JEE Main 2021 · 26 Feb 2021 · Shift 2

Q1.A cord is wound round the circumference of wheel of radius $r$. The axis of the wheel is horizontal and the moment of inertia about it is I . A weight mg is attached to the cord at the end. The weight falls from rest. After falling through a distance ' h ', the square of angular …

Q2.The trajectory of a projectile in a vertical plane is $\mathrm{y}=\alpha \mathrm{x}-\beta {\mathrm{x}}^2$, where $\alpha$ and $\beta$ are constants and $x\&y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of pro …

Q3.The length of metallic wire is ${\ell }_1$ when tension in it is $T_1$. It is ${\ell }_2$ when the tension is $T_2$. The original length of the wire will be :

Q4.A radioactive sample is undergoing $\alpha$ decay. At any time $t_1$, its activity is $A$ and another time $t_2$, the activity is $\frac{\mathrm{A}}{5}$. What is the average life time for the sample?

Q5.Find the peak current and resonant frequency of the following circuit (as shown in figure).

Q6.A tuning fork A of unknown frequency produces 5 beats/s with a fork of known frequency 340 Hz. When fork A is filed, the beat frequency decreases to 2 beats/s. What is the frequency of fork A?

Q7.An inclined plane making an angle of $30^{\circ }$ with the horizontal is placed in a uniform horizontal electric field $200\frac{\mathrm{N}}{\mathrm{C}}$ as shown in the figure. A body of mass 1 kg and charge 5 mC is allowed to slide down from rest at a height of 1 m. If the co …

Q8.The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors $\vec{a},\vec{b}$ and $\vec{c}$ respectively. Then choose the correct relation for these vectors.

Q9.Two masses $A$ and $B$, each of mass $M$ are fixed together by a massless spring. A force acts on the mass B as shown in figure. If the mass $A$ starts moving away from mass $B$ with acceleration '$a$', then the acceleration of mass $B$ will be:

Q10.A particle executes S.H.M., the graph of velocity as a function of displacement is:

Q11.A scooter accelerates from rest for time $t_1$ at constant rate $a_1$ and then retards at constant rate $a_2$ for time $t_2$ and comes to rest. The correct value of $\frac{t_1}{t_2}$ will be:

Q12.If '$C$' and '$V$' represent capacity and voltage respectively then what are the dimensions of $\lambda$ where $\mathrm{C}/\mathrm{V}=\lambda$?

Q13.The internal energy ($U$), pressure ($P$) and volume ($V$) of an ideal gas are related as $\mathrm{U}=3\mathrm{PV}+4$. The gas is:

Q14.An aeroplane, with its wings spread 10 m, is flying at a speed of $180\mathrm{km}/\mathrm{h}$ in a horizontal direction. The total intensity of earth's field at that part is $2.5\times 10^{-4}\mathrm{Wb}/{\mathrm{m}}^2$ and the angle of dip is $60^{\circ }$. The emf induc …

Q15.Given below are two statements: one is labeled as Assertion A and the other is labeled as Reason R.
Assertion A: For a simple microscope, the angular size of the object equals the angular size of the image.
Reason R: Magnification is achieved as the small object can be kept much …

Q16.A wire of $1\Omega$ has a length of 1 m. It is stretched till its length increases by $25\%$. The percentage change in resistance to the nearest integer is:

Q17.Given below are two statements:
Statement I: A second's pendulum has a time period of 1 second.
Statement II: It takes precisely one second to move between the two extreme positions.
In the light of the above statements, choose the correct answer from the options given below:

Q18.The recoil speed of a hydrogen atom after it emits a photon in going from $n=5$ state to $n=1$ state will be :

Q19.Given below are two statements :
Statement I : An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero but the electric field is not zero anywhere in the sphere.
Statement II: If R is the radius of a solid metallic sphe …

Q20.Draw the output signal Y in the given combination of gates.

Q21.Time period of a simple pendulum is T . The time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{\alpha }{\beta }\mathrm{T}$. The value of $\alpha$ is ____

Q22.The volume V of a given mass of monoatomic gas changes with temperature T according to the relation $\mathrm{V}={\mathrm{kT}}^{\frac{2}{3}}$. The work done when temperature changes by 90 K will be xR . The value of x is. ____
[R = universal gas constant]

Q23.Two stream of photons, possessing energies equal to twice and ten times the work function of metal are incident on the metal surface successively. The value of ratio of maximum velocities of the photoelectrons emitted in the two respective cases is $\mathrm{x}:\mathrm{y}$. The …

Q24.A point source of light S , placed at a distance 60 cm infront of the centre of a plane mirror of width 50 cm , hangs vertically on a wall. A man walks infront of the mirror along a line parallel to the mirror at a distance 1.2 m from it (see in the figure). The distance between …

Q25.If the highest frequency modulating a carrier is 5 kHz , then the number of AM broadcast stations accommodated in a 90 kHz bandwidth are ____

Q26.1 mole of rigid diatomic gas performs a work of $\frac{Q}{5}$ when heat $Q$ is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{xR}{8}$. The value of $x$ is ____ [ $\mathrm{R}=$ universal gas constant]

Q27.A particle executes S.H.M. with amplitude ' $a$ ' and time period ' $T$ '. The displacement of the particle when its speed is half of maximum speed is $\frac{\sqrt{x}a}{2}$. The value of $x$ is ____

Q28.The zener diode has a ${\mathrm{V}}_{\mathrm{z}}=30\mathrm{V}$. The current passing through the diode for the following circuit is ____ mA .

Q29.27 similar drops of mercury are maintained at 10 V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is ____ times that of a smaller drop.

Q30.In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of $OA:AB$ will be $x:y$. The value of $x$ is ____

Q1.Match List - I with List - II.

Choose the correct answer from the options given below :

Q2.Identify A in the given reaction.

Q3.Which of the following forms of hydrogen emits low energy $\beta$- particles ?

Q4.Identify A in the following chemical reaction.

Q5.The nature of charge on resulting colloidal particles when ${\mathrm{FeCl}}_3$ is added to excess of hot water is:

Q6.Ceric ammonium nitrate and ${\mathrm{CHCl}}_3$ /alc. KOH are used for the identification of functional groups present in $\_\_\_\_$ and $\_\_\_\_$ respectively.

Q7.Calgon is used for water treatment. Which of the following statement is NOT true about Calgon?

Q8.In $\stackrel{1}{\mathrm{C}}{\mathrm{H}}_2=\stackrel{2}{\mathrm{C}}=\stackrel{3}{\mathrm{C}}\mathrm{H}-\stackrel{4}{\mathrm{C}}{\mathrm{H}}_3$ molecule, the hybridization of carbon $1,2,3$ and 4 respectively are :

Q9.Match List-I with List-II.

List - I
(a) Sucrose
(b) Lactose
(c) Maltose

List - II
(i) $\beta$-D-Galactose and $\beta$-D-Glucose
(ii) $\alpha$-D-Glucose and $\beta$-D-Fructose
(iii) $\alpha$-D-Glucose and $\alpha$-D-Glucose

Choose the correct answer from the options giv …

Q10.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : In $Tl{I}_3$, isomorphous to $Cs{I}_3$, the metal is present in +1 oxidation state.
Reason R : Tl metal has fourteen f electrons in the electronic configuratio …

Q11.The correct order of electron gain enthalpy is :

Q12.A. Phenyl methanamine
B. N, N-Dimethylaniline
C. N-Methyl aniline
D. Benzenamine

Choose the correct order of basic nature of the above amines.

Q13.Identify A in the given chemical reaction.

Q14.Match List - I with List - II.

List - I
(a) Sodium Carbonate
(b) Titanium
(c) Chlorine
(d) Sodium hydroxide

List - II
(i) Deacon
(ii) Castner-Kellner
(iii) van-Arkel
(iv) Solvay

Choose the correct answer from the options given below :

Q15.2,4-DNP test can be used to identify :

Q16.1) Zn / HCl
2) Cr₂O₃, 773 K
10-20 atm

Considering the above reaction, the major product among the following is :

Q17.Match List-I with List-II.

List-I (Molecule)
(a) Ne₂
(b) N₂
(c) F₂
(d) O₂

List-II (Bond order)
(i) 1
(ii) 2
(iii) 0
(iv) 3

Choose the correct answer from the options given below :

Q18.Which pair of oxides is acidic in nature?

Q19.Seliwanoff test and Xanthoproteic test are used for the identification of.......and.....respectively.

Q20.The average S-F bond energy in kJ mol⁻¹ of SF₆ is ______ (Rounded off to the nearest integer) [Given : The values of standard enthalpy of formation of SF₆(g), S(g) and F(g) are -1100, 275 and 80 kJ mol⁻¹ respectively.]

Q21.The number of octahedral voids per lattice site in a lattice is ______ (Rounded off to the nearest integer)

Q22.In mildly alkaline medium, thiosulphate ion is oxidized by ${\mathrm{MnO}}_4^{-}$ to "A". The oxidation state of sulphur in 'A' is ____

Q23.When 12.2 g of benzoic acid is dissolved in 100 g of water, the freezing point of solution was found to be $-0.93^{\circ }\mathrm{C}$ $[{\mathrm{K}}_{\mathrm{f}}({\mathrm{H}}_2\mathrm{O})=1.86\mathrm{K}\mathrm{kg}{\mathrm{mol}}^{-1}]$. The number (n) of benzoic acid molecules …

Q24.If the activation energy of a reaction is $80.9\mathrm{kJ}{\mathrm{mol}}^{-1}$, the fraction of molecules at 700 K, having enough energy to react to form products is $e^{-x}$. The value of $x$ is ____ (Rounded-off to the nearest integer).
[use …

Q25.A ball weighing 10 g is moving with a velocity of $90{\mathrm{ms}}^{-1}$. If the uncertainty in its velocity is $5\%$, then the uncertainty in its position is ____ $\times 10^{-33}\mathrm{m}$. (Rounded off to the nearest integer)
[Given: …

Q26.The pH of ammonium phosphate solution, if ${\mathrm{pk}}_{\mathrm{a}}$ of phosphoric acid and ${\mathrm{pk}}_{\mathrm{b}}$ of ammonium hydroxide are 5.23 and 4.75 respectively, is ____

Q27.The number of stereoisomers possible for

____
[ox = oxalate]

Q28.Emf of the following cell at 298 K in V is $x\times 10^{-2}$.

The value of $x$ is ____ (Rounded off to the nearest integer)
[Given: …

Q29.The ${\mathrm{NaNO}}_3$ weighed out to make 50 mL of an aqueous solution containing $70.0\mathrm{mg}{\mathrm{Na}}^{+}$ per mL is ____ g. (Rounded off to the nearest integer)
[Given; Atomic weight in $\mathrm{g}{\mathrm{mol}}^{-1}$ - Na: 23; N: 14; O: 16]

Q1.Let $f(x)={\int }_0^x{e}^{t}f(t)dt+e^x$ be a differentiable function for all $x\in R$. Then $f(x)$ equals :

Q2.If vectors ${\vec{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\vec{a}}_2=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

Q3.If the locus of the mid-point of the line segment from the point $(3,2)$ to a point on the circle, $x^2+y^2=1$ is a circle of radius $r$, then $r$ is equal to :

Q4.Let $f(x)={\sin }^{-1}x$ and $g(x)=\frac{x^2-x-2}{2x^2-x-6}$. If $g(2)={\lim }_{x\to 2}g(x)$, then the domain of the function fog is:

Q5.A seven digit number is formed using digits $3,3,4,4,4,5,5$. The probability, that number so formed is divisible by 2 , is :

Q6.Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime }(a)=2$ and $f(a)=4$. Then ${\lim }_{x\to a}\frac{xf(a)-af(x)}{x-a}$ equals :

Q7.If the mirror image of the point ( $1,3,5$ ) with respect to the plane $4x-5y+2z=8$ is ( $\alpha ,\beta ,\gamma$ ) then $5(\alpha +\beta +\gamma )$ equals :

Q8.The sum of the series ${\sum }_{n=1}^{\infty }\frac{n^2+6n+10}{(2n+1)!}$ is equal to :

Q9.Let $A=\{1,2,3,\dots ,10\}$ and $f:A\to A$ be defined as
$f(k)=\{\begin{array}{cc}k+1& \text{ if }k\text{ is odd }\\ k& \text{ if }k\text{ is even }\end{array}$
Then the number of possible functions $g:A\to A$ such that $g$ o …

Q10.Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{x{y}^2+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of $y$, for which the point ( $3,y$ ) lies on the curve, is :

Q11.A natural number has prime factorization given by $n=2^x{3}^y{5}^z$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6},y>z$. Then the number of odd divisors of $n$, including 1 , is :

Q12.Consider the following system of equations:

where $a,b$ and $c$ are real constants. Then the system of equations :

Q13.Let $L$ be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from $(3,2,1)$ on $L$, then the value of $21(\alpha +\beta +\gamma )$ equals :

Q14.For $x>0$, if $f(x)={\int }_1^x\frac{{\log }_{e}t}{(1+t)}dt$, then $f(e)+f\left(\frac{1}{e}\right)$ is equal to :

Q15.Let $f:R\to R$ be defined as

If $f(x)$ is continuous on $R$, …

Q16.Let $A(1,4)$ and $B(1,-5)$ be two points. Let $P$ be a point on the circle $(x-1{)}^2+(y-1{)}^2=1$ such that $(PA{)}^2+(PB{)}^2$ have maximum value, then the points $P,A$ and $B$ lie on :

Q17.Let $A_1$ be the area of the region bounded by the curves $y=\sin x,y=\cos x$ and $y$-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y=\sin x,y=\cos x,x$-axis and $x=\frac{\pi }{2}$ in the first quadrant. Then,

Q18.If $0<a,b<1$, and ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\pi }{4}$, then the value of $(a+b)-\left(\frac{a^2+b^2}{2}\right)+\left(\frac{a^3+b^3}{3}\right)-\left(\frac{a^4+b^4}{4}\right)+\dots \dots \dots \dots \dots$ is

Q19.The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is :

Q20.Let $F_1(A,B,C)=(A\wedge \sim B)\vee [\sim C\wedge (A\vee B)]\vee \sim A$ and $F_2(A,B)=(A\vee B)\vee (B\to \sim A)$ be two logical expressions. Then :

Q21.Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and $(2x{)}^2+(2y{)}^2=31$. Then the square of the slope of the line $L$ is ______

Q22.If the arithmetic mean and geometric mean of the $p^{\text{th }}$ and $q^{\text{th }}$ terms of the sequence $-16,8,-4,2,\dots$ satisfy the equation $4x^2-9x+5=0$, then $p+q$ is equal to ______

Q23.The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is ______

Q24.Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let $P_n=(\alpha {)}^n+(\beta {)}^n,P_{n-1}=11$ and $P_{n+1}=29$ for some integer $n\ge 1$. Then, the value of $p_n^2$ is ______

Q25.Let z be those complex numbers which satisfy
$|z+5|\le 4$ and $z(1+i)+\bar{z}(1-i)\ge -10,i=\sqrt{-1}$.
If the maximum value of $|z+1{|}^2$ is $\alpha +\beta \sqrt{2}$, then the value of $(\alpha +\beta )$ is ____

Q26.Let $a$ be an integer such that all the real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a,a+1)$. Then $|a|$ is equal to ____

Q27.Let $X_1,X_2,\dots X_{18}$ be eighteen observations such that ${\sum }_{i=1}^{18}\left(X_i-\alpha \right)=36$ and ${\sum }_{i=1}^{18}{\left(X_i-\beta \right)}^2=90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is 1, the …

Q28.If $I_{m,n}={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx$, for $m,n\ge 1$, and ${\int }_0^1\frac{x^{m-1}+x^{n-1}}{(1+x{)}^{m+n}}dx=\alpha I_{m,n},\alpha \in R$, then $\alpha$ equals.....

Q29.Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through $(3,-3)$ and $(4,-2\sqrt{2})$ and given that $a-2\sqrt{2}b=3$, then $\left(a^2+b^2+ab\right)$ is equal to ____

Q30.If the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 3& 0& -1\end{array}\right]$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\left[\begin{array}{lll}1& 0& 0\\ 0& 4& 0\\ 0& 0& 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then …