JEE Main 2021 · 22 Jul 2021 · Shift 2

Q1.A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height $h$ is ______ s.

Q2.An electron of mass $m_e$ and a proton of mass $m_p$ are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is :

Q3.What will be the projection of vector $\overrightarrow{\mathrm{A}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ on vector $\overrightarrow{\mathrm{B}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ ?

Q4.A ray of light passes from a denser medium to a rarer medium at an angle of incidence i. The reflected and refracted rays make an angle of $90^{\circ }$ with each other. The angle of reflection and refraction are respectively $r$ and $r^{\prime }$. The critical angle is given by :

Q5.A nucleus with number 184 initially at rest emits an $\alpha$-particle. If the Q value of the reaction is 5.5 MeV, calculate the kinetic energy of the $\alpha$-particle.

Q6.Intensity of sunlight is observed as $0.092{\mathrm{Wm}}^{-2}$ at a point in free space. What will be the peak value of magnetic field at that point? $\left({\epsilon }_0=8.85\times 10^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}\right)$

Q7.In a circuit consisting of a capacitance and a generator with alternating emf ${\mathrm{E}}_{\mathrm{g}}={\mathrm{E}}_{{\mathrm{g}}_0}\sin \omega \mathrm{t},{\mathrm{V}}_{\mathrm{C}}$ and ${\mathrm{I}}_{\mathrm{C}}$ are the voltage and current. Correct phase diagram for such circuit is …

Q8.Consider a situation in which reverse biased current of a particular $\mathrm{P}-\mathrm{N}$ junction increase when it is exposed to a light of wavelength $\le 621\mathrm{nm}$. During this process, enhancement in carrier concentration takes place due to generation of hole - el …

Q9.A porter lifts a heavy suitcase of mass 80 kg and at the destination lowers it down by a distance of 80 cm with a constant velocity. Calculate the work done by the porter in lowering the suitcase.
(take $\mathrm{g}=9.8{\mathrm{ms}}^{-2}$)

Q10.Match List - I with List - II :

List - I
(a) $\omega \mathrm{L}>\frac{1}{\omega \mathrm{C}}$
(b) $\omega \mathrm{L}=\frac{1}{\omega \mathrm{C}}$
(c) $\omega L<\frac{1}{\omega C}$
(d) Resonant frequency

List - II
(i) Current is in phase with emf
(ii) Current lags behind the appl …

Q11.${\mathrm{T}}_0$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, the modified time period is :

Q12.Statement I : The ferromagnetic property depends on temperature. At high temperature, ferromagnet becomes paramagnet.

Statement II : At high temperature, the domain wall area of a ferromagnetic substance increases.
In the light of the above statements, choose the most appropriat …

Q13.A bullet of ' 4 g ' mass is fired from a gun of mass 4 kg . If the bullet moves with the muzzle speed of $50{\mathrm{ms}}^{-1}$, the impulse imparted to the gun velocity of recoil of gun are :

Q14.What should be the height of transmitting antenna and the population covered if the television telecast is to cover a radius of 150 km ? The average population density around the tower is $200/{\mathrm{km}}^2$ and the value of $R_e=6.5\times 10^6\mathrm{m}$.

Q15.Choose the correct option :

Q16.The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by $x(t)=A\sin \omega t+B\cos \omega t$ with $\omega =\sqrt{\frac{K}{m}}$. Suppose that at time $t=0$, the position of mass is $x(0)$ and velocity $v(0)$, then its dis …

Q17.What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature T ?

Q18.Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is :

Q19.A Copper( Cu ) rod of length 25 cm and cross sectional area $3{\mathrm{mm}}^2$ is joined with a similar aluminium(AI)rod as shown in figure. Find the resistance of the combination between the ends $A$ and B .
(Take Resistivity of Copper $=1.7\times 10^{-8}\Omega \mathrm{m}$ …

Q20.An electric dipole is placed on $x$ - axis in proximity to a line charge of linear density $3.0\times 10^{-6}\mathrm{C}/\mathrm{m}$. Line charge is placed on z -axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. I …

Q21.The position of the centre of mass of a uniform semi - circular wire of radius ' R ' placed in $x-y$ plane with its centre at the origin and the line joining its ends as $x$-axis is given by $\left(0,\frac{xR}{\pi }\right)$. Then, the value of $|x|$ is ______.

Q22.The total charge enclosed in an incremental volume of $2\times 10^{-9}{\mathrm{m}}^3$ located at the origin is ______ nC, if electric flux density of its field is found as …

Q23.A ray of light passing through a prism $(\mu =\sqrt{3})$ surface minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ______ (in degrees).

Q24.The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt{x}{v}_0$. Then the value of $x$ is ______.

Q25.Three students $S_1,S_2$ and $S_3$ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

| …

Q26.In 5 minutes, a body cools from $75^{\circ }\mathrm{C}$ to $65^{\circ }\mathrm{C}$ at room temperature of $25^{\circ }\mathrm{C}$. The temperature of body at the end of next 5 minutes is ______ ${}^{\circ }\mathrm{C}$.

Q27.The area of cross - section of a railway track is $0.01{\mathrm{m}}^2$. The temperature variation is $10^{\circ }\mathrm{C}$, Coefficient of liner expansion of material of track is $10^{-5}/{}^{\circ }\mathrm{C}$. The energy stored per meter in the track is ______ …

Q28.In an electric circuit, a cell of certain emf provides a potential difference of 1.25 V across a load resistance of $5\Omega$. However, it provides a potential difference of 1 V across a load resistance of $2\Omega$. The emf of the cell is given by $\frac{x}{10}\mathrm{V}$. T …

Q29.In a given circuit diagram, a 5 V zener diode along with a series resistance is connected across a 50 V power supply. The minimum value of the resistance required, if the maximum zener current is 90 mA be ______ $\Omega$.

Q30.Three particles $P,Q$ and $R$ are moving along the vectors $\vec{A}=\hat{i}+\hat{j},\vec{B}=\hat{j}+\hat{k}$ and $\overrightarrow{\mathrm{C}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}$ respectively. They strike on point and start to move in different directions. Now particle P …

Q1.Which one of the following compounds will provide a tertiary alcohol on reaction with excess of ${\mathrm{CH}}_3\mathrm{MgBr}$ followed by hydrolysis?

Q2.When silver nitrate solution is added to potassium iodide solution then the sol produced is:

Q3.Isotope(s) of hydrogen which emits low energy ${\beta }^{-}$ particles with ${\mathrm{t}}_{1/2}$ value $>12$ years is/are:

Q4.Which one of the following 0.06 M aqueous solutions has lowest freezing point?

Q5.Match List - I with List - II:

List - I (Species)
(a) ${\mathrm{SF}}_4$
(b) ${\mathrm{IF}}_5$
(c) ${\mathrm{NO}}_2^{+}$
(d) ${\mathrm{NH}}_4^{+}$

List - II (Hybrid Orbitals)
(i) ${\mathrm{sp}}^3{\mathrm{d}}^2$
(ii) ${\mathrm{d}}^2{\mathrm{sp}}^3$
(iii) …

Q6.Which one of the following compounds does not exhibit resonance?

Q7.The set having ions which are coloured and paramagnetic both is:

Q8.An organic compound $\mathrm{A}({\mathrm{C}}_6{\mathrm{H}}_6\mathrm{O})$ gives dark green colouration with ferric chloride. On treatment with ${\mathrm{CHCl}}_3$ and KOH, followed by acidification gives compound B. Compound B can also be obtained from compound C on reaction with …

Q9.Given below are the statements about diborane.
(a) Diborane is prepared by the oxidation of ${\mathrm{NaBH}}_4$ with ${\mathrm{I}}_2$.
(b) Each boron atom is in ${\mathrm{sp}}^2$ hybridized state.
(c) Diborane has one bridged 3 centre-2-electron bond.
(d) Diborane is a planar mol …

Q10.Match List - I with List - II:

List - I
(a) Chloroprene
(b) Neoprene
(c) Acrylonitrile
(d) Isoprene

List - II
(i) (see structure)
(ii) (see structure)
(iii) (see structure)
(iv) ${\mathrm{CH}}_2=\mathrm{CH}-\mathrm{CN}$

Choose the correct answer from the options given below:

Q11.Which one of the following molecules does not show stereo isomerism?

Q12.Sulphide ion is soft base and its ores are common for metals.
(a) Pb
(b) Al
(c) Ag
(d) Mg

Choose the correct answer from the options given below:

Q13.In the chemical reactions given above A and B respectively are:

Q14.Match List - I with List - II :

List - I (Elements)
(a) Ba
(b) Ca
(c) Li
(d) Na

List - II (Properties)
(i) Organic solvent soluble compounds
(ii) Outer electronic configuration $6{\mathrm{s}}^2$
(iii) Oxalate insoluble in water
(iv) Formation of very strong monoacidic base

C …

Q15.Which one of the following group - 15 hydride is the strongest reducing agent?

Q16.Thiamine and pyridoxine are also known respectively as:

Q17.Which one of the following reactions does not occur?

Q18.Which one of the following statements for D.I. Mendeleeff, is incorrect?

Q19.The water having more dissolved ${\mathrm{O}}_2$ is:

Q20.Which purification technique is used for high boiling organic liquid compound (decomposes near its boiling point)?

Q21.The total number of unpaired electrons present in $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]{\mathrm{Cl}}_2$ and $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]{\mathrm{Cl}}_3$ is ___.

Q22.${\mathrm{N}}_2{\mathrm{O}}_{5(\mathrm{g})}\to 2{\mathrm{NO}}_{2(\mathrm{g})}+\frac{1}{2}{\mathrm{O}}_{2(\mathrm{g})}$

In the above first order reaction the initial concentration of ${\mathrm{N}}_2{\mathrm{O}}_5$ is $2.40\times 10^{-2}\mathrm{mol}{\mathrm{L}}^{-1}$ a …

Q23.Number of electrons that Vanadium ($\mathrm{Z}=23$) has in p-orbitals is equal to _____

Q24.If the standard molar enthalpy change for combustion of graphite powder is $-2.48\times 10^2\mathrm{kJ}{\mathrm{mol}}^{-1}$, the amount of heat generated on combustion of 1 g of graphite powder is _____ kJ. (Nearest integer)

Q25.Value of $K_p$ for the equilibrium reaction $\mathrm{N}{}_2\mathrm{O}{}_4(\mathrm{g})\rightleftharpoons 2\mathrm{NO}{}_2(\mathrm{g})$ at 288 K is 47.9. The $K_c$ for this reaction at same temperature is ______. (Nearest integer)
($R=0.083{\text{ L bar K}}^{-1}{\text{ mol}}^{-1}$)

Q26.A copper complex crystallizing in a CCP lattice with a cell edge of 0.4518 nm has been revealed by employing X-ray diffraction studies. The density of a copper complex is found to be $7.26{\text{ g cm}}^{-3}$. The molar mass of copper complex is ______ ${\text{g mol}}^{-1}$.
(Neares …

Q27.The number of acyclic structural isomers (including geometrical isomers) for pentene is ______.

Q28.Assume a cell with the following reaction


$E_{\text{cell}}$ for the above reaction is ______ V. (Nearest integer)
[Given: $\log 2.5=0.3979$, $T=298\text{ K}$]

Q29.Methylation of 10 g of benzene give 9.2 g of toluene. Calculate the percentage yield of toluene ______. (Nearest integer)

Q30.If the concentration of glucose ($\mathrm{C}{}_6\mathrm{H}{}_{12}\mathrm{O}{}_6$) in blood is $0.72{\text{ g L}}^{-1}$, the molarity of glucose in blood is ______ $\times 10^{-3}\text{ M}$. (Nearest integer)
(Given: Atomic mass of $\mathrm{C}=12$, $\mathrm{H}=1$, $\mathrm{O}=16$ u)

Q1.Let the circle $S:36x^2+36y^2-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle S, then :

Q2.If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda )=(z-3),\lambda \in R$ is $\frac{1}{\sqrt{38}}$, then the integral value of $\lambda$ is equal to:

Q3.Let $n$ denote the number of solutions of the equation $z^2+3\bar{z}=0$, where $z$ is a complex number. Then the value of ${\sum }_{\mathrm{k}=0}^{\infty }\frac{1}{{\mathrm{n}}^{\mathrm{k}}}$ is equal to :

Q4.If ${\int }_0^{100\pi }{e}^{-\left(\frac{x}{\pi }-\left[\frac{x}{\pi }\right]\right)}{\sin }^{2}xdx=\frac{\alpha {\pi }^3}{1+4{\pi }^2},\alpha \in R$, where $[x]$ is the greatest integer less than or equal to $x$, then the value of $\alpha$ is :

Q5.Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in R$ satisfying the equation ${\left[e^x\right]}^2+\left[e^x+1\right]-3=0$ lie in the interval :

Q6.Which of the following Boolean expressions is not a tautology?

Q7.Let a line $L:2x+y=k,k>0$ be a tangent to the hyperbola $x^2-y^2=3$. If $L$ is also a tangent to the parabola $y^2=\alpha x$, then $\alpha$ is equal to :

Q8.Let $y=y(x)$ be the solution of the differential equation ${\mathrm{cosec}}^{2}xdy+2dx=(1+y\cos 2x){\mathrm{cosec}}^{2}xdx$, with $y\left(\frac{\pi }{4}\right)=0$. Then, the value of $(y(0)+1{)}^2$ is equal to :

Q9.Let $f:R\to R$ be defined as
$f(x)=\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x& ,x>0\\ 3x{e}^x& ,x\le 0\end{array}$
Then $f$ is increasing function in the interval.

Q10.Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\vec{d}=3\hat{i}+2\hat{j}+6\hat{k}$ and $|\vec{a}|=\sqrt{10}$. Then a possible value of …

Q11.Let $S_n$ denote the sum of the first $n$-terms of an arithmetic progression. If $S_{10}=530,S_5=140$, then $S_{20}-S_6$ is equal to :

Q12.Let $f:R\to R$ be defined as $f(x)=\{\begin{array}{cl}\frac{x^3}{(1-\cos 2x{)}^2}{\log }_e\left(\frac{1+2x{e}^{-2x}}{{\left(1-x{e}^{-x}\right)}^2}\right)& ,x\ne 0\\ \alpha & ,x=0\end{array}$. If $f$ is continuous at $x=0$, then $\alpha$ i …

Q13.The number of solutions of ${\sin }^{7}x+{\cos }^{7}x=1$, $x\in [0,4\pi ]$ is equal to:

Q14.Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times \vec{b}=\vec{c}$, $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the following is not true?

Q15.Let $L$ be the line of intersection of planes $\vec{r}\cdot (\hat{i}-\hat{j}+2\hat{k})=2$ and $\vec{r}\cdot (2\hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular on $L$ from the point $(1,2,0)$, then the value of …

Q16.Let $A=|a_{ij}|$ be a real matrix of order $3\times 3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3$. Then, the sum of all the entries of the matrix $A^3$ is equal to:

Q17.If the domain of the function $f(x)=\frac{{\cos }^{-1}\sqrt{x^2-x+1}}{\sqrt{{\sin }^{-1}\left(\frac{2x-1}{2}\right)}}$ is the interval $(\alpha ,\beta ]$, then $\alpha +\beta$ is equal to:

Q18.The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are:

Q19.Let $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1$. If $E_1$ and $E_2$ have same eccentricities, then its value is:

Q20.Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times 2$ matrices. The probability that such formed matrices have all different entries and are non-singular, is:

Q21.Let $A=\{0,1,2,3,4,5,6,7\}$. Then the number of bijective functions $f:A\to A$ such that $f(1)+f(2)=3-f(3)$ is equal to ____

Q22.Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $f(x)=\{\begin{array}{ll}3\left(1-\frac{|x|}{2}\right)& \text{if }|x|\le 2\\ 0& \text{if }|x|>2\end{array}$

Let $g:\mathbb{R}\to \mathbb{R}$ be given by $g(x)=f(x+2)-f(x-2)$. If $n$ and …

Q23.Let $y=y(x)$ be the solution of the differential equation

$\left((x+2)e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right)dx=(x+2)dy$, $y(1)=1$.

If the domain of $y=y(x)$ is an open interval $(\alpha ,\beta )$, then $|\alpha +\beta |$ is equal to ____

Q24.The number of elements in the set $\{n\in \{1,2,3,\dots ,100\}\mid (11{)}^n>(10{)}^n+(9{)}^n\}$ is ____

Q25.The area (in sq. units) of the region bounded by the curves $x^2+2y-1=0,y^2+4x-4=0$ and $y^2-4x-4=0$, in the upper half plane is ____

Q26.Let $\mathrm{A}=\left(\begin{array}{lll}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$. Then the number of $3\times 3$ matrices B with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is ____

Q27.If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8$, then the number of all numbers greater than 10,000 is equal to ____

Q28.The sum of all the elements in the set $\{n\in \{1,2,\dots ,100\}\mid$ H.C.F. of $n$ and 2040 is 1$\}$ is equal to ____

Q29.If the constant term, in binomial expansion of ${\left(2x^r+\frac{1}{x^2}\right)}^{10}$ is 180, then $r$ is equal to ____

Q30.Consider the following frequency distribution :

\begin{tabular}{lccccc}
Class : & $0-6$ & $6-12$ & $12-18$ & $18-24$ & $24-30$ \\
Frequency : & a & b & 12 & 9 & 5
\end{tabular}

If mean $=\frac{309}{22}$ and median $=14$, then the value $(a-b{)}^2$ is equal to