JEE Main 2021 · 25 Jul 2021 · Shift 2

Q1.A ray of light entering from air into a denser medium of refractive index $\frac{4}{3}$, as shown in figure. The light ray suffers total internal reflection at the adjacent surface as shown. The maximum value of angle $\theta$ should be equal to :

Q2.In the given potentiometer circuit arrangement, the balancing length AC is measured to be 250 cm. When the galvanometer connection is shifted from point (1) to point (2) in the given diagram, the balancing length becomes 400 cm. The ratio of the emf of two cells, …

Q3.A balloon was moving upwards with a uniform velocity of $10m/s$. An object of finite mass is dropped from the balloon when it was at a height of 75 m from the ground level. The height of the balloon from the ground when object strikes the ground was around :
(takes the …

Q4.An electron moving with speed v and a photon moving with speed c, have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is :

Q5.Two vectors $\vec{X}$ and $\vec{Y}$ have equal magnitude. The magnitude of $(\vec{X}-\vec{Y})$ is $n$ times the magnitude of $(\vec{X}+\vec{Y})$. The angle between $\vec{X}$ and $\vec{Y}$ is :

Q6.Two ideal electric dipoles A and B, having their dipole moment $p_1$ and $p_2$ respectively are placed on a plane with their centres at O as shown in the figure. At point C on the axis of dipole $A$, the resultant electric field is making an angle of $37^{\circ }$ with the axi …

Q7.A $10\Omega$ resistance is connected across $220\mathrm{V}-50\mathrm{Hz}$ AC supply. The time taken by the current to change from its maximum value to the rms value is :

Q8.The relation between time $t$ and distance $x$ for a moving body is given as $t=m{x}^2+nx$, where $m$ and $n$ are constants. The retardation of the motion is : (Where $v$ stands for velocity)

Q9.A prism of refractive index $\mu$ and angle of prism $A$ is placed in the position of minimum angle of deviation. If minimum angle of deviation is also $A$, then in terms of refractive index value of $A$ is :

Q10.Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is W, the weight of the same object on that planet will be :

Q11.A force $\vec{F}=(40\hat{i}+10\hat{j})\mathrm{N}$ acts on a body of mass 5 kg. If the body starts from rest, its position vector $\overrightarrow{\mathrm{r}}$ at time $\mathrm{t}=10\mathrm{s}$, will be :

Q12.Two ions having same mass have charges in the ratio $1:2$. They are projected normally in a uniform magnetic field with their speeds in the ratio $2:3$. The ratio of the radii of their circular trajectories is :

Q13.If $q_f$ is the free charge on the capacitor plates and $q_b$ is the bound charge on the dielectric slab of dielectric constant $k$ placed between the capacitor plates, then bound charge $q_b$ can be expressed as :

Q14.Two spherical soap bubbles of radii $r_1$ and $r_2$ in vacuum combine under isothermal conditions. The resulting bubble has radius equal to :

Q15.A heat engine has an efficiency of $\frac{1}{6}$. When the temperature of sink is reduced by $62^{\circ }\mathrm{C}$, its efficiency get doubled. The temperature of the source is :

Q16.The instantaneous velocity of a particle moving in a straight line is given as $\mathrm{v}=\alpha \mathrm{t}+\beta {\mathrm{t}}^2$, where $\alpha$ and $\beta$ are constants. The distance travelled by the particle between 1 s and 2 s is :

Q17.The force is given in terms of time $t$ and displacement $x$ by the equation
$\mathrm{F}=\mathrm{A}\cos \mathrm{Bx}+\mathrm{C}\sin \mathrm{Dt}$
The dimensional formula of $\frac{AD}{B}$ is :

Q18.When radiation of wavelength $\lambda$ is incident on a metallic surface, the stopping potential of ejected photoelectrons is 4.8 V. If the same surface is illuminated by radiation of double the previous wavelength, then the stopping potential becomes 1.6 V. The threshold wavelen …

Q19.The given potentiometer has its wire of resistance $10\Omega$. When the sliding contact is in the middle of the potentiometer wire, the potential drop across $2\Omega$ resistor is:

Q20.In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

Q21.From the given data, the amount of energy required to break the nucleus of aluminium ${}_{13}^{27}\mathrm{Al}$ is $x\times 10^{-3}$ J.
Mass of neutron $=1.00866$ u
Mass of proton $=1.00726$ u
Mass of Aluminium nucleus $=27.18846$ u
(Assume 1 u corresponds to xJ of energy)
(Round o …

Q22.A force of $F=(5y+20)\hat{j}$ N acts on a particle. The work done by this force when the particle is moved from $y=0$ m to $y=10$ m is _____ J.

Q23.A solid disc of radius 20 cm and mass 10 kg is rotating with an angular velocity of 600 rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10s is _____ $\pi \times 10^{-1}$ Nm.

Q24.A $16\Omega$ wire is bend to form a square loop. A 9 V supply having internal resistance of $1\Omega$ is connected across one of its sides. The potential drop across the diagonals of the square loop is _____ $\times 10^{-1}$ V.

Q25.Two circuits are shown in the figure (a) & (b). At a frequency of _____ rad/s the average power dissipated in one cycle will be same in both the circuits.

Q26.A light beam of wavelength 500 nm is incident on a metal having work function of 1.25 eV, placed in a magnetic field of intensity $B$. The electrons emitted perpendicular to the magnetic field $B$, with maximum kinetic energy are bent into circular arc of radius 30 cm. The value …

Q27.A message single of frequency 20 kHz and peak voltage of 20 volt is used to modulate a carrier wave of frequency 1 MHz and peak voltage of 20 volt. The modulation index will be _____.

Q28.A system consists of two types of gas molecules $A$ and $B$ having same number density $2\times 10^{25}/{\mathrm{m}}^3$. The diameter of A and B are $10\text{ A˚}$ and $5\text{ A˚}$ respectively. They suffer collision at room temperature. The ratio of average distance covered by t …

Q29.In a semiconductor, the number density of intrinsic charge carriers at $27^{\circ }\mathrm{C}$ is $1.5\times 10^{16}/{\mathrm{m}}^3$. If the semiconductor is doped with impurity atom, the hole density increases to $4.5\times 10^{22}/{\mathrm{m}}^3$. The electron density in the d …

Q30.The nuclear activity of a radioactive element becomes ${\left(\frac{1}{8}\right)}^{\text{th }}$ of its initial value in 30 years. The half-life of radioactive element is ____ years.

Q1.Which one of the following metals forms interstitial hydride easily?

Q2.Which one of the following is correct structure for cytosine?

Q3.Identify the species having one $\pi$-bond and maximum number of canonical forms from the following:

Q4.A biodegradable polyamide can be made from:

Q5.Match List - I with List - II:

List - I
(a) Concentration of Ag ore
(b) Blast furnace
(c) Blister copper
(d) Froth floatation method

List - II
(i) Reverberatory furnace
(ii) Pig iron
(iii) Leaching with dilute NaCN solution
(iv) Sulfide ores

Choose the correct answer from the …

Q6.Consider the above reaction, the product "P" is:

Q7.The correct decreasing order of densities of the following compounds is:
(A) (see structure)
(B) (see structure)
(C) (see structure)
(D) (see structure)

Q8.What is the major product "P" of the following reaction?
(i) ${\mathrm{NaNO}}_2,\mathrm{HCl},278\mathrm{K}$
(ii) ${\mathrm{H}}_2\mathrm{O}$ (major product)
P

Q9.Identify the process in which change in the oxidation state is five:

Q10.A reaction of benzonitrile with one equivalent ${\mathrm{CH}}_3\mathrm{MgBr}$ followed by hydrolysis produces a yellow liquid "P". The compound "P" will give positive ______.

Q11.The number of significant figure in 0.00340 is ____.

Q12.An LPG cylinder contains gas at a pressure of 300 kPa at $27^{\circ }\mathrm{C}$. The cylinder can withstand the pressure of $1.2\times 10^6\mathrm{Pa}$. The room in which the cylinder is kept catches fire. The minimum temperature at which the bursting of cylinder will t …

Q13.An accelerated electron has speed of $5\times 10^6{\mathrm{ms}}^{-1}$ with an uncertainty of $0.02\%$. The uncertainty in finding its location while in motion is $x\times 10^{-9}\mathrm{m}$. The value of $x$ is ____.
(Nearest integer)
[Use mass of electron …

Q14.Maleic anhydride can be prepared by: [structure of maleic anhydride shown in source]

Q15.The ionic radii of F⁻ and O²⁻ respectively are 1.33 Å and 1.4 Å, while the covalent radius of N is 0.74 Å. The correct statement for the ionic radius of N³⁻ from the following is:

Q16.The spin only magnetic moments (in BM) for free Ti³⁺, V²⁺ and Sc³⁺ ions respectively are (At. No. Sc: 21; Ti: 22; V: 23)

Q17.Which one of the following metal complexes is most stable? [coordination-complex formulas shown as options in source]

Q18.Match List - I with List - II:
List - I (Elements): (a) Li (b) Na (c) K (d) Cs
List - II (Properties): (i) Poor water solubility of I⁻ salt (ii) Most abundant element in cell fluid (iii) Bicarbonate salt used in fire extinguisher (iv) Carbonate salt decomposes easily on h …

Q19.In the following the correct bond order sequence is:

Q20.Match List - I with List - II:
List - I (Example of Colloids): (a) Cheese (b) Pumice stone (c) Hair cream (d) Cloud
List - II (Classification): (i) dispersion of liquid in liquid (ii) dispersion of liquid in gas (iii) dispersion of gas in solid (iv) dispersion of liquid i …

Q21.Br-C₆H₄-CHO reacts with EtOH (excess)/dry HCl gas to give major product 'A', which on treatment with t-BuO⁻K⁺ gives major product 'B'. Consider the above reaction sequence; products 'A' and 'B' formed respectively are: [structural options shown in source]

Q22.For the reaction of (E)-but-2-ene type alkene [H₃C-CH=CH-CH₃, structure shown] with Br₂ in CCl₄ giving product 'P', the total number of stereoisomers possible for product 'P' is ______.

Q23.Assuming that Ba(OH)₂ is completely ionized in aqueous solution under the given conditions, the concentration of H₃O⁺ ions in 0.005 M aqueous solution of Ba(OH)₂ at 298 K is ______ × 10⁻¹² mol L⁻¹. (nearest integer)

Q24.For a chemical reaction A → B, it was found that concentration of B is increased by 0.2 mol L⁻¹ in 30 min. The average rate of the reaction is ______ × 10⁻¹ mol L⁻¹ h⁻¹. (in nearest integer)

Q25.When 3.00 g of substance 'X' is dissolved in 100 g of CCl₄, it raises the boiling point by 0.60 K. The molar mass of the substance 'X' is ______ g mol⁻¹. (nearest integer) [Given K_b for CCl₄ is 5.0 K kg mol⁻¹]

Q26.A system does 200 J of work and at the same time absorbs 150 J of heat. The magnitude of the change in internal energy is ______ J. (Nearest integer)

Q27.0.8 g of an organic compound was analysed by Kjeldahl's method for the estimation of nitrogen. If the percentage of nitrogen in the compound was found to be 42%, then ______ ml of 1 M H₂SO₄ would have been neutralized by the ammonia evolved during the analysis.

Q1.If $[x]$ be the greatest integer less than or equal to $x$, then ${\sum }_{n=8}^{100}\left[\frac{(-1{)}^{n}n}{2}\right]$ is equal to :

Q2.The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is:

Q3.Let $a,b$ and $c$ be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are coplanar, then c is equal to :

Q4.If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $BC$ as diameter passes through the point :

Q5.If $\mathrm{P}=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$, then ${\mathrm{P}}^{50}$ is :

Q6.The sum of all those terms which are rational numbers in the expansion of ${\left(2^{\frac{1}{3}}+3^{\frac{1}{4}}\right)}^{12}$ is :

Q7.Let the equation of the pair of lines, $y=px$ and $y=qx$, can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of the angle bisectors of the lines $x^2-4xy-5y^2=0$ is :

Q8.The number of real solutions of the equation, $x^2-|x|-12=0$ is :

Q9.The lowest integer which is greater than ${\left(1+\frac{1}{10^{100}}\right)}^{10^{100}}$ is .........

Q10.If $|\vec{a}|=2,|\vec{b}|=5$ and $|\vec{a}\times \vec{b}|=8$, then $|\vec{a}\cdot \vec{b}|$ is equal to :

Q11.The value of $\cot \frac{\pi }{24}$ is :

Q12.Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2},P(X=j)=\frac{1}{3^j}(j=1,2,3,\dots ,\infty )$. Then the mean of the distribution and $P(X$ is positive and even) respectively are :

Q13.Consider functions $f:A\to B$ and $g:B\to C(A,B,C\subseteq R)$ such that $(g\circ f{)}^{-1}$ exists, then :

Q14.The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$, then the standard deviation of the second sample is :

Q15.Let $y=y(x)$ be the solution of the differential equation $xdy=\left(y+x^3\cos x\right)dx$ with $y(\pi )=0$, then $y\left(\frac{\pi }{2}\right)$ is equal to :

Q16.The value of the integral ${\int }_{-1}^1\log \left(x+\sqrt{x^2+1}\right)dx$ is :

Q17.If the greatest value of the term independent of ' $x$ ' in the expansion of ${\left(x\sin \alpha +a\frac{\cos \alpha }{x}\right)}^{10}$ is $\frac{10!}{(5!{)}^2}$, then the value of ' $a$ ' is equal to :

Q18.If $\mathrm{f}(\mathrm{x})=\{\begin{array}{cc}{\int }_0^{\mathrm{x}}(5+|1-\mathrm{t}|)\mathrm{dt},& x>2\\ 5\mathrm{x}+1,& x\le 2\end{array}$, then

Q19.If ${}^n{P}_r={}^n{P}_{r+1}$ and ${}^n{C}_r={}^n{C}_{r-1}$, then the value of $r$ is equal to:

Q20.Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

Q21.Consider the function $f(x)=\frac{P(x)}{\sin (x-2)},x\ne 2$

where $P(x)$ is a polynomial such that $P^{\prime \prime }(x)$ is always a constant and $P(3)=9$. If $f(x)$ is continuous at $x=2$, then $P(5)$ is equal to...... .

Q22.If $a+b+c=1$, $ab+bc+ca=2$ and $abc=3$, then the value of $a^4+b^4+c^4$ is equal to $\dots$...

Q23.If the co-efficients of $x^7$ and $x^8$ in the expansion of ${\left(2+\frac{x}{3}\right)}^n$ are equal, then the value of $n$ is equal to ____

Q24.If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is ____

Q25.If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is ____

Q26.If $(\vec{a}+3\vec{b})$ is perpendicular to $(7\vec{a}-5\vec{b})$ and $(\vec{a}-4\vec{b})$ is perpendicular to $(7\vec{a}-2\vec{b})$, then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is ____

Q27.Let $n\in N$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of ( $n+1$ ) terms ${}^{\mathrm{n}}{\mathrm{C}}_0,3.{}^{\mathrm{n}}{\mathrm{C}}_1,5.{}^{\mathrm{n}}{\mathrm{C}}_2,7.{}^{\mathrm{n}}{\mathrm{C}}_3,\dots .$. is equal …

Q28.A fair coin is tossed n -times such that the probability of getting at least one head is at least 0.9 . Then the minimum value of $n$ is ____

Q29.Let a curve $y=f(x)$ pass through the point $\left(2,{\left({\log }_{e}2\right)}^2\right)$ and have slope $\frac{2y}{x{\log }_{e}x}$ for all positive real value of $x$. Then the value of $f(e)$ is equal to ____

Q30.The equation of a circle is $\mathrm{Re}\left(z^2\right)+2(\mathrm{Im}(z){)}^2+2\mathrm{Re}(z)=0$, where $z=x+iy$. A line which passes through the center of the given circle and the vertex of the parabola, $x^2-6x-y+13=0$, has $y$ intercept equal …