JEE Main 2021 · 01 Sep 2021 · Shift 2

Q1.In the given figure, each diode has a forward bias resistance of $30\Omega$ and infinite resistance in reverse bias. The current ${\mathrm{I}}_1$ will be:

Q2.There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductor is 1:1. The magnetic field at point $P$ is

Q3.The half life period of radioactive element x is same as the mean life time of another radioactive element y. Initially they have the same number of atoms. Then:

Q4.A glass tumbler having inner depth of 17.5 cm is kept on a table. A student starts pouring water ($\mu =4/3$) into it while looking at the surface of water from the above. When he feels that the tumbler is half filled, he stops pouring water. Up to what height, the tumbler is actu …

Q5.A capacitor is connected to a 20 V battery through a resistance of $10\Omega$. It is found that the potential difference across the capacitor rises to 2 V in $1\mu \mathrm{s}$. The capacitance of the capacitor is ____ $\mu \mathrm{F}$.
Given …

Q6.Following plots show Magnetization (M) vs Magnetising field (H) and Magnetic susceptibility (χ) vs temperature (T) graph:

Which of the following combination will be represented by a diamagnetic material?

Q7.Four particles each of mass M, move along a circle of radius R under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is:

Q8.The temperature of an ideal gas in 3-dimensions is 300 K. The corresponding de-Broglie wavelength of the electron approximately at 300 K, is:
[$\mathrm{me}=$ mass of electron $=9\times 10^{-31}\mathrm{kg}$
$\mathrm{h}=$ Planck constant $=6.6\times 10^{-34}\mathrm{Js}$ …

Q9.A block of mass $m$ slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is: Given $\mathrm{m}=8\mathrm{kg},\mathrm{M}=$ 16 kg Assume all the surfaces shown in the figure to be frictionless …

Q10.Due to cold weather a 1 m water pipe of cross-sectional area $1{\mathrm{cm}}^2$ is filled with ice at $-10^{\circ }\mathrm{C}$. Resistive heating is used to melt the ice. Current of 0.5 A is passed through $4\mathrm{k}\Omega$ resistance. Assuming that all the heat produced is …

Q11.A body of mass '$m$' dropped from a height '$h$' reaches the ground with a sped of $0.8\sqrt{gh}$. The value of work done by the air-friction is:

Q12.Electric field of a plane electromagnetic wave propagating through a non-magnetic medium is given by $E=20\cos \left(2\times 10^{10}t-200x\right)V/m$. The dielectric constant of the medium is equal to: (Take ${\mu }_{\mathrm{r}}=1$)

Q13.The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ }$ and $48^{\circ }$ with the horizontal are ${\mathrm{R}}_1,{\mathrm{R}}_2$ and ${\mathrm{H}}_1,{\mathrm{H}}_2$ respectively. Choose the correct option:

Q14.A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravi …

Q15.An object of mass '$m$' is being moved with a constant velocity under the action of an applied force of 2 N along a frictionless surface with following surface profile.

The correct applied force vs distance graph will be:

Q16.A square loop of side 20 cm and resistance $1\Omega$ is moved towards right with a constant speed vo.The right arm of the loop is in a uniform magnetic field of 5T. The field is perpendicular to the plane of the loop and is going into it. The loop is connected to a network of re …

Q17.For the given circuit the current $i$ through the battery when the key in closed and the steady state has been reached is ______.

Q18.A cube is placed inside an electric field, $\vec{E}=150y^2\hat{j}$. The side of the cube is 0.5 m and is placed in the field as shown in the given figure. The charge inside the cube is:

Q19.A student determined Young's Modulus of elasticity using the formula $Y=\frac{{\mathrm{MgL}}^3}{4{\mathrm{bd}}^3\delta }$. The value of g is taken to be $9.8\mathrm{m}/{\mathrm{s}}^2$, without any significant error, his observation are as follows:

| Physical Quantity | Leas …

Q20.Two resistors ${\mathrm{R}}_1=(4\pm 0.8)\Omega$ and ${\mathrm{R}}_2=(4\pm 0.4)\Omega$ are connected in parallel. The equivalent resistance of their parallel combination will be:

Q21.The width of one of the two slits in a Young's double slit experiment is three times the other slit. If the amplitude of light coming from a slit is proportional to the slit-width, the ratio of minimum to maximum intensity in the interference pattern is $\mathrm{x}:4$ where x is …

Q22.A steel rod with $\mathrm{y}=2.0\times 10^{11}{\mathrm{Nm}}^{-2}$ and $\alpha =10^{-5}{}^{\circ }{\mathrm{C}}^{-1}$ of length 4 m and area of cross-section 10 ${\mathrm{cm}}^2$ is heated from $0^{\circ }\mathrm{C}$ to $400^{\circ }\mathrm{C}$ without being allowed to extend. The te …

Q23.A uniform heating wire of resistance $36\Omega$ is connected across a potential difference of 240 V. The wire is then cut into half and a potential difference of 240 V is applied across each half separately. The ratio of power dissipation in first case to the total power dissipa …

Q24.The temperature of 3.00 mol of an ideal diatomic gas is increased by $40.0^{\circ }\mathrm{C}$ without changing the pressure of the gas. The molecules in the gas rotate but do not oscillate. If the ratio of change in internal energy of the gas to the amount of work done by the ga …

Q25.The average translational kinetic energy of ${\mathrm{N}}_2$ gas molecules at ______ ${}^{\circ }\mathrm{C}$ becomes equal to the K.E. of an electron accelerated from rest through a potential difference of 0.1 volt. (Given …

Q26.The satellites revolve around a planet in coplanar circular orbits in anticlockwise direction. Their period of revolutions are 1 hour and 8 hours respectively. The radius of the orbit of nearer satellite is $2\times 10^3\mathrm{km}$. The angular speed of the farther satellit …

Q27.When a body slides down from rest along a smooth inclined plane making an angle of $30^{\circ }$ with the horizontal, it takes time T. When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time …

Q28.A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when i …

Q29.An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of $40,000\mathrm{kg}$ is moving with a speed of $72{\mathrm{kmh}}^{-1}$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest, t …

Q30.A carrier wave with amplitude of 250 V is amplitude modulated by a sinusoidal base band signal of amplitude 150 V. The ratio of minimum amplitude to maximum amplitude for the amplitude modulated wave is 50 : $x$, then value of $x$ is ______.

Q1.Number of paramagnetic oxides among the following given oxides is ______.
${\mathrm{Li}}_2\mathrm{O},\mathrm{CaO},{\mathrm{Na}}_2{\mathrm{O}}_2,{\mathrm{KO}}_2,\mathrm{MgO}$ and ${\mathrm{K}}_2\mathrm{O}$

Q2.In the following sequence of reactions,
(see reaction scheme)
The compounds $\mathbf{B}$ and $\mathbf{C}$ respectively are:

Q3.The Crystal Field Stabilization Energy (CFSE) and magnetic moment (spin-only) of an octahedral aqua complex of a metal ion $\left({\mathrm{M}}^{\mathrm{Z}+}\right)$ are $-0.8{\Delta }_0$ and 3.87 BM, respectively. Identify $\left({\mathrm{M}}^{\mathrm{Z}+}\right)$:

Q4.Water sample is called cleanest on the basis of which one of the BOD values given below:

Q5.The oxide without nitrogen-nitrogen bond is:

Q6.Given below are two statements:
Statement I: The nucleophilic addition of sodium hydrogen sulphite to an aldehyde or a ketone involves proton transfer to form a stable ion.
Statement II: The nucleophilic addition of hydrogen cyanide to an aldehyde or a ketone yields amine as fina …

Q7.The potassium ferrocyanide solution gives a Prussian blue colour, when added to:

Q8.Identify the element for which electronic configuration in +3 oxidation state is $[\mathrm{Ar}]3{\mathrm{d}}^5$:

Q9.Calamine and Malachite, respectively, are the ores of:

Q10.Match List-I with List-II

List - I (Colloid Preparation method)
(a) Hydrolysis
(b) Reduction
(c) Oxidation
(d) Double Decomposition

List - II (Chemical Reaction)
(i) …

Q11.Identify $A$ in the following reaction.
(see reaction scheme)

Q12.Which one of the following gives the most stable Diazonium salt?

Q13.In the following sequence of reactions a compound $\mathbf{A}$, (molecular formula ${\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_2$ ) with a straight chain structure gives a C4 carboxylic acid. A is …

Q14.Hydrogen peroxide reacts with iodine in basic medium to give:

Q15.Which one of the following compounds is aromatic in nature?

Q16.In the given chemical reaction, colors of the ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$ ions, are respectively:

Q17.Which one of the following given graphs represents the variation of rate constant (k) with temperature (T) for an endothermic reaction?

Q18.Monomer units of Dacron polymer are:

Q19.Experimentally reducing a functional group cannot be done by which one of the following reagents?

Q20.The stereoisomers that are formed by electrophilic addition of bromine to trans-but-2-ene is/are

Q21.An empty LPG cylinder weighs 14.8 kg . When full, it weighs 29.0 kg and shows a pressure of 3.47 atm . In the course of use at ambient temperature, the mass of the cylinder is reduced to 23.0 kg . The final pressure inside of the cylinder is ____ atm. (Nearest integer)
(Assume LP …

Q22.The sum of oxidation states of two silver ions in $\left[\mathrm{Ag}{\left({\mathrm{NH}}_3\right)}_2\right]\left[\mathrm{Ag}(\mathrm{CN}{)}_2\right]$ complex is ____.

Q23.The molar solubility of $\mathrm{Zn}(\mathrm{OH}{)}_2$ in 0.1 M NaOH solution is $x\times 10^{-18}\mathrm{M}$. The value of $x$ is ____. (Nearest integer)
(Given : The solubility product of $\mathrm{Zn}(\mathrm{OH}{)}_2$ is $2\times 10^{-20}$ )

Q24.If 80 g of copper sulphate ${\mathrm{CuSO}}_4\cdot 5{\mathrm{H}}_2\mathrm{O}$ is dissolved in deionised water to make 5 L of solution. The concentration of the copper sulphate solution is $x\times 10^{-3}\mathrm{mol}{\mathrm{L}}^{-1}$. The value of x is ____. [Atomic mas …

Q25.A 50 watt bulb emits monochromatic red light of wavelength of 795 nm. The number of photons emitted per second by the bulb is $x\times 10^{20}$. The value of $x$ is ______.
[Given: $\mathrm{h}=6.63\times 10^{-34}\mathrm{Js}$ and $\mathrm{c}=3.0\times 10^8{\mathrm{ms}}^{-1}$ …

Q26.A peptide synthesized by the reactions of one molecule each of Glycine, Leucine, Aspartic acid and Histidine will have ______ peptide linkages.

Q27.For the reaction $2{\mathrm{NO}}_2(\mathrm{g})\rightleftharpoons {\mathrm{N}}_2{\mathrm{O}}_4(\mathrm{g})$, when $\Delta \mathrm{S}=-176.0{\mathrm{JK}}^{-1}$ and $\Delta \mathrm{H}=-57.8\mathrm{kJ}{\mathrm{mol}}^{-1}$, the magnitude of $\Delta \mathrm{G}$ at 298 K for the re …

Q28.The number of atoms in 8 g of sodium is $\mathrm{x}\times 10^{23}$. The value of x is ______.
(Nearest integer)
[Given: ${\mathrm{N}}_{\mathrm{A}}=6.02\times 10^{23}{\mathrm{mol}}^{-1}$, Atomic mass of $\mathrm{Na}=23.0\mathrm{u}$]

Q29.If the conductivity of mercury at $0^{\circ }\mathrm{C}$ is $1.07\times 10^6\mathrm{S}{\mathrm{m}}^{-1}$ and the resistance of a cell containing mercury is $0.243\Omega$, then the cell constant of the cell is $\times \times 10^4{\mathrm{m}}^{-1}$. The value of $\times$ is …

Q1.If $n$ is the number of solutions of the equation $2\cos x\left(4\sin \left(\frac{\pi }{4}+x\right)\sin \left(\frac{\pi }{4}-x\right)-1\right)=1,x\in [0,\pi ]$ and $S$ is the sum of all these solutions, then the order pair $(n,S)$ is:

Q2.The distance of line $3y-2z-1=0=3x-z+4$ from the point $(2,-1,6)$ is:

Q3.${\cos }^{-1}(\cos (-5))+{\sin }^{-1}(\sin (6))-{\tan }^{-1}(\tan (12))$ is equal to:
(The inverse trigonometric functions take the principal values)

Q4.If $y=y(x)$ is the solution curve of the differential equation $x^{2}dy+\left(y-\frac{1}{x}\right)dx=0;x>0$, and $y(1)=1$, then $\mathrm{y}\left(\frac{1}{2}\right)$ is equal to:

Q5.Let $a_1,a_2,\dots \dots ,a_{21}$ be an AP such that ${\sum }_{n=1}^{20}\frac{1}{a_n{a}_{n+1}}=\frac{4}{9}$. If the sum of this AP is 189, then $a_6{a}_{16}$ is equal to:

Q6.Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point of intersection in the first quadrant. The $\tan \theta$ is equal to:

Q7.The range of the function
$f(x)={\log }_{\sqrt{5}}\left(3+\cos \left(\frac{3\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}+x\right)+\cos \left(\frac{\pi }{4}-x\right)-\cos \left(\frac{3\pi }{4}-x\right)\right)$ is:

Q8.Consider the system of linear equations
$-x+y+2z=0$
$3x-ay+5z=1$
$2x-2y-az=7$
Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the set of all $\mathrm{a}\in \mathrm{R}$ for which the system has infinitely many solut …

Q9.The area, enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0,x=\frac{\pi }{2}$, is:

Q10.Let $f:R\to R$ be a continuous function. The ${\lim }_{x\to \pi /4}\frac{\frac{\pi }{4}{\int }_2^{{\sec }^{2}x}f(x)dx}{x^2-\frac{{\pi }^2}{16}}$ is equal to:

Q11.Let ${\mathrm{P}}_1,{\mathrm{P}}_2\dots ,{\mathrm{P}}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points ${\mathrm{P}}_{\mathrm{i}},{\mathrm{P}}_{\mathrm{j}}$, ${\mathrm{p}}_{\mathrm{k}}$ such that $\mathrm{i}+\mathrm{j}+\mathrm{k}\ne 15$, is:

Q12.Let the acute angle bisector of the two planes $x-2y-2z+1=0$ and $2x-3y-6z+1=0$ be the plane $P$. Then which of the following points lies on $P$ ?

Q13.Let $J_{n,m}={\int }_0^{1/2}\frac{x^n}{x^m-1}dx,\forall n>m$ and $n,m\in N$. Consider a matrix $A=[a_{ij}{]}_{3\times 3}$ where $a_{ij}=\{\begin{array}{ll}{J}_{6+i,3}-J_{i+3,3},& i\le j\\ 0,& i>j\end{array}$. Then $|\text{adj }{A}^{-1}|$ is:

Q14.Let $S_n=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\dots +(n-1)\cdot 1,n\ge 4$. The sum ${\sum }_{n=4}^{\infty }\left(\frac{2S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to:

Q15.Two squares are chosen at random on a chessboard (see figure). The probability they have a side in common is:

Q16.The function $f(x)$, that satisfies the condition $f(x)=x+{\int }_0^{\pi /2}\sin x\cdot \cos yf(y)dy$ is:

Q17.The function $f(x)=x^3-6x^2+ax+b$ is such that $f(2)=f(4)=0$.
Consider two statements.
(S1) there exists $x_1,x_2\in (2,4),x_1<x_2$, such that $f^{\prime }(x_1)=-1$ and $f^{\prime }(x_2)=0$.
(S2) there exists $x_3,x_4\in (2,4),x_3<x_4$, such that $f$ is decreasing in …

Q18.Consider the parabola with vertex $\left(\frac{1}{2},\frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let P be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point Q, then $(PQ{)}^2$ i …

Q19.The number of pairs (a, b) of real numbers, such that whenever $\alpha$ is a root of the equation $x^2+ax+b=0$, ${\alpha }^2-2$ is also a root of this equation, is:

Q20.Which of the following is equivalent to the Boolean expression $p\wedge \sim q$?

Q21.Let X be a random variable with distribution.

If the mean of X is 2.3 and variance of X is ${\sigma }^2$, then $100{\sigma }^2$ is equal to:

Q22.If for the complex numbers z satisfying $|z-2-2i|\le 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to ______.

Q23.Let the points of intersections of the lines $x-y+1=0$, $x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is ______.

Q24.Let $f(x)=x^6+2x^4+x^3+2x+3,x\in R$. Then the natural number $n$ for which ${\lim }_{x\to 1}\frac{x^{n}f(1)-f(x)}{x-1}=44$ is ______.

Q25.Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is 19 units, the …

Q26.Let $[t]$ denote the greatest integer $\le t$. The number of points where the function $f(x)=[x]\left|x^2-1\right|+\sin \left(\frac{\pi }{[x]+3}\right)-[x+1],x\in (-2,2)$ is not continuous is

Q27.All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is

Q28.Let $f(x)$ be a polynomial of degree 3 such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10f(10)$ is equal to

Q29.A man starts walking from the point $\mathrm{P}(-3,4)$, touches the x -axis at R , and then turns to reach at the point $\mathrm{Q}(0,2)$. The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $50(PR{)}^2+(RQ{)}^2$ is equal to

Q30.If the sum of the coefficients in the expansion of $(x+y{)}^n$ is 4096 , then the greatest coefficient in the expansion is