JEE Main 2021 · 31 Aug 2021 · Shift 1

Q1.Choose the correct waveform that can represent the voltage across $R$ of the following circuit, assuming the diode is ideal one :

Q2.A coil having N turns is would tightly in the form of a spiral with inner and outer radii 'a' and 'b' respectively. Find the magnetic field at centre, when a current I passes through coil:

Q3.In an ac circuit, an inductor, a capacitor and a resistor are connected in series with $X_L=R=X_C$. Impedance of this circuit is:

Q4.Which of the following equations is dimensionally incorrect ?
Where $\mathrm{t}=$ time, $\mathrm{h}=$ height, $\mathrm{s}=$ surface tension, $\theta =$ angle, $\mathrm{p}=$ density, $\mathrm{a},\mathrm{r}=$ radius, $\mathrm{g}=$ acceleration due to gravity, …

Q5.A uniform heavy rod of weight $10\mathrm{kg}{\mathrm{ms}}^{-2}$, cross-sectional area $100{\mathrm{cm}}^2\mathrm{f}$ and length 20 cm is hanging from a fixed support. Young modulus of the material of the rod is $2\times 10^{11}{\mathrm{Nm}}^{-2}$. Neglecting the latera …

Q6.A small square look offside 'a' and one turn is placed inside a larger square loop of side b and one turn ( $b\gg a$ ). The two loops are coplanar with their centres coinciding. If a current I is passed in the square loop offside ' $b$ ', then the coefficient of mutual induc …

Q7.A helicopter is flying horizontally with a speed ' $v$ ' at an altitude ' $h$ ' has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?

Q8.The masses and radii of the earth and moon are ( ${\mathrm{M}}_1,{\mathrm{R}}_1$ ) and ( ${\mathrm{M}}_2,{\mathrm{R}}_2$ ) respectively. Their centres are at a distance ' $r$ ' apart. Find the minimum escape velocity for a particle of mass ' $m$ ' to be projected from t …

Q9.Two plane mirrors ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ are at right angle to each shown. A point source ' P ' is placed at ' a ' and ' 2 a ' meter away from $M_1$ and $M_2$ respectively. The shortest distance between the images thus formed is : (Take $\sqrt{5}=2.3$ …

Q10.A reversible engine has an efficiency of $\frac{1}{4}$. If the temperature of the sink is reduced by $58^{\circ }\mathrm{C}$, its efficiency becomes double. Calculate the temperature of the sink :

Q11.A moving proton and electron have the same de-Broglie wavelength. If K and P denote the K .E. and momentum respectively. Then choose the correct option :

Q12.Consider a galvanometer shunted with $5\Omega$ resistance and $2\%$ of current passes through it. What is the resistance of the given galvanometer?

Q13.A body of mass M moving at speed ${\mathrm{V}}_0$ collides elastically with a mass ' m ' at rest. After the collision, the two masses move at angles ${\theta }_1$ and ${\theta }_2$ with respect to the initial direction of motion of the body of mass M . The largest possible va …

Q14.In the following logic circuit the sequence of the inputs $A,B$ are $(0,0),(0,1),(1,0)$ and $(1,1)$. The output Y for this sequence will be :

Q15.A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B , which in turn disintegrates to some other stable nucleus C . Plot of a graph showing the variation of number of atoms of nucleus $B$ vesus time is :
(Assume that at $t=0$, there are no $B$ …

Q16.An object is placed at the focus of concave lens having focal length f . What is the magnification and distance of the image from the optical centre of the lens?

Q17.For an ideal gas the instantaneous change in pressure ' p ' with volume ' v ' is given by the equation $\frac{\mathrm{dp}}{\mathrm{dv}}=-\mathrm{ap}$. If $\mathrm{p}={\mathrm{p}}_0$ at $\mathrm{v}=0$ is the given boundary condition, then the maximum temperature one mole of …

Q18.Angular momentum of a single particle moving with constant speed along circular path :

Q19.Match List-I with List-II.

List-I:
(a) Torque
(b) Impulse
(c) Tension
(d) Surface Tension

List-II:
(i) $ML{T}^{-1}$
(ii) $M{T}^{-2}$
(iii) $M{L}^2{T}^{-2}$
(iv) $ML{T}^{-2}$

Choose the most appropriate answer from the option given below :

Q20.Two particles A and B having charges $20\mu C$ and $-5\mu C$ respectively are held fixed with a separation of 5 cm. At what position a third charged particle should be placed so that it does not experience a net electric force?

Q21.The voltage drop across $15\Omega$ resistance in the given figure will be _____ V.

Q22.A square shaped wire with resistance of each side $3\Omega$ is bent to form a complete circle. The resistance between two diametrically opposite points of the circle in unit of $\Omega$ will be _____.

Q23.A car is moving on a plane inclined at $30^{\circ }$ to the horizontal with an acceleration of $10{\mathrm{ms}}^{-2}$ parallel to the plane upward. A bob is suspended by a string from the roof of the car. The angle in degrees which the string makes with the vertical is _____. (Tak …

Q24.A capacitor of $50\mu F$ is connected in a circuit as shown in figure. The charge on the upper plate of the capacitor is _____ $\mu C$.

Q25.A wire having a linear mass density $9.0\times 10^{-4}kg/m$ is stretched between two rigid supports with a tension of 900 N. The wire resonates at a frequency of 500 Hz. The next higher frequency at which the same wire resonates is 550 Hz. The length of the wire is __ …

Q26.When a rubber ball is taken to a depth of _____ m in deep sea, its volume decreases by $0.5\%$.
(The bulk modulus of rubber $=9.8\times 10^8{\mathrm{Nm}}^{-2}$
Density of sea water $=10^3{\mathrm{kgm}}^{-3}$
$g=9.8{m/s}^2$)

Q27.A particle of mass 1 kg is hanging from a spring of force constant $100{\mathrm{Nm}}^{-1}$. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system wi …

Q28.A block moving horizontally on a smooth surface with a speed of $40{\mathrm{ms}}^{-1}$ splits into two equal parts. If one of the parts moves at $60{\mathrm{ms}}^{-1}$ in the same direction, then the fractional change in the kinetic energy will be $x:4$ where $x=$ _____.

Q29.The electric field in an electromagnetic wave is given by $E=(50{\mathrm{NC}}^{-1})\sin \omega (t-x/c)$
The energy contained in a cylinder of volume V is $5.5\times 10^{-12}\mathrm{J}$. The value of V is _____ ${\mathrm{cm}}^3$.
(given …

Q30.If the sum of the heights of transmitting and receiving antennas in the line of sight of communication is fixed at 160 m, then the maximum range of LOS communication is ______ km. (Take radius of Earth $=6400\mathrm{km}$)

Q1.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Metallic character decreases and non-metallic character increases on moving from left to right in a period.
Reason (R) : It is due to increase in ionisation …

Q2.The structure of product C , formed by the following sequence of reactions is :

Q3.Given below are two statements :
Statement-I : The process of producing syn-gas is called gasification of coal.
Statement-II : The composition of syn-gas is $\mathrm{CO}+{\mathrm{CO}}_2+{\mathrm{H}}_2(1:1:1)$
In the light of the above statements, choose the most appropriate ans …

Q4.Choose the correct name for compound given below :

Q5.Select the graph that correctly describes the adsorption isotherms at two temperatures $T_1$ and $T_2\left({\mathrm{T}}_1>{\mathrm{T}}_2\right)$ for a gas :
( x - mass of the gas adsorbed ; m - mass of adsorbent ; P - pressure)

Q6.Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Aluminium is extracted from bauxite by the electrolysis of molten mixture of ${\mathrm{Al}}_2{\mathrm{O}}_3$ with cryolite.
Reason (R): The oxidation state o …

Q7.The major component/ingredient of Portland Cement is :

Q8.The denticity of an organic ligand, biuret is :

Q9.BOD values (in ppm) for clean water (A) and polluted water (B) are expected respectively

Q10.The correct order of reactivity of the given chlorides with acetate in acetic acid is :

Q11.In the structure of the dichromate ion, there is a :

Q12.Which one of the following lanthanides exhibits +2 oxidation state with diamagnetic nature? (Given Z for $\mathrm{Nd}=60,\mathrm{Yb}=70,\mathrm{La}=57,\mathrm{Ce}=58$ )

Q13.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R) :
Assertion (A) : A simple distillation can be used to separate a mixture of propanol and propanone.
Reason (R): Two liquids with a difference of more than …

Q14.Monomer of Novolac is :

Q15.Which one of the following compounds contains $\beta -{\mathrm{C}}_1-{\mathrm{C}}_4$ glycosidic linkage?

Q16.Which one of the following 0.10 M aqueous solutions will exhibit the largest freezing point depression?

Q17.The major products A and B in the following set of reactions are :

Q18.Which one of the following is the correct PV vs P plot at constant temperature for an ideal gas? ( P and V stand for pressure and volume of the gas respectively)

Q19.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Treatment of bromine water with propene yields 1-bromopropan-2-ol.
Reason (R): Attack of water on bromonium ion follows Markovnikov rule and results in 1-bro …

Q20.${\mathrm{A}}_3{\mathrm{B}}_2$ is a sparingly soluble salt of molar mass $\mathrm{M}\left(\mathrm{g}{\mathrm{mol}}^{-1}\right)$ and solubility $x\mathrm{g}{\mathrm{L}}^{-1}$. The solubility product satisfies $K_{sp}=a{\left(\frac{x}{M}\right)}^5$. The value of $a$ is ______. (In …

Q21.The total number of reagents from those given below, that can convert nitrobenzene into aniline is ______. (Integer answer)
I. $\mathrm{Sn}-\mathrm{HCl}$
II. $\mathrm{Sn}-{\mathrm{NH}}_4\mathrm{OH}$
III. $\mathrm{Fe}-\mathrm{HCl}$
IV. $\mathrm{Zn}-\mathrm{HCl}$
V. …

Q22.The molarity of the solution prepared by dissolving 6.3 g of oxalic acid $\left({\mathrm{H}}_2{\mathrm{C}}_2{\mathrm{O}}_4\cdot 2{\mathrm{H}}_2\mathrm{O}\right)$ in 250 mL of water in $\mathrm{mol}{\mathrm{L}}^{-1}$ is $x\times 10^{-2}$. The value of $x$ is ______. (Nearest …

Q23.Consider the following cell reaction :
${\mathrm{Cd}}_{(\mathrm{s})}+{\mathrm{Hg}}_2{\mathrm{SO}}_{4(\mathrm{s})}+\frac{9}{5}{\mathrm{H}}_2{\mathrm{O}}_{(\mathrm{l})}+{\mathrm{CdSO}}_4\cdot \frac{9}{5}{\mathrm{H}}_2{\mathrm{O}}_{(\mathrm{s})}+2{\mathrm{Hg}}_{(\mathrm{l})}$
The valu …

Q24.Consider the sulphides $\mathrm{HgS},\mathrm{PbS},\mathrm{CuS},{\mathrm{Sb}}_2{\mathrm{S}}_3,{\mathrm{As}}_2{\mathrm{S}}_3$ and CdS . Number of these sulphides soluble in $50\%{\mathrm{HNO}}_3$ is ____ .

Q25.For a first order reaction, the ratio of the time for $75\%$ completion of a reaction to the time for 50% completion is ____ . (Integer answer)

Q26.According to the following figure, the magnitude of the enthalpy change of the reaction
$\mathrm{A}+\mathrm{B}\to \mathrm{M}+\mathrm{N}$ in kJ mol${}^{-1}$
is equal to ____ . (Integer answer)

Q27.The number of halogen/(s) forming halic (V) acid is ____ .

Q28.The number of hydrogen bonded water molecule(s) associated with stoichiometry ${\mathrm{CuSO}}_4\cdot 5{\mathrm{H}}_2\mathrm{O}$ is

Q29.$\mathrm{Ge}(\mathrm{Z}=32)$ in its ground state electronic configuration has $x$ completely filled orbitals with ${\mathrm{m}}_{\mathrm{l}}=$ 0 . The value of $x$ is ____ .

Q1.Let ${}^{\ast },\Delta \in \{\wedge ,\mathrm{v}\}$ be such that the Boolean expression $({\mathrm{p}}^{\ast }\sim \mathrm{q})\Rightarrow (\mathrm{p}\Delta \sim \mathrm{q})$ is a tautology. Then :

Q2.Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3x-2y+4z-7=0$ and $x+5y-2z+9=0$, be $\alpha x+\beta y+\gamma z+3=0$, then $\alpha +\beta +\gamma$ is equal to:

Q3.The number of real roots of the equation $e^{4x}+2e^{3x}-e^x-6=0$ is:

Q4.The function $\mathrm{f}(\mathrm{x})=\left|{\mathrm{x}}^2-2\mathrm{x}-3\right|\cdot {\mathrm{e}}^{|9{\mathrm{x}}^2-12\mathrm{x}+4|}$ is not differentiable at exactly:

Q5.If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines
$\mathrm{x}\mathrm{cosec}\alpha -\mathrm{y}\sec \alpha =\mathrm{k}\cot 2\alpha$ and $\mathrm{x}\sin \alpha +\mathrm{y}\cos \alpha =\mathrm{k}\sin 2\alpha$ respectively, then …

Q6.${\lim }_{x\to 0}\frac{{\sin }^2\left(\pi {\cos }^{4}x\right)}{x^4}$ is equal to:

Q7.If $a_r=\cos \frac{2r\pi }{9}+i\sin \frac{2r\pi }{9},r=1,2,3\dots ,i=\sqrt{-1}$, then the determinant $\left|\begin{array}{lll}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is equal to:

Q8.The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S(>R)$ respectively from the origin, is :

Q9.Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3r^2$, then $r^2-d$ is equal to:

Q10.The line $12x\cos \theta +5y\sin \theta =60$ is tangent to which of the following curves?

Q11.If the function $f(x)=\{\begin{array}{l}\frac{1}{x}{\log }_x\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right),x<0\\ k,x=0\\ \frac{{\cos }^{2}x-{\sin }^{2}x-1}{\sqrt{x^2+1}-1},x>0\end{array}$
is continuous at $x=0$, then …

Q12.The integral $\int \frac{1}{\sqrt[4]{(x-1{)}^3(x+2{)}^5}}\mathrm{dx}$ is equal to: (where C is a constant of integration)

Q13.The sum of 10 terms of the series
is :

Q14.Which of the following is not correct for relation on the set of real numbers?

Q15.If , then is equal to:

Q16.Let be a non-negative function in and twice differentiable in . If and , then :

Q17.Let and be two vectors such that and the angle between and is . If is a unit vector, then is equal to:

Q18. is a root of the equation:

Q19.A vertical pole fixed to the horizontal ground is divided in the ratio by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) …

Q20.If the following system of linear equations



has no solution, then:

Q21.If 'R' is the least value of 'a' such that the function is increasing on and 'S' is the greatest value of 'a' such that the function is decreasing on , then the value of is ______.

Q22.Let denote the greatest integer . Then the value of is ______.

Q23.If the variable line lies between the two circles and , without intercepting a chord on either circle, then the sum of all the integral values of is ______.

Q24.The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ______.

Q25.The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2x-y+z=6$ from the point $(-1,-1,2)$ is ____.

Q26.If $\left(\frac{3^6}{4^4}\right)\mathrm{k}$ is the term, independent of x , in the binomial expansion of ${\left(\frac{\mathrm{x}}{4}-\frac{12}{{\mathrm{x}}^2}\right)}^{12}$, then k is equal to ____.

Q27.A point $z$ moves in the complex plane such that arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi }{4}$, then the minimum value of $|z-9\sqrt{2}-2i{|}^2$ is equal to ____.

Q28.The mean of 10 numbers $7\times 8,10\times 10,13\times 12,16\times 14,\dots$ is ____.

Q29.An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability t …

Q30.If $x\varphi (x)={\int }_5^x\left(3t^2-2{\varphi }^{\prime }(t)\right)dt,x>-2$, and $\varphi (0)=4$, then $\varphi (2)$ is ____.