JEE Main 2021 · 27 Aug 2021 · Shift 2

Q1.A player kicks a football with an initial speed of $25{\mathrm{ms}}^{-1}$ at an angle of $45^{\circ }$ from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion? (Take $\mathrm{g}=10{\mathrm{ms}}^{-2}$)

Q2.A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the centre is $\mathrm{V}\mathrm{kg}/\mathrm{m}$. The value of V is:

Q3.An antenna is mounted on a 400 m tall building. What will be the wavelength of signal that can be radiated effectively by the transmission tower upto a range of 44 km?

Q4.The colour coding on a carbon resistor is shown in the given figure. The resistance value of the given resistor is :

Q5.A coaxial cable consists of an inner wire of radius '$a$' surrounded by an outer shell of inner and outer radii '$b$' and '$c$' respectively. The inner wire carries an electric current ${\mathrm{i}}_0$, which is distributed uniformly across cross-sectional area. The outer shell c …

Q6.The boxes of masses 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (use $\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2$)

Q7.For full scale deflection of total 50 divisions, 50 mV voltage is required in galvanometer. The resistance of galvanometer if its current sensitivity is 2 div/mA will be :

Q8.For a transistor $\alpha$ and $\beta$ are given as $\alpha =\frac{{\mathrm{I}}_{\mathrm{C}}}{{\mathrm{I}}_{\mathrm{E}}}$ and $\beta =\frac{{\mathrm{I}}_{\mathrm{C}}}{{\mathrm{I}}_{\mathrm{B}}}$. Then the correct relation between $\alpha$ and $\beta$ will be:

Q9.A monochromatic neon lamp with wavelength of 670.5 nm illuminates a photo-sensitive material which has a stopping voltage of 0.48 V. What will be the stopping voltage if the source light is changed with another source of wavelength of 474.6 nm ?

Q10.Figure shows a rod AB, which is bent in a $120^{\circ }$ circular arc of radius $R$. A charge ($-Q$) is uniformly distributed over rod $AB$. What is the electric field $\vec{E}$ at the centre of curvature O ?

Q11.A constant magnetic field of 1 T is applied in the $\mathrm{x}>0$ region. A metallic circular ring of radius 1 m is moving with a constant velocity of $1\mathrm{m}/\mathrm{s}$ along the $x$-axis. At $t=0$, the centre of $O$ of the ring is at $x=-1\mathrm{m}$. What will be t …

Q12.The height of victoria falls is 63 m. What is the difference in temperature of water at the top and at the bottom of fall ?
[Given $1\mathrm{cal}=4.2\mathrm{J}$ and specific heat of water $=1{\mathrm{calg}}^{-1}{}^{\circ }{\mathrm{C}}^{-1}$]

Q13.Two discs have moments of intertia $I_1$ and $I_2$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, ${\omega }_1$ and ${\omega }_2$ respectively and are brought into contact face to face with their axe …

Q14.Curved surfaces of a plano-convex lens of refractive index ${\mu }_1$ and a planoconcave lens of refractive index ${\mu }_2$ have equal radius of curvature as shown in figure. Find the ratio of radius of curvature to the focal length of the combined lenses.

Q15.The light waves from two coherent sources have same intensity $I_1=I_2=I_0$. In interference pattern the intensity of light at minima is zero. What will be the intensity of light at maxima?

Q16.If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density :

Q17.If the rms speed of oxygen molecules at $0^{\circ }\mathrm{C}$ is $160\mathrm{m}/\mathrm{s}$, find the rms speed of hydrogen molecules at $0^{\circ }\mathrm{C}$.

Q18.Water drops are falling from a nozzle of a shower onto the floor, from a height of 9.8 m . The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop from the floor when t …

Q19.The ratio of the equivalent resistance of the network (shown in figure) between the points a and b when switch is open and switch is closed is $x:8$. The value of $x$ is ____.

Q20.A heat engine operates between a cold reservoir at temperature $T_2=400\mathrm{K}$ and a hot reservoir at temperature ${\mathrm{T}}_1$. It takes 300 J of heat from the hot reservoir and delivers 240 J of heat to the cold reservoir in a cycle. The minimum temperature of the ho …

Q21.A plane electromagnetic wave with frequency of 30 MHz travels in free space. At particular point in space and time, electric field is $6\mathrm{V}/\mathrm{m}$. The magnetic field at this point will be $x\times 10^{-8}\mathrm{T}$. The value of $x$ is ____.

Q22.A tuning fork is vibrating at 250 Hz . The length of the shortest closed organ pipe that will resonate with the tuning fork will be ____ cm .
(Take speed of sound in air as $340{\mathrm{ms}}^{-1}$)

Q23.$X$ different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are excited to states with principal quantum number $\mathrm{n}=6$ ? The value of $X$ is ____.

Q24.A zener diode of power rating 2 W is to be used as a voltage regulator. If the zener diode has a breakdown of 10 V and it has to regulate voltage fluctuated between 6 V and 14 V , the value of $R_s$ for safe operation should be ____ $\Omega$.

Q25.Two simple harmonic motion, are represented by the equations $y_1=10\sin \left(3\pi t+\frac{\pi }{3}\right)$ ${\mathrm{y}}_2=5(\sin 3\pi \mathrm{t}+\sqrt{3}\cos 3\pi \mathrm{t})$
Ratio of amplitude of $y_1$ to $y_2=x:1$. The value of $x$ is ____

Q26.A bullet of 10 g , moving with velocity v , collides head-on with the stationary bob of a pendulum and recoils with velocity $100\mathrm{m}/\mathrm{s}$. The length of the pendulum is 0.5 m and mass of the bob is 1 kg . The minimum value of $\mathrm{v}=$ ____ …

Q27.An ac circuit has an inductor and a resistor of resistance $R$ in series, such that $X_L=3R$. Now, a capacitor is added in series such that $X_C=2R$. The ratio of new power factor with the old power factor of the circuit is $\sqrt{5}:x$ The value of $x$ is ____.

Q28.Wires ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ are made of same material having the breaking stress of $1.25\times 10^9\mathrm{N}/{\mathrm{m}}^2$. ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ have cross-sectional area of $8\times 10^{-7}{\mathrm{m}}^2$ and …

Q1.The oxide that gives ${\mathrm{H}}_2{\mathrm{O}}_2$ most readily on treatment with ${\mathrm{H}}_2\mathrm{O}$ is :

Q2.The correct structures of A and B formed in the following reactions are:

Q3.Potassium permanganate on heating at 513 K gives a product which is :

Q4.In stratosphere most of the ozone formation is assisted by :

Q5.Lyophilic sols are more stable than lyophobic sols because :

Q6.Choose the correct statement from the following :

Q7.The compound/s which will show significant intermolecular H-bonding is/are :

(a)

(b)

(c)

Q8.Match list I with list II

List I (Name of ore / mineral) | List II (Chemical formula)
(a) Calamine | (i) ZnS
(b) Malachite | (ii) ${\mathrm{FeCO}}_3$
(c) Siderite | (iii) ${\mathrm{ZnCO}}_3$
(d) Sphalerite | (iv) ${\mathrm{CuCO}}_3\cdot \mathrm{Cu}(\mathrm{OH}{)}_2$

Choose the …

Q9.Which one of the following is formed (mainly) when red phosphorus is heated in a sealed tube at 803 K ?

Q10.The correct order of ionic radii for the ions, ${\mathrm{P}}^{3-},{\mathrm{S}}^{2-},{\mathrm{Ca}}^{2+},{\mathrm{K}}^{+},{\mathrm{Cl}}^{-}$ is:

Q11.Given below are two statements:
Statement I: Ethyl pent-4-yn-oate on reaction with ${\mathrm{CH}}_3\mathrm{MgBr}$ gives a $3^{\circ }$-alcohol.
Statement II: In this reaction one mole of ethyl pent-4-yn-oate utilizes two moles of ${\mathrm{CH}}_3\mathrm{MgBr}$.
In the light of t …

Q12.Hydrolysis of sucrose gives:

Q13.Which one of the following chemicals is responsible for the production of HCl in the stomach leading to irritation and pain?

Q14.Which one of the following is used to remove most of plutonium from spent nuclear fuel?

Q15.Which one of the following reactions will not yield propionic acid?

Q16.Which one of the following tests used for the identification of functional groups in organic compounds does not use copper reagent ?

Q17.The addition of dilute NaOH to ${\mathrm{Cr}}^{3+}$ salt solution will give:

Q18.The major product (A) formed in the reaction given below is

Q19.The number of optical isomers possible for ${\left[\mathrm{Cr}{\left({\mathrm{C}}_2{\mathrm{O}}_4\right)}_3\right]}^{3-}$ is ____.

Q20.The first order rate constant for the decomposition of ${\mathrm{CaCO}}_3$ at 700 K is $6.36\times 10^{-3}{\mathrm{s}}^{-1}$ and activation energy is $209\mathrm{kJ}{\mathrm{mol}}^{-1}$. Its rate constant (in ${\mathrm{s}}^{-1}$) at 600 K is $x\times 10^{-6}$. The value of x is …

Q21.When 5.1 g of solid ${\mathrm{NH}}_4\mathrm{HS}$ is introduced into a two litre evacuated flask at $27^{\circ }\mathrm{C}$, 20% of the solid decomposes into gaseous ammonia and hydrogen sulphide. The ${\mathrm{K}}_{\mathrm{p}}$ for the reaction at $27^{\circ }\mathrm{C}$ is x …

Q22.100 g of propane is completely reacted with 1000 g of oxygen. The mole fraction of carbon dioxide in the resulting mixture is $x\times 10^{-2}$. The value of $x$ is ____. (Nearest integer)
[Atomic weight : $\mathrm{H}=1.008;\mathrm{C}=12.00;\mathrm{O}=16.00$]

Q23.Data given for the following reaction is as follows:


| Substance | …

Q24.The number of species having non - pyramidal shape among the following is ____.
(A) ${\mathrm{SO}}_3$
(B) ${\mathrm{NO}}_3^{-}$
(C) ${\mathrm{PCl}}_3$
(D) ${\mathrm{CO}}_3^{2-}$

Q25.The resistance of a conductivity cell with cell constant $1.14{\mathrm{cm}}^{-1}$, containing 0.001 M KCl at 298 K is $1500\Omega$. The molar conductivity of 0.001 M KCl solution at 298 K in $\mathrm{S}{\mathrm{cm}}^2{\mathrm{mol}}^{-1}$ is ____. (Integer answer)

Q26.The number of photons emitted by a monochromatic (single frequency) infrared range finder of power 1 mW and wavelength of 1000 nm , in 0.1 second is $x\times 10^{13}$. The value of $x$ is ____. (Nearest integer)
( …

Q27.Two flasks I and II shown below are connected by a valve of negligible volume.

When the valve is opened, the final pressure of the system in bar is $x\times 10^{-2}$. The value of $x$ is ____. (Integer answer)
[Assume-Ideal gas; 1 bar $=10^5\mathrm{Pa}$; Molar mass of …

Q28.40 g of glucose (Molar mass $=180$ ) is mixed with 200 mL of water. The freezing point of solution is ____ K. (Nearest integer)
[Given : ${\mathrm{K}}_{\mathrm{f}}=1.86\mathrm{K}\mathrm{kg}{\mathrm{mol}}^{-1}$; Density of water $=1.00\mathrm{g}{\mathrm{cm}}^{-3}$; Freezing po …

Q29.The major product of the following reaction, if it occurs by SN2 mechanism, is : [reaction scheme + structure options — see source PDF]

Q1.The Boolean expression $(p\wedge q)\Rightarrow ((r\wedge q)\wedge p)$ is equivalent to:

Q2.Let $M$ and $m$ respectively be the maximum and minimum values of the function $f(x)={\tan }^{-1}(\sin x+\cos x)$ in $\left[0,\frac{\pi }{2}\right]$. Then the value of $\tan (M-m)$ is equal to

Q3.The angle between the straight lines, whose direction cosines are given by the equations $2\ell +2\mathrm{m}-\mathrm{n}=0$ and $\mathrm{mn}+\mathrm{nI}+\mathrm{Im}=0$, is:

Q4.The value of the integer ${\int }_0^1\frac{\sqrt{x}dx}{(1+x)(1+3x)(3+x)}$ is:

Q5.If ${\lim }_{x\to \infty }\left(\sqrt{x^2-x+1}-ax\right)=b$, then the ordered pair $(a,b)$ is :

Q6.The area of the region bounded by the parabola $(y-2{)}^2=(x-1)$, the tangent to it at the point whose ordinate is 3 and the $x$-axis is:

Q7.A differential equation representing the family of parabolas with axis parallel to $y$-axis and whose length of latus rectum is the distance of the point $(2,-3)$ form the line $3x+4y=5$, is given by :

Q8.The equation of the plane passing through the line of intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r}\cdot (2\hat{i}+3\hat{j}-\hat{k})+4=0$ and parallel to the $x-$ axis is

Q9.The set of all values of $k>-1$, for which the equation ${\left(3x^2+4x+3\right)}^2-(k+1)\left(3x^2+4x+3\right)\left(3x^2+4x+2\right)+k{\left(3x^2+4x+2\right)}^2=0$ has real roots is:

Q10.Let $A=\left(\left[\begin{array}{ccc}[x+1]& [x+2]& [x+3]\\ [x]& [x+3]& [x+3]\\ [x]& [x+2]& [x+4]\end{array}\right]\right)$ where $[t]$ denotes the greatest integer less than or equal to $t$. If $\det (A)=192$, then the set of values …

Q11.If $y(x)={\cot }^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right),x\in \left(\frac{\pi }{2},\pi \right)$, then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is:

Q12.It two tangents from a point $P$ to the parabola $y^2=16(x-3)$ are at right angles, then the locus of point $P$ is:

Q13.Let $A(a,0),B(b,2b+1)$ and $C(0,b),b\ne 0,|b|\ne 1$, be points such that the area of triangle $ABC$ is 1 sq. unit, then the sum of all possible values of $a$ is:

Q14.Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is:

Q15.Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is:

Q16.Two poles $AB$ of length a meter and $CD$ of length $a+b(b\ne a)$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $BD=x$ and $\tan \angle ACB=\frac{1}{2}$, then:

Q17.A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:

Q18.Let $Z$, be the set of all integers,
$A=\left\{(x,y)\in Z\times Z:(x-2{)}^2+y^2\le 4\right\}$
$B=\left\{(x,y)\in Z\times Z:x^2+y^2\le 4\right\}$
$C=\left\{(x,y)\in Z\times Z:(x-2{)}^2+(y-2{)}^2\le 4\right\}$
If the total number of relations from $A\cap B$ t …

Q19.If the solution curve of the differential equation $\left(2x-10y^3\right)dy+ydx=0$, passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation:

Q20.If $0<x<1$ and $y=\frac{1}{2}{x}^2+\frac{2}{3}{x}^3+\frac{3}{4}{x}^4+\dots$, then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is:

Q21.Let $S$ be the sum of all solutions (in radians) of the equation ${\sin }^4\theta +{\cos }^4\theta -\sin \theta \cos \theta =0$ in $[0,4\pi ]$. Then $\frac{8S}{\pi }$ is equal to ______.

Q22.Let $S=[1,2,3,4,5,6,9]$. Then the number of elements in the set $T=\{A\subseteq S:A\ne \varphi$ and the sum of all the elements of $A$ is not a multiple of 3$\}$ is ______.

Q23.Two circles each of the radius 5 units touch each other at the point ( 1,2 ). If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha ,\beta )$ and $C_2(\gamma ,\delta ),C_1\ne C_2$ are their centres, then $|(\alpha +\beta )(\gamma +\delta )|$ is equal to __ …

Q24.$3\times 7^{22}+2\times 10^{22}-44$ when divided by 18 leaves the remainder ______.

Q25.Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x-y+z+3=0$ and let $\mathrm{R}(3,5,\gamma )$ be a point of this plane. Then the square of the length of the line segment SR is

Q26.The probability distribution of random variable $X$ is given by:

Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda \cdot k$ then $\lambda$ is equal to

Q27.Let $z_1$ and $z_2$ be two complex numbers such that $\arg \left(z_1-z_2\right)=\frac{\pi }{4}$ and $z_1,z_2$ satisfy the equation $|z-3|=\mathrm{Re}(z)$. Then the imaginary part of $z_1+z_2$ is equal to

Q28.An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $\mu$ is the average marks of girls and …

Q29.Let $\mathrm{A}(\sec \theta ,2\tan \theta )$ and $\beta (\sec \varphi ,2\tan \varphi )$, where $\theta +\varphi =\pi /2$ be two points on the hyperbola $2x^2-y^2=2$. If $(\alpha ,\beta )$ is the point of the intersection of the normals to the hyperbola at $A$ and $B$, then …

Q30.If $\int \frac{2e^x+3e^{-x}}{4e^x+7e^{-x}}dx=\frac{1}{14}\left(ux+v{\log }_e\left(4e^x+7e^{-x}\right)\right)+C$ where $C$ is a constant of integration, then $u+v$ is equal to