JEE Main 2021 · 26 Aug 2021 · Shift 1

Q1.A solid metal sphere of radius R having charge q is enclosed inside the concentric spherical shell of inner radius a and outer radius $b$ as shown in figure. The approximate variation electric field $\vec{E}$ as a function of distince $r$ from centre $O$ is given by:

Q2.The rms speed of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are ${\mathrm{V}}_{\mathrm{H}}$, ${\mathrm{V}}_{\mathrm{O}}$ and ${\mathrm{V}}_{\mathrm{C}}$ respectively then:

Q3.In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation the reading on the main scale …

Q4.The magnitude of vectors $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$ in the given figure are equal. The direction of $\overrightarrow{OA}+\overrightarrow{OB}-\overrightarrow{OC}$ with $x$-axis will be :-

Q5.The fractional change in the magnetic field intensity at a distance ' $r$ ' from centre on the axis of current carrying coil of radius ' $a$ ' to the magnetic field intensity at the centre of the same coil is : (Take $\mathrm{r}<\mathrm{a}$ )

Q6.In the given figure, the emf of the cell is 2.2 V and if internal resistance is $0.6\Omega$. Calculate the power dissipated in the whole circuit :

Q7.An inductor coil stores 64 J of magnetic field energy and dissipates energy at the rate of 640 W when a current of 8 A is passed through it. If this coil is joined across an ideal battery, find the time constant of the circuit in seconds :

Q8.If $\mathrm{E},\mathrm{L},\mathrm{M}$ and G denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of $P$ in the formula $P=E{L}^2{M}^{-5}{G}^{-2}$ are :-

Q9.A series LCR circuit driven by 300 V at a frequency of 50 Hz contains a resistance $\mathrm{R}=3\mathrm{k}\Omega$, an inductor of inductive reactances ${\mathrm{X}}_{\mathrm{L}}=250\pi \Omega$ and an unknown capacitor. The value of capacitance to maximize the average power s …

Q10.Inside a uniform spherical shell :
(a) the gravitational field is zero
(b) the gravitational potential is zero
(c) the gravitational field is same everywhere
(d) the gravitation potential is same everywhere
(e) all the above

Choose the most appropriate answer from the options gi …

Q11.In a photoelectric experiment ultraviolet light of wavelength 280 nm is used with lithium cathode having work function $\varphi =2.5\mathrm{eV}$. If the wavelength of incident light is switched to 400 nm , find out the change in the stopping potential. …

Q12.Car B overtakes another car A at a relative speed of $40{\mathrm{ms}}^{-1}$. How fast will the image of car B appear to move in the mirror of focal length 10 cm fitted in car $A$, when the car $B$ is 1.9 m away from the car A ?

Q13.An electric appliance supplies $6000\mathrm{J}/\min$ heat to the system. If the system delivers a power of 90 W . How long it would take to increase the internal energy by $2.5\times 10^3\mathrm{J}$ ?

Q14.A particular hydrogen like ion emits radiation of frequency $2.92\times 10^{15}\mathrm{Hz}$ when it makes transition from $n=3$ to $n=1$. The frequency in Hz of radiation emitted in transition from $n=2$ to $n=1$ will be :

Q15.The material filled between the plates of a parallel plate capacitor has resistivity $200\Omega \mathrm{m}$. The value of capacitance of the capacitor is 2 pF . If a potential difference of 40 V is applied across the plates of the capacitor, then the value of leakage current …

Q16.Two narrow bores of diameter 5.0 mm and 8.0 mm are joined together to form a U-shaped tube open at both ends. If this U-tube contains water, what is the difference in the level of two limbs of the tube.
[Take surface tension of water …

Q17.Identify the logic operation carried out by the given circuit :-

Q18.The initial mass of a rocket is 1000 kg. Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of $20{\mathrm{ms}}^{-2}$. The gases come out at a relative speed of $500{\mathrm{ms}}^{-1}$ with respect to the rocket :[Use …

Q19.Statement-I: By doping silicon semiconductor with pentavalent material, the electrons density increases.
Statement-II: The n-type semiconductor has net negative charge.
In the light of the above statements, choose the most appropriate answer from the options given below:

Q20.What equal length of an iron wire and a copper-nickel alloy wire, each of 2 mm diameter connected parallel to give an equivalent resistance of $3\Omega$?
(Given resistivities of iron and copper-nickel alloy wire are $12\mu \Omega \mathrm{cm}$ and $51\mu \Omega \mathrm{cm}$ r …

Q21.Two spherical balls having equal masses with radius of 5 cm each are thrown upwards along the same vertical direction at an interval of 3s with the same initial velocity of $35\mathrm{m}/\mathrm{s}$, then these balls collide at a height of ______ m. (Take …

Q22.A source and a detector move away from each other in absence of wind with a speed of $20\mathrm{m}/\mathrm{s}$ with respect to the ground. If the detector detects a frequency of 1800 Hz of the sound coming from the source, then the original frequency of source considering spe …

Q23.Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle an …

Q24.Two short magnetic dipoles m1 and m2 each having magnetic moment of $1{\mathrm{Am}}^2$ are placed at point O and P respectively. The distance between OP is 1 meter. The torque experienced by the magnetic dipole m2 due to the presence of ${\mathrm{m}}_1$ is ______ …

Q25.White light is passed through a double slit and interference is observed on a screen 1.5 m away. The separation between the slits is 0.3 mm. The first violet and red fringes are formed 2.0 mm and 3.5 mm away from the central white fringes. The difference in wavelengths of red and …

Q26.A soap bubble of radius 3 cm is formed inside the another soap bubble of radius 6 cm. The radius of an equivalent soap bubble which has the same excess pressure as inside the smaller bubble with respect to the atmospheric pressure is ______ cm.

Q27.A uniform chain of length 3 meter and mass 3 kg overhangs a smooth table with 2 meter laying on the table. If k is the kinetic energy of the chain in joule as it completely slips off the table, then the value of $k$ is ______ (Take $\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2$)

Q28.The electric field in a plane electromagnetic wave is given by …

Q29.An amplitude modulated wave is represented by
$C_m(t)=10(1+0.2\cos 12560t)\sin \left(111\times 10^{4}t\right)$ volts.
The modulating frequency in kHz will be .......

Q30.Two travelling waves produces a standing wave represented by equation, $y=1.0\mathrm{mm}\cos \left(1.57{\mathrm{cm}}^{-1}\right)x\sin \left(78.5{\mathrm{s}}^{-1}\right)t$. The node closest to the origin in the region $x>0$ will be at $x=$____ cm.

Q1.The polymer formed on heating Novolac with formaldehyde is :

Q2.The correct options for the products A and B of the following reactions are

Q3.What are the products formed in sequence when excess of ${\mathrm{CO}}_2$ is passed in slaked lime?

Q4.Which one of the following methods is most suitable for preparing deionized water?

Q5.Given below are two statements.
Statement I: In the titration between strong acid and weak base methyl orange is suitable as an indicator.
Statement II: For titration of acetic acid with NaOH phenolphthalein is not a suitable indicator. In the light of the above statements, choos …

Q6.Given below are two statements.
Statement I: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus.
Statement II: According to …

Q7.Which one of the following is correct for the adsorption of a gas at a given temperature on a solid surface?

Q8.Among the following compounds I-IV, which one forms a yellow precipitate on reacting sequentially with (i) NaOH (ii) dil. ${\mathrm{HNO}}_3$ (iii) ${\mathrm{AgNO}}_3$ ?

Q9.The conversion of hydroxyapatite occurs due to presence of ${\mathrm{F}}^{-}$ions in water. The correct formula of hydroxyapatite is:

Q10.The correct sequential addition of reagents in the preparation of 3-nitrobenzoic acid from benzene is:

Q11.Given below are two statements:
Statement I : The limiting molar conductivity of KCl (strong electrolyte) is higher compared to that of ${\mathrm{CH}}_3\mathrm{COOH}$ (weak electrolyte).
Statement II: Molar conductivity decreases with decrease in concentration of electrolyte.
In …

Q12.Which one of the following when dissolved in water gives coloured solution in nitrogen atmosphere?

Q13.Which one of the following complexes is violet in colour?

Q14.Excess of isobutane on reaction with $\mathrm{Br}{}_2$ in presence of light at $125^{\circ }\mathrm{C}$ gives which one of the following, as the major product?

Q15.Given below are two statements.
Statement I: Frenkel defects are vacancy as well as interstitial defects.
Statement II: Frenkel defect leads to colour in ionic solids due to presence of F-centres.
Choose the most appropriate answer for the statements from the options given below:

Q16.The major products formed in the following reaction sequence $\mathbf{A}$ and $\mathbf{B}$ are :

Q17.Given below are two statements.
Statement I: The choice of reducing agents for metals extraction can be made by using Ellingham diagram, a plot of $\Delta G$ vs temperature.
Statement II: The value of $\Delta G$ increases from left to right in Ellingham diagram.
In the light of t …

Q18.The incorrect statement is:

Q19.The Born-Haber cycle for KCl is evaluated with the following data:
${\Delta }_{\mathrm{f}}{\mathrm{H}}^{\Theta }$ for $\mathrm{KCl}=-436.7{\mathrm{kJmol}}^{-1}$ ; ${\Delta }_{\text{sub }}{\mathrm{H}}^{\Theta }$ for $\mathrm{K}=89.2{\mathrm{kJmol}}^{-1}$; …

Q20.Of the following four aqueous solutions, total number of those solutions whose freezing point is lower than that of $0.10\mathrm{M}{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}$ is ____
(Integer answer)
(A) $0.10\mathrm{M}{\mathrm{Ba}}_3{\left({\mathrm{PO}}_4\right)}_2$
(B) …

Q21.The ratio of number of water molecules in Mohr's salt and potash alum is ____ $\times 10^{-1}$
(Integer answer)

Q22.The following data was obtained for chemical reaction given below at 975 K .

\begin{tabular}{lclc}
& $2{\mathrm{NO}}_{(\mathrm{g})}+$ & $2{\mathrm{H}}_{2(\mathrm{g})}\to$ & ${\mathrm{N}}_{2(\mathrm{g})}+2{\mathrm{H}}_2{\mathrm{O}}_{(\mathrm{g})}$ \\
& {$[\mathrm{NO}]$} & { …

Q23.The ${\mathrm{OH}}^{-}$ concentration in a mixture of 5.0 mL of $0.0504\mathrm{M}{\mathrm{NH}}_4\mathrm{Cl}$ and 2 mL of $0.0210\mathrm{M}{\mathrm{NH}}_3$ solution is $x\times 10^{-6}\mathrm{M}$. The value of $x$ is ______. (Nearest integer)
[Given …

Q24.An aqueous KCl solution of density $1.2\mathrm{g}{\mathrm{mL}}^{-1}$ has a molality of $3.30\mathrm{mol}{\mathrm{kg}}^{-1}$. The molarity of the solution in $\mathrm{mol}{\mathrm{L}}^{-1}$ is ______ (Nearest integer)
[Molar mass of KCl = 74.5]

Q25.The total number of negative charge in the tetrapeptide, Gly-Glu-Asp-Tyr, at pH 12.5 will be ______. (Integer answer)

Q26.The number of $4f$ electrons in the ground state electronic configuration of ${\mathrm{Gd}}^{2+}$ is ______.
[Atomic number of Gd = 64]

Q27.These are physical properties of an element
(A) Sublimation enthalpy
(B) Ionisation enthalpy
(C) Hydration enthalpy
(D) Electron gain enthalpy

The total number of above properties that affect the reduction potential is ______ (Integer answer)

Q28.$A{B}_3$ is an interhalogen T-shaped molecule. The number of lone pairs of electrons on A is ______. (Integer answer)

Q1.The sum of solutions of the equation $\frac{\cos x}{1+\sin x}=|\tan 2x|,x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)-\left\{\frac{\pi }{4},-\frac{\pi }{4}\right\}$ is:

Q2.Let $f(x)=\cos \left(2{\tan }^{-1}\sin \left({\cot }^{-1}\sqrt{\frac{1-x}{x}}\right)\right),0<x<1$. Then:

Q3.Out of all the patients in a hospital $89\%$ are found to be suffering from heart ailment and $98\%$ are suffering from lungs infection. If $\mathrm{K}\%$ of them are suffering from both ailments, then K can not belong to the set:

Q4.The equation $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$ represents a circle with:

Q5.Let $\theta \in \left(0,\frac{\pi }{2}\right)$. If the system of linear equations.
$\left(1+{\cos }^2\theta \right)x+{\sin }^2\theta y+4\sin 3\theta z=0$
${\cos }^2\theta x+\left(1+{\sin }^2\theta \right)y+4\sin 3\theta z=0$ …

Q6.A plane $P$ contains the line $x+2y+3z+1=0=x-y-z-6$, and is perpendicular to the plane $-2x+y+z+8=0$. Then which of the following points lies on $P$ ?

Q7.The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is:

Q8.If ${}^{20}{\mathrm{C}}_{\mathrm{r}}$ is the co-efficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansion of $(1+\mathrm{x}{)}^{20}$, then the value of ${\sum }_{\mathrm{r}=0}^{20}{\mathrm{r}}^2{}^{20}{\mathrm{C}}_{\mathrm{r}}$ is equal to

Q9.If $\mathrm{A}=\left(\begin{array}{cc}\frac{1}{\sqrt{5}}& \frac{2}{\sqrt{5}}\\ \frac{-2}{\sqrt{5}}& \frac{1}{\sqrt{5}}\end{array}\right),\mathrm{B}=\left(\begin{array}{cc}1& 0\\ \mathrm{i}& 1\end{array}\right),\mathrm{i}=\sqrt{-1}$, and …

Q10.The value of ${\int }_{\frac{-1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}{\left({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2\right)}^{1/2}dx$ is:

Q11.On the ellipse $\frac{x^2}{8}+\frac{y^2}{4}=1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line $x+2y=0$. Let $S$ and $S^{\prime }$ be the foci of the ellipse and e be its eccentricity. If $A$ is the area of th …

Q12.Let $A$ and $B$ be independent events such that $P(A)=p,P(B)=2p$. The largest value of $p$, for which $P$ (exactly one of $A,B$ occurs) $=\frac{5}{9}$, is:

Q13.If a line along a chord of the circle $4x^2+4y^2+120x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^2=30x$, then the length of this chord is:

Q14.If the truth value of the Boolean expression $((p\vee q)\wedge (q\to r)\wedge (\sim r))\to (p\wedge q)$ is false, then the truth values of the statements $p,q,r$ respectively can be:

Q15.The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If $\alpha$ and $\sqrt{\beta }$ are the mean and standard deviation respectively for correct data, then …

Q16.Let ABC be a triangle with $\mathrm{A}(-3,1)$ and $\angle \mathrm{ACB}=\theta ,0<\theta <\frac{\pi }{2}$. If the equation of the median through $B$ is $2x+y-3=0$ and the equation of angle bisector of $C$ is $7x-4y-1=0$, then $\tan \theta$ is equal to:

Q17.Let $y=y(x)$ be a solution curve of the differential equation $(y+1){\tan }^{2}xdx+\tan xdy+ydx=0,x\in \left(0,\frac{\pi }{2}\right)$. If ${\lim }_{x\to 0^{+}}xy(x)=1$, then the value of $y\left(\frac{\pi }{4}\right)$ is:

Q18.Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\times \vec{c}=\vec{b}$ and $\vec{a}\cdot \vec{c}=3$, then $\vec{a}\cdot (\vec{b}\times \vec{c})$ is equal to

Q19.If the sum of an infinite GP $\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2,{\mathrm{ar}}^3,\dots \dots$ is 15 and the sum of the squares of its each term is 150, then the sum of $a{r}^2,a{r}^4,a{r}^6,\dots$. is:

Q20.The value of ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{r=0}^{2n-1}\frac{n^2}{n^2+4r^2}$ is:

Q21.A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left(\frac{4}{\pi }+1\right)k$ is …

Q22.The sum of all integral values of $k(k\ne 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is ______.

Q23.The area of the region $S=\left\{(x,y):3x^2\le 4y\le 6x+24\right\}$ is ______.

Q24.If ${}^1{P}_1+2\cdot {}^2{P}_2+3\cdot {}^3{P}_3+\dots \dots \dots +15\cdot {}^{15}{P}_{15}={}^q{P}_r-s,0\le s\le 1$, then ${}^{q+s}{C}_{r-s}$ is equal to ______.

Q25.Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2y-z=3$. If $d$ is the distance of the point $(0,0,6)$ from $L$, then $d^2$ is equal to ____.

Q26.If $y=y(x)$ is an implicit function of $x$ such that ${\log }_e(x+y)=4xy$, then $\frac{d^{2}y}{d{x}^2}$ at $x=0$ is equal to ____.

Q27.The locus of a point which moves such that the sum of squares of its distances from the points $(0,0),(1,0),(0,1),(1,1)$ is 18 units, is a circle of diameter $d$. Then $d^2$ is equal to ____.

Q28.Let $z=\frac{1-i\sqrt{3}}{2},i=\sqrt{-1}$. Then the value of $21+{\left(z+\frac{1}{z}\right)}^3+{\left(z^2+\frac{1}{z^2}\right)}^3+{\left(z^3+\frac{1}{z^3}\right)}^3+\dots \dots +{\left(z^{21}+\frac{1}{z^{21}}\right)}^3$ is ____.

Q29.Let $a,b\in R,b\ne 0$. Define a function $f(x)=\{\begin{array}{ll}a\sin \frac{\pi }{2}(x-1),& \text{ for }x\le 0\\ \frac{\tan 2x-\sin 2x}{b{x}^3},& \text{ for }x>0\end{array}$ If $f$ is continuous at $x=0$, then $10-ab$ is equal to ____.

Q30.The number of three digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is ____.