JEE Main 2023 · 30 Jan 2023 · Shift 1

Q1.The charge flowing in a conductor changes with time as $\mathrm{Q}(\mathrm{t})=\alpha \mathrm{t}-\beta {\mathrm{t}}^2+\gamma {\mathrm{t}}^3$. Where $\alpha ,\beta$ and $\gamma$ are constants. Minimum value of current is :

Q2.The pressure $(\mathrm{P})$ and temperature $(\mathrm{T})$ relationship of an ideal gas obeys the equation ${\mathrm{PT}}^2=$ constant. The volume expansion coefficient of the gas will be:

Q3.A person has been using spectacles of power- 1.0 diopter for distant vision and a separate reading glass of power 2.0 diopters. What is the least distance of distinct vision for this person:

Q4.As per the given figure, a small ball P slides down the quadrant of a circle and hits the other ball Q of equal mass which is initially at rest. Neglecting the effect of friction and assume the collision to be elastic, the velocity of ball Q after collision will be : …

Q5.Choose the correct relationship between Poisson ratio ( $\sigma$ ). bulk modulus ( K ) and modulus of rigidity $(\eta )$ of a given solid object:

Q6.The magnetic moments associated with two closely wound circular coils $A$ and $B$ of radius $r_A=10\mathrm{cm}$ and $r_B=20\mathrm{cm}$ respectively are equal if: (Where ${\mathrm{N}}_{\mathrm{A}},{\mathrm{I}}_{\mathrm{A}}$ and …

Q7.A small object at rest, absorbs a light pulse of power 20 mW and duration 300 ns . Assuming speed of light as $3\times 10^8\mathrm{m}/\mathrm{s}$. the momentum of the object becomes equal to :

Q8.Speed of an electron in Bohr's $7^{\text{th }}$ orbit for Hydrogen atom is $3.6\times 10^6\mathrm{m}/\mathrm{s}$. The corresponding speed of the electron in $3^{\text{rd }}$ orbit, in $\mathrm{m}/\mathrm{s}$ is :

Q9.A massless square loop, of wire of resistance $10\Omega$. supporting a mass of I g. hangs vertically with one of its sides in a uniform magnetic field of $10^3\mathrm{G}$, directed outwards in the shaded region. A dc voltage V is applied to the loop. For what value of V . …

Q10.Two isolated metallic solid spheres of radii $R$ and 2R are charged such that both have same charge density $\sigma$. The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is ${\sigma }^{\prime }$. The ratio …

Q11.Heat is given to an ideal gas in an isothermal process.
A. Internal energy of the gas will decrease.
B. Internal energy of the gas will increase.
C. Internal energy of the gas will not change.
D. The gas will do positive work.
E. The gas will do negative work.

Choose the correct …

Q12.Electric field in a certain region is given by $\vec{E}=\left(\frac{A}{x^2}\hat{i}+\frac{B}{y^3}\hat{j}\right)$. The SI unit of $A$ and $B$ are :

Q13.The output waveform of the given logical circuit for the following inputs A and B as shown below, is :

Q14.The height of liquid column raised in a capillary tube of certain radius when dipped in liquid A vertically is, 5 cm. If the tube is dipped in a similar manner in another liquid B of surface tension and density double the values of liquid A, the height of liquid column raised in …

Q15.A sinusoidal carrier voltage is amplitude modulated. The resultant amplitude modulated wave has maximum and minimum amplitude of 120 V and 80 V respectively. The amplitude of each sideband is :

Q16.In a series LR circuit with $X_L=R$, power factor is $P_1$. If a capacitor of capacitance C with $X_C=X_L$ is added to the circuit the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be :

Q17.If the gravitational field in the space is given as $\left(-\frac{K}{r^2}\right)$. Taking the reference point to be at $r=2$ cm with gravitational potential $V=10$ J/kg. Find the gravitational potential at $r=3$ cm in SI unit (Given, that $K=6$ J cm/kg)

Q18.A ball of mass 200 g rests on a vertical post of height 20 m. A bullet of mass 10 g, travelling in horizontal direction, hits the centre of the ball. After collision both travels independently. The ball hits the ground at a distance 30 m and the bullet at a distance of 120 m from …

Q19.Match Column-I with Column-II :

Choose the correct answer from the options given below:

Q20.The figure represents the momentum time (p-t) curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively?
If $(t_3-t_2)<t_1$.

Q21.The general displacement of a simple harmonic oscillator is $x=A\sin \omega t$. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when $t=\frac{T}{\beta }$. The value of $\beta$ is ______.

Q22.A capacitor of capacitance $900\mu$F is charged by a 100 V battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capaci …

Q23.In Young's double slit experiment, two slits $S_1$ and $S_2$ are 'd' distance apart and the separation from slits to screen is D (as shown in figure). Now if two transparent slabs of equal thickness 0.1 mm but refractive index 1.51 and 1.55 are introduced in the path of beam ( …

Q24.In the following circuit, the value of $2I_1$ (twice the magnitude of the current $I_1$) is ______ A.

Q25.A horse rider covers half the distance with $5\mathrm{m}/\mathrm{s}$ speed. The remaining part of the distance was travelled with speed $10\mathrm{m}/\mathrm{s}$ for half the time and with speed $15\mathrm{m}/\mathrm{s}$ for other half of the time. The mean speed of t …

Q26.A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of 24 W. The radius of curvature of hemisphere is 10 cm and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is …

Q27.As per the given figure, if $\frac{\mathrm{dI}}{\mathrm{dt}}=-1\mathrm{A}/\mathrm{s}$ then the value of ${\mathrm{V}}_{\mathrm{AB}}$ at this instant will be $\_\_\_\_$ V.

Q28.In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by 0.5 mm on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a …

Q29.In an experiment for estimating the value of focal length of converging mirror, image of an object placed at 40 cm from the pole of the mirror is formed at distance 120 cm from the pole of the mirror. These distances are measured with a modified scale in which there are 20 small …

Q30.A thin uniform rod of length 2 m, cross sectional area 'A' and density 'd' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $\omega$. If value of $\omega$ in terms of its rotational kinetic energy $E$ is …

Q1.Which of the following compounds would give the following set of qualitative analysis?
(i) Fehling's Test : Positive
(ii) Na fusion extract upon treatment with sodium nitroprusside gives a blood red colour but not

Q2.What is the correct order of acidity of the protons marked A-D in the given compounds?

Q3.Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Ketoses give Seliwanoff's test faster than Aldoses.

Reason (R) : Ketoses undergo β-elimination followed by formation of furfural.

In the light of the above …

Q4.In the extraction of copper, its sulphide ore is heated in a reverberatory furnace after mixing with silica to :

Q5.Amongst the following compounds, which one is an antacid?

Q6.The major products 'A' and 'B', respectively, are

Q7.Benzyl isocyanide can be obtained by :
(A) (see reaction)
(B) (see reaction)
(C) (see reaction)
(D) (see reaction)

Choose the correct answer from the options given below :

Q8.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In expensive scientific instruments, silica gel is kept in watch-glasses or in semipermeable membrane bags.

Reason (R) : Silica gel adsorbs moisture from a …

Q9.Caprolactam when heated at high temperature in presence of water, gives

Q10.The alkaline earth metal sulphate(s) which are readily soluble in water is/are:
(A) ${\mathrm{BeSO}}_4$
(B) ${\mathrm{MgSO}}_4$
(C) ${\mathrm{CaSO}}_4$
(D) ${\mathrm{SrSO}}_4$
(E) ${\mathrm{BaSO}}_4$

Choose the correct answer from the options given below:

Q11.Which of the following is correct order of ligand field strength?

Q12.Formation of photochemical smog involves the following reaction in which $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are respectively.
(i) ${\mathrm{NO}}_2\stackrel{h\nu }{\to }\mathrm{A}+\mathrm{B}$
(ii) $\mathrm{B}+{\mathrm{O}}_2\to \mathrm{C}$
(iii) …

Q13.During the qualitative analysis of ${\mathrm{SO}}_3^{2-}$ using dilute ${\mathrm{H}}_2{\mathrm{SO}}_4,{\mathrm{SO}}_2$ gas is evolved which turns ${\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$ solution (acidified with dilute ${\mathrm{H}}_2{\mathrm{SO}}_4$ ):

Q14.To inhibit the growth of tumours, identify the compounds used from the following:
(A) EDTA
(B) Coordination Compounds of Pt
(C) D - Penicillamine
(D) Cis-Platin

Choose the correct answer from the option given below:

Q15.In the wet tests for identification of various cations by precipitation, which transition element cation doesn't belong to group IV in qualitative inorganic analysis?

Q16.For ${\mathrm{OF}}_2$ molecule consider the following:
(A) Number of lone pairs on oxygen is 2.
(B) FOF angle is less than $104.5^{\circ }$.
(C) Oxidation state of O is -2.
(D) Molecule is bent 'V' shaped.
(E) Molecular geometry is linear.

Correct options are:

Q17.Lithium aluminium hydride can be prepared from the reaction of

Q18.Consider the cell
${\mathrm{Pt}}_{(\mathrm{s})}\left|{\mathrm{H}}_2(\mathrm{g},1\mathrm{atm})\right|{\mathrm{H}}^{+}(\mathrm{aq},1\mathrm{M})||{\mathrm{Fe}}^{3+}(\mathrm{aq}),{\mathrm{Fe}}^{2+}(\mathrm{aq})\mid \mathrm{Pt}(\mathrm{s})$
When the potential of the cell is 0.712 …

Q19.A 300 mL bottle of soft drink has $0.2\mathrm{M}{\mathrm{CO}}_2$ dissolved in it. Assuming ${\mathrm{CO}}_2$ behaves as an ideal gas, the volume of the dissolved ${\mathrm{CO}}_2$ at STP is ______ mL. (Nearest integer)

Given: At STP, molar volume of an ideal gas is …

Q20.A solution containing 2 g of a non-volatile solute in 20 g of water boils at 373.52 K. The molecular mass of the solute is ______ $\mathrm{g}{\mathrm{mol}}^{-1}$. (Nearest integer)

Given, water boils at $373\mathrm{K},{\mathrm{K}}_{\mathrm{b}}$ for water …

Q21.If compound A reacts with B following first order kinetics with rate constant $2.011\times 10^{-3}{\mathrm{s}}^{-1}$. The time taken by A (in seconds) to reduce from 7 g to 2 g will be ______. (Nearest Integer)
$[\log 5=0.698,\log 7=0.845,\log 2=0.301]$

Q22.The energy of one mole of photons of radiation of frequency $2\times 10^{12}\mathrm{Hz}$ in $\mathrm{J}{\mathrm{mol}}^{-1}$ is ______.
(Nearest integer)
(Given: $\mathrm{h}=6.626\times 10^{-34}\mathrm{Js}$ …

Q23.The number of electrons involved in the reduction of permanganate to manganese dioxide in acidic medium is ______.

Q24.When 2 litre of ideal gas expands isothermally into vacuum to a total volume of 6 litre, the change in internal energy is ______ J. (Nearest integer)

Q25.600 mL of 0.01 M HCl is mixed with 400 mL of $0.01\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$. The pH of the mixture is ______ $\times 10^{-2}$. (Nearest integer)
[Given $\log 2=0.30,\log 3=0.48$
$\log 5=0.69$
$\log 7=0.84$
$\log 11=1.04]$

Q26.A trisubstituted compound 'A', ${\mathrm{C}}_{10}{\mathrm{H}}_{12}{\mathrm{O}}_2$ gives neutral ${\mathrm{FeCl}}_3$ test positive. Treatment of compound 'A' with NaOH and ${\mathrm{CH}}_3\mathrm{Br}$ gives ${\mathrm{C}}_{11}{\mathrm{H}}_{14}{\mathrm{O}}_2$, with hydroiodic a …

Q27.Some amount of dichloromethane $\left(\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{l}}_2\right)$ is added to 671.141 mL of chloroform $\left(\mathrm{CHC}{\mathrm{l}}_3\right)$ to prepare a $2.6\times 10^{-3}m$ (molal) solution of $\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{l}}_2$ (DCM). The concentration of DCM is ______ ppm (by mass).

Given: Atomic …

Q28.Match List I with List II:
LIST-I (molecules/ions): A. IF₇ B. ICl₄⁻ C. XeF₆ D. XeF₂
LIST-II (No. of lone pairs of e⁻ on central atom): I. Three II. One III. Two IV. Zero
Choose the correct answer from the options given below:

Q1.Let $A=\left(\begin{array}{cc}m& n\\ p& q\end{array}\right),d=|A|\ne 0,|A-d(\mathrm{Adj}A)|=0$. Then

Q2.If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

Q3.Let the system of linear equations
$x+y+kz=2$
$2x+3y-z=1$
$3x+4y+2z=k$
have infinitely many solutions. Then the system
$(k+1)x+(2k-1)y=7$
$(2k+1)x+(k+5)y=10$ has:

Q4.If $\tan 15^{\circ }+\frac{1}{\tan 75^{\circ }}+\frac{1}{\tan 105^{\circ }}+\tan 195^{\circ }=2a$, then the value of $\left(a+\frac{1}{a}\right)$ is:

Q5.Suppose $f:\mathbb{R}\to (0,\infty )$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y),\forall x,y\in \mathbb{R}$. If $f(3)=320$, then ${\sum }_{n=0}^{5}f(n)$ is equal to:

Q6.If $a_n=\frac{-2}{4n^2-16n+15}$, then $a_1+a_2+\dots +a_{25}$ is equal to:

Q7.If the coefficient of $x^{15}$ in the expansion of ${\left(a{x}^3+\frac{1}{b{x}^{\frac{1}{3}}}\right)}^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of ${\left(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^3}\right)}^{15}$, where $a$ and $b$ are positive real numbers, th …

Q8.If $\vec{a},\vec{b},\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a}=\alpha \vec{b}-\hat{n}$, $(\alpha \ne 0)$ and $\vec{b}\cdot \vec{c}=12$, then $|\vec{c}\times (\vec{a}\times \vec{b})|$ is equal …

Q9.The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is:

Q10.Let $y=x+2,4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h{)}^2+(y-k{)}^2=r^2$. Then $h+k$ is equal to:

Q11.Let the solution curve $y=y(x)$ of the differential equation

pass through the origin. Then $y(1)$ is equal to:

Q12.Let a unit vector $\overrightarrow{\mathrm{OP}}$ make angle $\alpha ,\beta ,\gamma$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\beta \in \left(0,\frac{\pi }{2}\right)$. $\overrightarrow{\mathrm{OP}}$ is perpendicular to the plane t …

Q13.If $[t]$ denotes the greatest integer $\le t$, then the value of $\frac{3(\mathrm{e}-1)}{\mathrm{e}}{\int }_1^2{\mathrm{x}}^2{\mathrm{e}}^{[\mathrm{x}]+[{\mathrm{x}}^3]}\mathrm{dx}$ is:

Q14.If $\mathrm{P}(\mathrm{h},\mathrm{k})$ be point on the parabola $\mathrm{x}=4{\mathrm{y}}^2$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of P from the directrix of the parabola $y^2=4(x+y)$ is equal to:

Q15.The coefficient of $x^{301}$ in $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\dots +x^{500}$ is:

Q16.Among the statements:
$(S1):((p\vee q)\Rightarrow r)\iff (p\Rightarrow r)$
$(S2):((p\vee q)\Rightarrow r)\iff ((p\Rightarrow r)\vee (q\Rightarrow r))$

Q17.The minimum number of elements that must be added to the relation $\mathrm{R}=\{(\mathrm{a},\mathrm{b}),(\mathrm{b},\mathrm{c})\}$ on the set $\{\mathrm{a},\mathrm{b},\mathrm{c}\}$ so that it becomes symmetric and transitive is:

Q18.If the solution of the equation ${\log }_{\cos x}\cot x+4{\log }_{\sin x}\tan x=1,x\in \left(0,\frac{\pi }{2}\right)$, is ${\sin }^{-1}\left(\frac{\alpha +\sqrt{\beta }}{2}\right)$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to:

Q19.Let $S=\{1,2,3,4,5,6\}$. Then the number of one-one functions $f:S\to P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n)\subset f(m)$ where $\mathrm{n}<\mathrm{m}$ is ______.

Q20.If ${\lambda }_1$ and ${\lambda }_2$ are two values of $\lambda$ such that the angle between the planes $P_1:\vec{r}\cdot (3\hat{i}-5\hat{j}+\hat{k})=7$ and $P_2:\vec{r}\cdot (\lambda \hat{i}+\hat{j}-3\hat{k})=9$ is ${\sin }^{-1}\left(\frac{2\sqrt{6}}{5}\right)$, the …

Q21.Let $\mathrm{z}=1+\mathrm{i}$ and ${\mathrm{z}}_1=\frac{1+\mathrm{i}\overline{\mathrm{z}}}{\overline{\mathrm{z}}(1-\mathrm{z})+\frac{1}{\mathrm{z}}}$. Then $\frac{12}{\pi }\arg \left({\mathrm{z}}_1\right)$ is equal to ______.

Q22.${\lim }_{x\to 0}\frac{48}{x^4}{\int }_0^x\frac{t^3}{t^6+1}dt$ is equal to ____.

Q23.The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and $a$ and $b$ are respectively mean and variance of remaining 6 observation, then $a+3b-5$ is equal to ____.

Q24.If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3y+2z-1=0$, $4x-y+z=0$ is $Ax+By+Cz=1$, then $140(C-B+A)$ is equal to ____.

Q25.Let ${\sum }_{n=0}^{\infty }\frac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}=ae+\frac{b}{e}+c$,
where $a,b,c\in \mathbb{Z}$ and $e={\sum }_{n=0}^{\infty }\frac{1}{n!}$. Then $a^2-b+c$ is equal to ____.

Q26.Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ____.

Q27.Let $f^1(x)=\frac{3x+2}{2x+3},x\in \mathbb{R}-\left\{\frac{-3}{2}\right\}$

For $n\ge 2$, define $f^n(x)=f^1\circ f^{n-1}(x)$.
If $f^5(x)=\frac{ax+b}{bx+a}$, $\gcd (a,b)=1$, then $a+b$ is equal to ____.