{\top }$ is 9, is equal to

Q29.Let $\mathrm{P}=\left[\begin{array}{ccc}-30& 20& 56\\ 90& 140& 112\\ 120& 60& 14\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ccc}2& 7& {\omega }^2\\ -1& -\omega & 1\\ 0& -\omega & -\omega +1\end{array}\right]$ where $\omega =\frac{-1+\mathrm{i}\sqrt{3}}{2}$ …

Q30.Let ABCD be a square of side of unit length. Let a circle ${\mathrm{C}}_1$ centered at A with unit radius is drawn. Another circle $C_2$ which touches $C_1$ and the lines $AD$ and $AB$ are tangent to it, is also drawn. Let a tangent line from the point C to the circle …