JEE Main 2021 · 27 Aug 2021 · Shift 1

Q1.A balloon carries a total load of 185 kg at normal pressure and temperature of $27^{\circ }\mathrm{C}$. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is $-7^{\circ }\mathrm{C}$?

Q2.A uniformly charged disc of radius R having surface charge density $\sigma$ is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :-

Q3.For an ideal gas ${\mathrm{PT}}^3=$ constant. Find the coefficient of volume expansion of the gas is:

Q4.Calculate the amount of charge on capacitor of $4\mu \mathrm{F}$. The internal resistance of battery is $1\Omega$

Q5.There are $10^{10}$ radioactive nuclei in a given radioactive element, Its half-life time is 1 minute. How many nuclei will remain after 30 seconds?
$(\sqrt{2}=1.414)$

Q6.Two ions of masses 4 amu and 16 amu have charges +2 e and +3 e respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then:

Q7.In a photoelectric experiment, increasing the intensity of incident light:

Q8.The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure. The potential energy $\mathrm{U}(\mathrm{x})$ versus time (t) plot of the particle is correctly shown in figure :

Q9.If E and H represents the intensity of electric field and magnetising field respectively, then the unit of $E/H$ will be:

Q10.The resultant of these forces $\overrightarrow{\mathrm{OP}},\overrightarrow{\mathrm{OQ}},\overrightarrow{\mathrm{OR}},\overrightarrow{\mathrm{OS}}$ and $\overrightarrow{\mathrm{OT}}$ is approximately ..... N [Take $\sqrt{3}=1.7,\sqrt{2}=1.4$. Given $\hat{i}$ and …

Q11.Five identical cells each of internal resistance $1\Omega$ and emf 5 V are connected in series and in parallel with an external resistance ' R '. For what value of ' R ', current in series and parallel combination will remain the same?

Q12.In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius 2.0 $\times 10^{-5}\mathrm{m}$ and density $1.2\times 10^3{\mathrm{kgm}}^{-3}$ ? Take viscosity of liquid $=1.8\times 10^{-5}{\mathrm{Nsm}}^{-2}$. (Neglect buoyancy due to air …

Q13.A huge circular arc of length 4.4 ly subtends an angle ' 4 s ' at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second?
Given : $1\mathrm{ly}=9.46\times 10^{15}\mathrm{m}$ …

Q14.Moment of inertia of a square plate of side / about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

Q15.Which of the following is not a dimensionless quantity?

Q16.For a transistor in CE mode to be used as an amplifier, it must be operated in :

Q17.Electric field in a plane electromagnetic wave is given by

The velocity of electromagnetic wave in this medium is :
(Given C = speed of light in vacuum)

Q18.An object is placed beyond the centre of curvature C of the given concave mirror. If the distance of the object is $d_1$ from C and the distance of the image formed is $d_2$ from C, the radius of curvature of this mirror is :

Q19.Find the distance of the image from object O, formed by the combination of lenses in the figure :

Q20.A circuit is arranged as shown in figure. The output voltage $V_O$ is equal to ______ V.

Q21.A transmitting antenna has a height of 320 m and that of receiving antenna is 2000 m. The maximum distance between them for satisfactory communication in line of sight mode is 'd'. The value of 'd' is ______ km.

Q22.A rod CD of thermal resistance $10.0{\mathrm{KW}}^{-1}$ is joined at the middle of an identical rod AB as shown in figure. The end A, B and D are maintained at $200^{\circ }\mathrm{C}$, $100^{\circ }\mathrm{C}$ and $125^{\circ }\mathrm{C}$ respectively. The heat current in CD is P …

Q23.First, a set of n equal resistors of $10\Omega$ each are connected in series to a battery of emf 20 V and internal resistance $10\Omega$. A current I is observed to flow. Then, the n resistors are connected in parallel to the same battery. It is observed that the current is inc …

Q24.The alternating current is given by

The r.m.s. value of this current is ______ A

Q25.A uniform conducting wire of length is 24 a, and resistance R is wound up as a current carrying coil in the shape of an equilateral triangle of side 'a' and then in the form of a square of side 'a'. The coil is connected to a voltage source $V_0$. The ratio of magnetic moment o …

Q26.Two cars X and Y are approaching each other with velocities $36\mathrm{km}/\mathrm{h}$ and $72\mathrm{km}/\mathrm{h}$ respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is 1320 Hz. If the velocity of sound in …

Q27.A body of mass (2M) splits into four masses $\{m,M-m,m,M-m\}$, which are rearranged to form a square as shown in the figure. The ratio of $\frac{M}{m}$ for which, the gravitational potential energy of the system becomes maximum is $x:1$. The value of $x$ is ______

Q28.If the velocity of a body related to displacement $x$ is given by $v=\sqrt{5000+24x}\mathrm{m}/\mathrm{s}$, then the acceleration of the body is ______

Q29.Two persons A and B perform same amount of work in moving a body through a certain distance d with application of forces acting at angle $45^{\circ }$ and $60^{\circ }$ with the direction of displacement respectively. The ratio of force applied by person $A$ to the force appl …

Q1.In the following sequence of reactions, the final product D is :

Q2.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Synthesis of ethyl phenyl ether may be achieved by Williamson synthesis.
Reason (R): Reaction of bromobenzene with sodium ethoxide yields ethyl phenyl ether. …

Q3.Out of following isomeric forms of uracil, which one is present in RNA?

Q4.The gas 'A' is having very low reactivity reaches to stratosphere. It is non-toxic and nonflammable but dissociated by UV-radiations in stratosphere. The intermediates formed initially from the gas 'A' are :

Q5.The correct statement about (A), (B), (C) and (D) is:

Q6.The number of water molecules in gypsum, dead burnt plaster and plaster of paris, respectively are:

Q7.The unit of the van der Waals gas equation parameter 'a' in $\left(P+\frac{a{n}^2}{V^2}\right)(V-nb)=nRT$ is :

Q8.The structure of the starting compound $\mathbf{P}$ used in the reaction given below is:

Q9.Deuterium resembles hydrogen in properties but :

Q10.Match List-I with List-II:
List-I (Species)
List-II (Number of lone pairs of electrons on the central atom)
(a) ${\mathrm{XeF}}_2$
(b) ${\mathrm{XeO}}_2{\mathrm{F}}_2$
(c) ${\mathrm{XeO}}_3{\mathrm{F}}_2$
(d) ${\mathrm{XeF}}_4$
(i) 0
(ii) 1
(iii) 2
(iv) 3
Choose the most approp …

Q11.In the following sequence of reactions the P is:

Q12.The nature of oxides ${\mathrm{V}}_2{\mathrm{O}}_3$ and CrO is indexed as ' X ' and ' Y ' type respectively. The correct set of $X$ and $Y$ is:

Q13.In polythionic acid, ${\mathrm{H}}_2{\mathrm{S}}_{\mathrm{x}}{\mathrm{O}}_6(\mathrm{x}=3$ to 5$)$ the oxidation state(s) of sulphur is/are :

Q14.Which of the following is not a correct statement for primary aliphatic amines?

Q15.In which one of the following molecules strongest back donation of an electron pair from halide to boron is expected?

Q16.Tyndall effect is more effectively shown by:

Q17.Match items of List-I with those of List-II :

List-I (Property)
(a) Diamagnetism
(b) Ferrimagnetism
(c) Paramagnetism
(d) Antiferromagnetism

List-II (Example)
(i) MnO
(ii) ${\mathrm{O}}_2$
(iii) NaCl
(iv) ${\mathrm{Fe}}_3{\mathrm{O}}_4$

Choose the most appropriate answer …

Q18.Acidic ferric chloride solution on treatment with excess of potassium ferrocyanide gives a Prussian blue coloured colloidal species. It is:

Q19.Which refining process is generally used in the purification of low melting metals?

Q20.1 mol of an octahedral metal complex with formula ${\mathrm{MCl}}_3\cdot 2\mathrm{L}$ on reaction with excess of ${\mathrm{AgNO}}_3$ gives 1 mol of AgCl . The denticity of Ligand L is ____ (integer answer)

Q21.In Carius method for estimation of halogens, 0.2 g of an organic compound gave 0.188 g of AgBr . The percentage of bromine in the compound is ____ . (Nearest integer) [Atomic mass : $\mathrm{Ag}=108,\mathrm{Br}=80$ ]

Q22.The reaction that occurs in a breath analyser, a device used to determine the alcohol level in a person's blood stream is …

Q23.1 kg of 0.75 molal aqueous solution of sucrose can be cooled up to $-4^{\circ }\mathrm{C}$ before freezing. The amount of ice (in g) that will be separated out is ____ . (Nearest integer)
[Given : …

Q24.The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to $\frac{h^2}{xm{a}_0^2}$. The value of $10x$ is ______. ($a_0$ is radius of Bohr's orbit) (Nearest integer)
[Given : $\pi =3.14$]

Q25.The number of moles of CuO, that will be utilized in Dumas method for estimating nitrogen in a sample of 57.5 g of $N,N$-dimethylaminopentane is ______ $\times 10^{-2}$. (Nearest integer).

Q26.The number of $f$ electrons in the ground state electronic configuration of $\mathrm{Np}(Z=93)$ is ______. (Nearest integer)

Q27.When 10 mL of an aqueous solution of $\mathrm{KMnO}{}_4$ was titrated in acidic medium, equal volume of 0.1 M of an aqueous solution of ferrous sulphate was required for complete discharge of colour. The strength of $\mathrm{KMnO}{}_4$ in grams per litre is ______ $\times 10^{-2}$. (Nearest in …

Q28.200 mL of 0.2 M HCl is mixed with 300 mL of 0.1 M NaOH. The molar heat of neutralization of this reaction is -57.1 kJ. The increase in temperature in ${}^{\circ }\mathrm{C}$ of the system on mixing is $x\times 10^{-2}$. The value of $x$ is ______. (Nearest integer)
[Given : Specifi …

Q29.The number of moles of $\mathrm{NH}{}_3$, that must be added to 2 L of $0.80\mathrm{M}\mathrm{AgNO}{}_3$ in order to reduce the concentration of $\mathrm{Ag}{}^{+}$ ions to $5.0\times 10^{-8}\mathrm{M}$ ($K_{\text{formation}}$ for $[\mathrm{Ag}(\mathrm{NH}{}_3){}_2{]}^{+}=1.0\times 10^8$) is ______. (Nearest intege …

Q1.A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:

Q2.Let $\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}$, where $A,B,C$ are angles of a triangle $ABC$. If the lengths of the sides opposite these angles are $a,b,c$ respectively, then :

Q3.A tangent and a normal are drawn at the point $P(2,-4)$ on the parabola $y^2=8x$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a,b)$ is a point such that $AQBP$ is a square, then $2a+b$ is equal to :

Q4.The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to a line, whose direction ratios are $2,3,-6$ is

Q5.When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x$. All other faces occur with probability $\frac{1}{6}$. Note that opposite faces sum to 7 in any die. If $0<x<\frac{1}{6}$, …

Q6.If $U_n=\left(1+\frac{1}{n^2}\right){\left(1+\frac{2^2}{n^2}\right)}^2\cdots {\left(1+\frac{n^2}{n^2}\right)}^n$, then ${\lim }_{n\to \infty }{\left(U_n\right)}^{\frac{-4}{n^2}}$ is equal to

Q7.Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t,0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is :

Q8.Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x,f(x))$ is given by $f(x)+x{f}^{\prime }(x)=x^2$. Then:

Q9.The statement $(p\wedge (p\to q)\wedge (q\to r))\to r$ is :

Q10.If $S=\left\{z\in C:\frac{z-i}{z+2i}\in R\right\}$, then

Q11.Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2x+y-3z+4=0$, is :

Q12.If $\alpha ,\beta$ are the distinct roots of $x^2+bx+c=0$, then ${\lim }_{x\to \beta }\frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{(x-\beta {)}^2}$ is equal to

Q13.If for $x,y\in R,x<0,y={\log }_{10}x+{\log }_{10}{x}^{1/3}+{\log }_{10}{x}^{1/9}+\dots$. upto $\infty$ terms and $\frac{2+4+6+\dots +2y}{3+6+9+\dots +3y}=\frac{4}{{\log }_{10}x}$, then the ordered pair $(x,y)$ is equal to

Q14.Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that that $y(0)=7$. Then $y(\pi )$ is equal to

Q15.If ${\left({\sin }^{-1}x\right)}^2-{\left({\cos }^{-1}x\right)}^2=a,0<x<1,a\ne 0$, then the value of $2x^2-1$ is

Q16.If the matrix $A=\left(\begin{array}{cc}0& 2\\ K& -1\end{array}\right)$ satisfied $A\left(A^3+3I\right)=2I$, then the value of $K$ is

Q17.${\sum }_{\mathrm{k}=0}^{20}{\left({}^{20}{\mathrm{C}}_{\mathrm{k}}\right)}^2$ is equal to

Q18.${\int }_6^{16}\frac{{\log }_e{x}^2}{{\log }_e{x}^2+{\log }_e\left(x^2-44x+484\right)}dx$ is equal to

Q19.If $0<x<1$, then $\frac{3}{2}{x}^2+\frac{5}{3}{x}^3+\frac{7}{4}{x}^4+\dots$, is equal to

Q20.If $x^2+9y^2-4x+3=0,x,y\in R$, then $x$ and $y$ respectively lie in the intervals:

Q21.If $A=\{x\in R;|x-2|>1\},B=\left\{x\in R:\sqrt{x^2-3}>1\right\},C=\{x\in R;|x-4|\ge 2\}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^C\cap Z$ is

Q22.Let the equation $x^2+y^2+px+(1-p)y+5=0$ represent circles of varying radius $r\in (0,5]$. Then the number of elements in the set $S=\{q:q=p^2$ and $q$ is an integer $\}$ is

Q23.The number of distinct real roots of the equation $3x^4+4x^3-12x^2+4=0$ is

Q24.Let $\vec{a}=\hat{i}+5\hat{j}+\alpha \hat{k},\vec{b}=\hat{i}+3\hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\vec{b}\times \vec{c}|=5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}$. Then the greatest among the …

Q25.If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4a^2}=1$ and the coordinate axis is kab , then k is equal to

Q26.A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55 , is

Q27.If the system of linear equations
$2x+y-z=3$
$\mathrm{x}-\mathrm{y}-\mathrm{z}=\alpha$
$3x+3y+\beta z=3$
has infinitely many solution, then $\alpha +\beta -\alpha \beta$ is equal to

Q28.Let n be an odd natural number such that the variance of $1,2,3,4,\dots ,\mathrm{n}$ is 14 . Then n is equal to

Q29.If $\int \frac{dx}{{\left(x^2+x+1\right)}^2}=a{\tan }^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+b\left(\frac{2x+1}{x^2+x+1}\right)+C,x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3}a+b)$ is equal to

Q30.If $y^{1/4}+y^{-1/4}=2x$ and $\left(x^2-1\right)\frac{d^{2}y}{d{x}^2}+\alpha x\frac{dy}{dx}+\beta y=0$, then $|\alpha -\beta |$ is equal to