JEE Main 2021 · 20 Jul 2021 · Shift 1

Q1.The amount of heat needed to raise the temperature of 4 moles of a rigid diatomic gas from $0^{\circ }\mathrm{C}$ to $50^{\circ }\mathrm{C}$ when no work is done is ______. ($R$ is the universal gas constant)

Q2.A deuteron and an alpha particle having equal kinetic energy enter perpendicularly into a magnetic field. Let $r_d$ and $r_{\alpha }$ be their respective radii of circular path. The value of $\frac{r_d}{r_{\alpha }}$ is equal to:

Q3.If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\vec{A}\cdot \vec{B}=|\vec{A}\times \vec{B}|$. Then the value of $|\vec{A}-\vec{B}|$ will be:

Q4.Consider a mixture of gas molecule of types A, B and C having masses ${\mathrm{m}}_{\mathrm{A}}<{\mathrm{m}}_{\mathrm{B}}<{\mathrm{m}}_{\mathrm{C}}$. The ratio of their root mean square speeds at normal temperature and pressure is:

Q5.A nucleus of mass M emits $\gamma$-ray photon of frequency 'v'. The loss of internal energy by the nucleus is: [Take 'c' as the speed of electromagnetic wave]

Q6.A radioactive material decays by simultaneous emissions of two particles with half lives of 1400 years and 700 years respectively. What will be the time after which one third of the material remains? (Take $\ln 3=1.1$)

Q7.The value of current in the $6\Omega$ resistance is :

Q8.The radiation corresponding to $3\to 2$ transition of a hydrogen atom falls on a gold surface to generate photoelectrons. These electrons are passed through a magnetic field of $5\times 10^{-4}\mathrm{T}$. Assume that the radius of the largest circular circular path f …

Q9.For the circuit shown below, calculate the value of ${\mathrm{I}}_2$ :

Q10.The arm PQ of a rectangular conductor is moving from $x=0$ to $x=2b$ outwards and then inwards from $x=2b$ to $x=0$ as shown in the figure. A uniform magnetic field perpendicular to the plane is acting from $\mathrm{x}=0$ to $\mathrm{x}=\mathrm{b}$. Identify the graph the varia …

Q11.Region I and II are separated by a spherical surface of radius 25 cm. An object is kept in region I at a distance of 40 cm from the surface. The distance of the image from the surface is :

Q12.The normal reaction 'N' for a vehicle of 800 kg mass, negotiating a turn on a $30^{\circ }$ banked road at maximum possible speed without skidding is $\_\_\_\_$ $\times 10^3\mathrm{kg}\mathrm{m}/{\mathrm{s}}^2$.
[Given $\cos 30^{\circ }=0.87,{\mu }_5=0.2$]

Q13.The entropy of any system is given by $\mathrm{S}={\alpha }^2\beta \ell \mathrm{n}\left[\frac{\mu \mathrm{kR}}{\mathrm{J}{\beta }^2}+3\right]$ Where $\alpha$ and $\beta$ are the constants. $\mu ,\mathrm{J},\mathrm{k}$ and R no. of moles, mechanical equivalent of heat, Boltzmann …

Q14.A current of 5 A is passing through a non-linear magnesium wire of cross-section $0.04{\mathrm{m}}^2$. At every point the direction of current density is at an angle of $60^{\circ }$ with unit vector of area of cross-section. The magnitude of electric field at every point of the …

Q15.The value of tension in a long thin metal wire has been changed from $T_1$ to $T_2$. The lengths of the metal wire at two different values of tension $T_1$ and $T_2$ are ${\ell }_1$ and ${\ell }_2$ respectively. The actual length of the metal wire is :

Q16.AC voltage $\mathrm{V}(\mathrm{t})=20\sin \omega \mathrm{t}$ of frequency 50 Hz is applied to a parallel plate capacitor. The separation between the plates is 2 mm and the area is $1{\mathrm{m}}^2$. The amplitude of the oscillating displacement current for the applied Ac volta …

Q17.A butterfly is flying with a velocity $4\sqrt{2}\mathrm{m}/\mathrm{s}$ in North-East direction. Wind is slowly blowing at $1\mathrm{m}/\mathrm{s}$ from North to South. The resultant displacement of the butterfly in 3 seconds is :

Q18.A certain charge $Q$ is divided into parts $q$ and $(Q-q)$. How should the charges $Q$ and $q$ be divided so that q and $(Q-q)$ placed at a certain distance apart experience maximum electrostatic repulsion ?

Q19.A steel block of 10 kg rests on a horizontal floor as shown. When three iron cylinders are placed on it as shown, the block and cylinders go down with an acceleration $0.2\mathrm{m}/{\mathrm{s}}^2$. The normal reaction ${\mathrm{R}}^{\prime }$ by the floor if mass of the iro …

Q20.A person whose mass is 100 kg travels from Earth to Mars in a spaceship. Neglect all other object in sky and take acceleration due to gravity on the surface of the Earth and mars as $10\mathrm{m}/{\mathrm{s}}^2$ and $4\mathrm{m}/{\mathrm{s}}^2$ respectively. Identify …

Q21.The amplitude of wave disturbance propagating in the positive $x$-direction is given by $\mathrm{y}=\frac{1}{1+(\mathrm{x}{)}^2}$ at time $\mathrm{t}=0$ and $\mathrm{y}=\frac{1}{1+(\mathrm{x}-2{)}^2}$ at $\mathrm{t}=1\mathrm{s}$, where x and y are in metres. The shape …

Q22.In the reported figure, heat energy absorbed by a system in going through a cyclic process is ____ $\pi \mathrm{J}$.

Q23.In a spring gun having spring constant $100\mathrm{N}/\mathrm{m}$ a small ball 'B' of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m. There should be a box placed at a distance 'd' on the ground so that the ball falls in it. If …

Q24.The frequency of a car horn encountered a change from 400 Hz to 500 Hz, when the car approaches a vertical wall. If the speed of sound is $330\mathrm{m}/\mathrm{s}$. Then the speed of car is ____ $\mathrm{km}/\mathrm{h}$.

Q25.A body having specific charge $8\mu \mathrm{C}/\mathrm{g}$ is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of $100\mathrm{V}/\mathrm{m}$ is applied horizontal …

Q26.A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass 'm' travelling along the surface hits at one end of the rod with a velocity 'u' in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle c …

Q27.An object viewed from a near point distance of 25 cm, using a microscopic lens with magnification '6', gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube …

Q28.In an LCR series circuit, an inductor 30 mH and a resistor $1\Omega$ are connected to an AC source of angular frequency $300\mathrm{rad}/\mathrm{s}$. The value of capacitance for which, the current leads the voltage by $45^{\circ }$ is …

Q29.A circular disc reaches from top to bottom of an inclined plane of length 'L' when it slips down the plane, it takes time '${\mathrm{t}}_1$'. When it rolls down the plane, it takes time ${\mathrm{t}}_2$. The value of $\frac{t_1}{t_2}$ is $\sqrt{\frac{3}{x}}$. The valu …

Q30.A carrier wave $V_C(t)=160\sin \left(2\pi \times 10^{6}t\right)$ volts is made to vary between $V_{\max }=200\mathrm{V}$ and $V_{\text{min }}=120\mathrm{V}$ by a message signal $V_m(t)=A_m\sin \left(2\pi \times 10^{3}t\right)$ volts. The peak voltage $A_m$ of t …

Q1.Chemical nature of the nitrogen oxide compound obtained from a reaction of concentrated nitric acid and ${\mathrm{P}}_4{\mathrm{O}}_{10}$ (in 4:1 ratio) is:

Q2.Compound $A$ is converted to $B$ on reaction with ${\mathrm{CHCl}}_3$ and KOH . The compound $B$ is toxic and can be decomposed by C . $\mathrm{A},\mathrm{B}$ and C respectively are:

Q3.Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason $\mathbf{R}$.
Assertion A: The dihedral angles in ${\mathrm{H}}_2{\mathrm{O}}_2$ in gaseous phase is $90.2^{\circ }$ and in solid phase is $111.5^{\circ }$.
Reason R: The ch …

Q4.Among the given species the Resonance stabilized carbocations are:

Q5.A s-block element (M) reacts with oxygen to form an oxide of the formula ${\mathrm{MO}}_2$. The oxide is pale yellow in colour and paramagnetic. The element ( M ) is:

Q6.Green chemistry in day- to day life is in the use of:

Q7.Identify the incorrect statement from the following:

Q8.The metal that can be purified economically by fractional distillation method is:

Q9.The set in which compounds have different nature is:

Q10.The conditions given below are in the context of observing Tyndall effect in colloidal solutions:
(a) The diameter of the colloidal particles is comparable to the wavelength of light used.
(b) The diameter of the colloidal particles in much smaller than the wavelength of light us …

Q11.According to the valence bond theory the hybridization of central metal atom is ${\mathrm{dsp}}^2$ for which one of the following compounds?

Q12.In the given reaction, 3-bromo-2,2-dimethylbutane $\stackrel{{\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}}{\to }$ 'A' (major product). Product A is:

Q13.Orlon fibres are made up of:

Q14.The species given below that does NOT show disproportionation reaction is:

Q15.For above chemical reactions, identify the correct statement from the following

Q16.An inorganic compound 'X' on treatment with concentrated ${\mathrm{H}}_2{\mathrm{SO}}_4$ produces brown fumes and gives dark brown ring with ${\mathrm{FeSO}}_4$ in presence of concentrated ${\mathrm{H}}_2{\mathrm{SO}}_4$. Also compound 'X' gives precipitate 'Y', when its solution …

Q17.Which among the above compound/s does / do not form silver mirror when treated with Tollen's reagent?

Q18.Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Sharp glass edge becomes smooth on heating it upto its melting point.
Reason R: The viscosity of glass decreases on melting.
Choose the most appropriate answer from …

Q19.The correct structure of Rhumann's Purple, the compound formed in the reaction of ninhydrin with proteins is:

Q20.The correct order of intensity of colors of the compounds is:

Q21.The number of lone pairs of electrons on the central I atom in ${\mathrm{I}}_3^{-}$ is ____.

Q22.An average person needs abut 10000 KJ energy per day. The amount of glucose (molar mass $=180.0\mathrm{g}{\mathrm{mol}}^{-1}$) needed to meet this energy requirement is ____ g,
(Nearest integer)
(Use: ${\Delta }_{\mathrm{c}}\mathrm{H}$ (glucose) …

Q23.To synthesise 1.0 mole of 2-methylpropan-2-ol from Ethylethanoate ____ equivalents of ${\mathrm{CH}}_3\mathrm{MgBr}$ reagent will be required. (Integer value)

Q24.The spin only magnetic moment value for the complex $[\mathrm{Co}(\mathrm{CN}{)}_6{]}^{4-}$ is ____ BM.
[At. No. of $\mathrm{Co}=27$]

Q25.At $20^{\circ }\mathrm{C}$, the vapour pressure of benzene is 70 torr and that of methyl benzene is 20 torr. The mole fraction of benzene in the vapour phase at $20^{\circ }\mathrm{C}$ above an equimolar mixture of benzene and methyl benzene is ____ $\times 10^{-2}$. (N …

Q26.$2{\mathrm{SO}}_2(\mathrm{g})+{\mathrm{O}}_2(\mathrm{g})\rightleftharpoons 2{\mathrm{SO}}_3(\mathrm{g})$

In an equilibrium mixture, the partial pressures are
${\mathrm{P}}_{{\mathrm{SO}}_3}=43\mathrm{kPa};{\mathrm{P}}_{{\mathrm{O}}_2}=530\mathrm{Pa}$ and …

Q27.250 mL of 0.5 M NaOH was added to 500 mL of 1 M HCl. The number of unreacted HCl molecules in the solution after complete reaction is ____ $\times 10^{21}$ (Nearest integer)
$\left({\mathrm{N}}_{\mathrm{A}}=6.022\times 10^{23}\right)$

Q28.The number of nitrogen atoms in a semicarbazone molecule of acetone is ____.

Q29.The inactivation rate of a viral preparation in proportional to the amount of virus. In the first minute after preparation, $10\%$ of the virus is inactivated. The rate constant for viral inactivation is ____ $\times 10^{-3}{\min }^{-1}$. (Nearest integer)
[ Use: …

Q30.The Azimuthal quantum number for the valence electrons of ${\mathrm{Ga}}^{+}$ion is ____.
(Atomic number of Ga=31)

Q1.Let 'a' be a real number such that the function $f(x)=a{x}^2+6x-15,x\in R$ is increasing in $\left(-\infty ,\frac{3}{4}\right)$ and decreasing in $\left(\frac{3}{4},\infty \right)$. Then the function …

Q2.Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a $3\times 3$ matrix, where …

Q3.If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1\mathrm{and}\arg (z)-\arg (w)=\frac{3\pi }{2}$, then $\arg \left(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega }\right)$ is :
(Here $\arg (\mathrm{z})$ denotes the principal argument of complex num …

Q4.If in a triangle $\mathrm{ABC},\mathrm{AB}=5$ units, $\angle \mathrm{B}={\cos }^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $△\mathrm{ABC}$ is 5 units, then the area (in sq. units) of $△ABC$ is :

Q5.Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :

Q6.The coefficient of $x^{256}$ in the expansion of $(1-x{)}^{101}{\left(x^2+x+1\right)}^{100}$ is :

Q7.The probability of selecting integers $a\in [-5,30]$ such that $x^2+2(a+4)x-5a+64>0$, for all $x\in R$ is :

Q8.The Boolean expression $(p\wedge \sim q)\Rightarrow (q\vee \sim p)$ is equivalent to :

Q9.Let a be a positive real numbers such that
${\int }_0^a{e}^{x-[x]}dx=10e-9$
where $[x]$ is the greatest integer less than or equal to $x$. Then a is equal to :

Q10.Let $A=\left[\begin{array}{ll}2& 3\\ a& 0\end{array}\right],a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\det (Q)=9$, then the modulus of the sum of all possible values of determinant of P is equal t …

Q11.If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+(3{)}^{1/4}x+3^{1/2}=0$, then the value of ${\alpha }^{96}\left({\alpha }^{12}-1\right)+{\beta }^{96}\left({\beta }^{12}-1\right)$ is equal to :

Q12.The number of real roots of the equation ${\tan }^{-1}\sqrt{x(x+1)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{4}$ is :

Q13.The value of the integral ${\int }_{-1}^1{\log }_e(\sqrt{1-x}+\sqrt{1+x})dx$ is equal to :

Q14.The mean of 6 distinct observations is 6.5 and their variance is 10.25 . If 4 out of 6 observations are $2,4,5$ and 7 , then the remaining two observations are :

Q15.Let the tangent to the parabola $\mathrm{S}:{\mathrm{y}}^2=2\mathrm{x}$ at the point $\mathrm{P}(2,2)$ meet the x -axis at Q and normal at it meet the parabola S at the point R . Then the area (in sq. units) of the triangle PQR is equal to :

Q16.Let a function $f:R\to R$ be defined as $f(x)=\{\begin{array}{ccc}\sin x-e^x& \text{ if }& x\le 0\\ a+[-x]& \text{ if }& 0<x<1\\ 2x-b& \text{ if }& x\ge 1\end{array}$
where $[x]$ is the greatest integer less than or equal to $x$. If …

Q17.Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the val …

Q18.Let $y=y(x)$ be the solution of the differential equation
$e^x\sqrt{1-y^2}dx+\left(\frac{y}{x}\right)dy=0,y(1)=-1$
Then the value of $(y(3){)}^2$ is equal to :

Q19.Let $y=y(x)$ be the solution of the differential equation
$x\tan \left(\frac{y}{x}\right)dy=\left(y\tan \left(\frac{y}{x}\right)-x\right)dx,-1\le x\le 1,y\left(\frac{1}{2}\right)=\frac{\pi }{6}$
Then the area of the region bounded by the curves …

Q20.Let $A=\left(\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right)$ and $B=7A^{20}-20A^7+2I$, where $I$ is an identity matrix of order $3\times 3$. If $B=\left[b_{ij}\right]$, then $b_{13}$ is equal to........

Q21.Let $y=mx+c,m>0$ be the focal chord of $y^2=-64x$, which is tangent to $(x+10{)}^2+y^2=4$. Then, the value of $4\sqrt{2}(m+c)$ is equal to ____

Q22.Let $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ be in arithmetic progression with common difference $\lambda$. If $\left|\begin{array}{ccc}x+a-c& x+b& x+a\\ x-1& x+c& x+b\\ x-b+d& x+d& x+c\end{array}\right|=2$, then the value of ${\lambda }^2$ is equal to..... …

Q23.Let $\vec{a},\vec{b},\vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36{\cos }^{2}2\theta$ is equal to ____

Q24.Let $T$ be the tangent to the ellipse $E:x^2+4y^2=5$ at the point $P(1,1)$. If the area of the region bounded by the tangent $T$, ellipse $E$, lines $x=1$ and $x=\sqrt{5}$ is $\alpha \sqrt{5}+\beta +\gamma {\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)$, then …

Q25.There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ______.

Q26.The number of rational terms in the binomial expansion of ${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$ is ______.

Q27.Let P be a plane passing through the points $(1,0,1)$, $(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ be such that $\overrightarrow{\mathrm{a}}$ is parallel to the plane P, per …

Q28.If the value of ${\lim }_{x\to 0}(2-\cos x\sqrt{\cos 2x}{)}^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then a is equal to ______.

Q29.If the shortest distance between the lines ${\vec{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k}),\lambda \in R$, $\alpha >0$ and ${\vec{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k}),\mu \in R$ is 9, then $\alpha$ is equal to _ …