JEE Main 2021 · 31 Aug 2021 · Shift 2

Q1.A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the end of light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoin …

Q2.If $R_E$ be the radius of Earth, then the ratio between the acceleration due to gravity at a depth '$r$' below and a height '$r$' above the earth surface is:

Q3.Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to:

Q4.Two thin metallic spherical shells of radii $r_1$ and $r_2$ ($r_1<r_2$) are placed with their centres coinciding. A material of thermal conductivity K is filled in the space between the shells. The inner shell is maintained at temperature ${\theta }_1$ and the outer shell at tempe …

Q5.For a body executing S.H.M.:
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time interval are always equal.
(c) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E. in one time period …

Q6.The equivalent resistance of the given circuit between the terminals $A$ and $B$ is :

Q7.If VA and VB are the input voltages (either 5 V or 0 V ) and Vo is the output voltage then the two gates represented in the following circuit (A) and (B) are:-

Q8.Four identical hollow cylindrical columns of mild steel support a big structure of mass $50\times 10^3\mathrm{kg}$, The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each col …

Q9.If velocity $[\mathrm{V}]$, time $[\mathrm{T}]$ and force $[\mathrm{F}]$ are chosen as the base quantities, the dimensions of the mass will be :

Q10.The magnetic field vector of an electromagnetic wave is given by $B=B_0=\frac{\hat{i}+\hat{j}}{\sqrt{2}}\cos (kz-\omega t)$; where $\hat{i},\hat{j}$ represents unit vector along $x$ and $y$-axis respectively. At $t=0\mathrm{s}$, two electric charges q 1 of …

Q11.A mixture of hydrogen and oxygen has volume $500{\mathrm{cm}}^3$, temperature 300 K , pressure 400 kPa and mass 0.76 g . The ratio of masses of oxygen to hydrogen will be :-

Q12.Statement-1: If three forces ${\vec{F}}_1,{\vec{F}}_2$ and ${\vec{F}}_3$ are represented by three sides of a triangle and, ${\vec{F}}_1+{\vec{F}}_2=-{\vec{F}}_3$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement-II: A trian …

Q13.A free electron of 2.6 eV energy collides with a ${\mathrm{H}}^{+}$ion. This results in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon. ( $\mathrm{h}=6.6\times 10^{-34}\mathrm{Js}$ )

Q14.A block moving horizontally on a smooth surface with a speed of $40\mathrm{m}/\mathrm{s}$ splits into two parts with masses in the ratio of $1:2$. If the smaller part moves at $60\mathrm{m}/\mathrm{s}$ in the same direction, then the fractional change in kinetic en …

Q15.Statement-I: Two forces $(\vec{P}+\vec{Q})$ and $(\vec{P}-\vec{Q})$ where $\vec{P}\perp \vec{Q}$, when act at an angle ${\theta }_1$ to each other, the magnitude of their resultant is $\sqrt{3\left({\mathrm{P}}^2+{\mathrm{Q}}^2\right)}$, when they act at an angle …

Q16.A bob of mass ' $m$ ' suspended by a thread of length I undergoes simple harmonic oscillations with time period $T$. If the bob is immersed in a liquid that has density $\frac{1}{4}$ times that of the bob and the length of the thread is increased by $1/3^{\text{rd }}$ …

Q17.A coil is placed in a magnetic field as shown below : $B$
A current is induced in the coil because $\vec{B}$ is:

Q18.Statement-I : To get a steady dc output from the pulsating voltage received from a full wave rectifier we can connect a capacitor across the output parallel to the load RL.

Statement-II : To get a steady dc output from the pulsating voltage received from a full wave rectifier we …

Q19.A current of 1.5 A is flowing through a triangle, of side 9 cm each. The magnetic field at the centroid of the triangle is :
(Assume that the current is flowing in the clockwise direction.)

Q20.Choose the incorrect statement :
(a) The electric lines of force entering into a Gaussian surface provide negative flux.
(b) A charge 'q' is placed at the centre of a cube. The flux through all the faces will be the same.
(c) In a uniform electric field net flux through a closed …

Q21.A particle is moving with constant acceleration 'a'. Following graph shows $v^2$ versus $x$ (displacement) plot. The acceleration of the particle is ______ $\mathrm{m}/{\mathrm{s}}^2$.

Q22.The diameter of a spherical bob is measured using a vernier callipers. 9 divisions of the main scale, in the vernier callipers, are equal to 10 divisions of vernier scale. One main scale division is 1 mm. The main scale reading is 10 mm and $8^{\text{th}}$ division of vernier sca …

Q23.A resistor dissipates 192 J of energy in 1 s when a current of 4 A is passed through it. Now, when the current is doubled, the amount of thermal energy dissipated in 5 s in ______ J.

Q24.A bandwidth of 6 MHz is available for A.M. transmission. If the maximum audio signal frequency used for modulating the carrier wave is not to exceed 6 kHz. The number of stations that can be broadcasted within this band simultaneously without interfering with each other will be _ …

Q25.A long solenoid with $1000\mathrm{turns}/\mathrm{m}$ has a core material with relative permeability 500 and volume $10^3{\mathrm{cm}}^3$. If the core material is replaced by another material having relative permeability of 750 with same volume maintaining same current of 0 …

Q26.In a Young's double slit experiment, the slits are separated by 0.3 mm and the screen is 1.5 m away from the plane of slits. Distance between fourth bright fringes on both sides of central bright is 2.4 cm. The frequency of light used is ______ $\times 10^{14}\mathrm{Hz}$.

Q27.At very high frequencies, the effective impendance of the given circuit will be ______ $\Omega$.

Q28.Cross-section view of a prism is the equilateral triangle $ABC$ in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from P (midpoint of BC) to A is ______ …

Q29.A parallel plate capacitor of capacitance $200\mu \mathrm{F}$ is connected to a battery of 200 V. A dielectric slab of dielectric constant 2 is now inserted into the space between plates of capacitor while the battery remain connected. The change in the electrostatic energy in …

Q30.A sample of gas with $\gamma =1.5$ is taken through an adiabatic process in which the volume is compressed from $1200{\mathrm{cm}}^3$ to $300{\mathrm{cm}}^3$. If the initial pressure is 200 kPa. The absolute value of the work done by the gas in the process = $\_\_\_\_$ J.

Q1.Identify correct A, B and C in the reaction sequence given below :

Q2.Match List-I with List-II.

List - I (Parameter)
(a) Cell constant
(b) Molar conductivity
(c) Conductivity
(d) Degree of dissociation of electrolyte

List - II (Unit)
(i) S cm² mol⁻¹
(ii) Dimensionless
(iii) m⁻¹
(iv) Ω⁻¹ m⁻¹

Choose the most appropriate answer from the options gi …

Q3.For following sequence of reactions, the correct products are:

Q4.Which one of the following correctly represents the order of stability of oxides, X₂O; (X = halogen)?

Q5.Arrange the following conformational isomers of n-butane in order of their increasing potential energy :

I (see structure)
II (see structure)
III (see structure)
IV (see structure)

Q6.The Eu²⁺ ion is a strong reducing agent in spite of its ground state electronic configuration (outermost) : [Atomic number of Eu = 63]

Q7.For the reaction given below :

(see structure)

The compound which is not formed as a product in the reaction is a :

Q8.Which of the following is NOT an example of fibrous protein ?

Q9.Spin only magnetic moment in BM of [Fe(CO)₄(C₂O₄)]⁺ is :

Q10.The incorrect expression among the following is:

Q11.The structures of A and B formed in the following reaction are : [Ph = -C₆H₅]

(see structure)

Q12.Which among the following is not a polyester ?

Q13.Match List-I with List-II.

List - I (Metal ion)
(a) ${\mathrm{Mn}}^{2+}$
(b) ${\mathrm{As}}^{3+}$
(c) ${\mathrm{Cu}}^{2+}$
(d) ${\mathrm{Al}}^{3+}$

List - II (Group in Qualitative Analysis)
(i) Group - III
(ii) Group - IIA
(iii) Group - IV
(iv) Group - IIB

Choose the most appropriate …

Q14.The deposition of $X$ and $Y$ on ground surfaces is referred as wet and dry depositions, respectively, X and Y are :

Q15.The major products A and B formed in the following reaction sequence are :

Q16.The number of $\mathrm{S}=\mathrm{O}$ bonds present in sulphurous acid, peroxodisulphuric acid and pyrosulphuric acid, respectively are :

Q17.In which one of the following sets all species show disproportionation reaction ?

Q18.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Lithium salts are hydrated.
Reason (R) : Lithium has higher polarising power than other alkali metal group members.
In the light of the above statements, cho …

Q19.The transformation occurring in Duma's method is given below : …

Q20.${\mathrm{CH}}_4$ is adsorbed on 1 g charcoal at $0^{\circ }\mathrm{C}$ following the Freundlich adsorption isotherm. 10.0 mL of ${\mathrm{CH}}_4$ is adsorbed at 100 mm of Hg , whereas 15.0 mL is adsorbed at 200 mm of Hg . The volume of ${\mathrm{CH}}_4$ adsorbed at 300 mm of Hg …

Q21.According to molecular orbital theory, the number of unpaired electron(s) in ${\mathrm{O}}_2^{2-}$ is:

Q22.The pH of a solution obtained by mixing 50 mL of 1 M HCl and 30 mL of 1 M NaOH is $x\times 10^{-4}$. The value of $x$ is ____. (Nearest integer)
[log $2.5=0.3979$ ]

Q23.The empirical formula for a compound with a cubic close packed arrangement of anions and with cations occupying all the octahedral sites is ${\mathrm{A}}_{\mathrm{x}}\mathrm{B}$. The value of x is ______.

Q24.1.22 g of an organic acid is separately dissolved in 100 g of benzene (${\mathrm{K}}_{\mathrm{b}}=2.6\mathrm{K}\mathrm{kg}{\mathrm{mol}}^{-1}$) and 100 g of acetone (${\mathrm{K}}_{\mathrm{b}}=1.7\mathrm{K}\mathrm{kg}{\mathrm{mol}}^{-1}$). The acid is known to dimerize in ben …

Q25.For the reaction $\mathrm{A}\to \mathrm{B}$, the rate constant $\mathrm{K}$ (in ${\mathrm{s}}^{-1}$) is given by

The energy of activation in $\mathrm{kJ}{\mathrm{mol}}^{-1}$ is ______. (Nea …

Q26.The value of magnetic quantum number of the outermost electron of ${\mathrm{Zn}}^{+}$ ion is ______.

Q27.Sodium oxide reacts with water to produce sodium hydroxide. 20.0 g of sodium oxide is dissolved in 500 mL of water. Neglecting the change in volume, the concentration of the resulting NaOH solution is ______ $\times 10^{-1}\mathrm{M}$. (Nearest integer)
[Atomic mass : …

Q28.In the electrolytic refining of blister copper, the total number of main impurities, from the following, removed as anode mud is ______.
Pb, Sb, Se, Te, Ru, Ag, Au and Pt

Q1.The domain of the function $f(x)={\sin }^{-1}\left(\frac{3x^2+x-1}{(x-1{)}^2}\right)+{\cos }^{-1}\left(\frac{x-1}{x+1}\right)$ is :

Q2.If $y\frac{dy}{dx}=x\left[\frac{y^2}{x^2}+\frac{\varphi \left(\frac{y^2}{x^2}\right)}{{\varphi }^{\prime }\left(\frac{y^2}{x^2}\right)}\right],x>0,\varphi >0$, and $y(1)=-1$, then $\varphi \left(\frac{y^2}{4}\right)$ is equal to :

Q3.Let $f:N\to N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in N$. If $f(6)=18$, then $f(2).f(3)$ is equal to :

Q4.The number of solutions of the equation $32^{{\tan }^{2}x}+32^{{\sec }^{2}x}=81,0\le x\le \frac{\pi }{4}$ is:

Q5.If $\alpha ={\lim }_{x\to \frac{\pi }{4}}\frac{{\tan }^{3}x-\tan x}{\cos \left(x+\frac{\pi }{4}\right)}$ and $\beta ={\lim }_{x\to 0}(\cos x{)}^{\cot x}$ are the roots of the equation, $a{x}^2+bx-4=0$, then the ordered pair $(\mathrm{a},\mathrm{b})$ is:

Q6.Let $\vec{a},\vec{b},\vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies …

Q7.Let A be the set of all points $(\alpha ,\beta )$ such that the area of triangle formed by the points $(5,6),(3,2)$ and $(\alpha ,\beta )$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in A , is :

Q8.The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2x+3y+2z=0$ and $x-2y+z=0$ is :

Q9.An angle of intersection of the curves, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2+y^2=ab,a>b$ is:

Q10.Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $\mathrm{g}(3)=2\mathrm{g}(1)$ is :

Q11.Negation of the statement $(p\vee r)\Rightarrow (q\vee r)$ is:

Q12.If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3i)|$ is:

Q13.If $\alpha +\beta +\gamma =2\pi$, then the system of equations
$x+(\cos \gamma )y+(\cos \beta )z=0$
$(\cos \gamma )x+y+(\cos \alpha )z=0$
$(\cos \beta )x+(\cos \alpha )y+z=0$
has:

Q14.Let $a_1,a_2,a_3,\dots$ be an A.P. If $\frac{a_1+a_2+\dots +a_{10}}{a_1+a_2+\dots +a_p}=\frac{100}{p^2},p\ne 10$, then $\frac{a_{11}}{a_{10}}$ is equal to:

Q15.The sum of the roots of the equation, $x+1-2{\log }_2\left(3+2^x\right)+2{\log }_4\left(10-2^{-x}\right)=0$, is:

Q16.Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $f(0)=0,f(1)=1$ and $f(2)=2$, then

Q17.If $[x]$ is the greatest integer $\le x$, then ${\pi }^2{\int }_0^2\left(\sin \frac{\pi x}{2}\right)(x-[x]{)}^{[x]}dx$ is equal to:

Q18.If $\frac{dy}{dx}=\frac{2^{x}y+2^y\cdot 2^x}{2^x+2^{x+y}{\log }_{e}2},y(0)=0$, then for $y=1$, the value of $x$ lies in the interval:

Q19.The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is:

Q20.The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

Q21.Suppose the line $\frac{x-2}{\alpha }=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3y-2z+\beta =0$. Then $(\alpha +\beta )$ is equal to ____.

Q22.Let $f(x)$ be a cubic polynomial with $f(1)=-10,f(-1)=6$, and has a local minima at $x=1$, and $f^{\prime }(x)$ has a local minima at $x=-1$. Then $f(3)$ is equal to ____.

Q23.If $\int \frac{\sin x}{{\sin }^{3}x+{\cos }^{3}x}dx=\alpha {\log }_e|1+\tan x|+\beta {\log }_e\left|1-\tan x+{\tan }^{2}x\right|+\gamma {\tan }^{-1}\left(\frac{2\tan x-1}{\sqrt{3}}\right)+C$, when $C$ is constant of integration, then the value of …

Q24.If the line $y=mx$ bisects the area enclosed by the lines $x=0,y=0,x=\frac{3}{2}$ and the curve $y=1+4x-x^2$, then $12m$ is equal to ____.

Q25.A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^2=8x$. If the line $L$ is also tangent to the circle $x^2+y^2=a$, then ' $a$ ' is equal to ____.

Q26.Let $B$ be the centre of the circle $x^2+y^2-2x+4y+1=0$. Let the tangents at two points $P$ and $Q$ on the circle intersect at the point $A(3,1)$. Then $8\cdot \left(\frac{\text{ area }△APQ}{\text{ area }△BPQ}\right)$ is equal to ____.

Q27.The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____.

Q28.If the coefficient of $a^7{b}^8$ in the expansion of $(a+2b+4ab{)}^{10}$ is $K\cdot 2^{16}$, then $K$ is equal to ____.

Q29.If $\mathrm{S}=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\dots .$. , then 160 S is equal to ____.

Q30.The number of elements in the set
$\left\{\mathrm{A}=\left(\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ 0& \mathrm{d}\end{array}\right):\mathrm{a},\mathrm{b},\mathrm{d}\in \{-1,0,1\}\mathrm{and}(1-\mathrm{A}{)}^3=\mathrm{I}-{\mathrm{A}}^3\right\}$, where I is …