JEE Main 2022 July 25 Shift 1 Question Paper with Solution PDF- Download Here

The total number of functions, f : {1, 2, 3, 4} \(\rightarrow\) {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to

• (1) 60
• (2) 90
• (3) 108
• (4) 126

If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x ^4+ x ^3+ x ^2+ x +1=0\), then \(\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}\) is equal to

• (1) \(-4\)
• (2) \(-1\)
• (3) 1
• (4) 4

For \(n \in N\), let \(S _{ n }=\left\{ z \in C :| z -3+2 i |=\frac{ n }{4}\right\}\) and \(T_n=\left\{z \in C:|z-2+3 i|=\frac{1}{n}\right\}\) Then the number of elements in the set \(\left\{ n \in N : S _{ n } \cap T _{ n }=\phi\right\}\) is :

• (1) 0
• (2) 2
• (3) 3
• (4) 4

• (1) 6
• (2) 7
• (3) 8
• (4) 9

If \(\lim_{n\rightarrow \infty}\)(\(\sqrt{n^2-n-1}\) + nα+β)=0 then 8(α + β) is equal to

• (1) 4
• (2) -8
• (3) -4
• (4) 8

If the absolute maximum value of the function
f(x)=(x²–2x+7)e(4x³−12x²−180x+31) in the interval [–3, 0] is f(α), then

• (1) α = 0
• (2) α = –3
• (3) α∈ (–1, 0)
• (4) α∈ (–3, –1]

• (1) \(\frac{27}{4}\)
• (2) \(\frac{29}{4}\)
• (3) \(\frac{37}{4}\)
• (4) \(\frac{9}{2}\)

• (1) -4
• (2) -1
• (3) 1
• (4) 4

• (1) 4
• (2) 2
• (3) 1
• (4) 0

The slope of the tangent to a curve C: y = y(x) at any point (x, y) on it is \(\frac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}\).If C passes through the points (0,\(\frac{1}{2}\)+\(\frac{\pi}{2\sqrt2}\)) and (α,\(\frac{1}{2e^{2\alpha}}\)), then e^α is equal to

• (1) \(\frac{3+√2}{3-√2}\)
• (2) \(\frac{3}{\sqrt2}\)(\(\frac{3+√2}{3-√2}\))
• (3) \(\frac{1}{\sqrt2}\)(\(\frac{\sqrt2+1}{\sqrt2-1}\))
• (4) (\(\frac{\sqrt2+1}{\sqrt2-1}\))

The general solution of the differential equation (x – y²)dx + y(5x + y²)dy = 0 is:

• (1) (y²+x)4=C|(y²+2x)³|
• (2) (y²+2x)4=C|(y²+x)³|
• (3) |(y²+x)³|=C(2y²+x)⁴
• (4) |(y²+2x)³|=C(2y²+x)⁴

A line, with the slope greater than one, passes through the point A (4,3) and intersects the line x-y-2=0 at the point B If the length of the line segment AB is \(\frac{\sqrt{29}}{3},\) then B also lies on the line :

• (1) 2 x+y=9
• (2) 3 x-2 y=7
• (3) x+2 y=6
• (4) 2 x -3 y =3

Let the locus of the centre \((\alpha, \beta), \) \(\beta>0\), of the circle which touches the circle \(x ^2+( y -1)^2=1\) externally and also touches the x-axis be L Then the area bounded by L and the line y=4 is :

• (1) \(\frac{32 \sqrt{2}}{3}\)
• (2) \(\frac{40 \sqrt{2}}{3}\)
• (3) \(\frac{64}{3}\)
• (4) \(\frac{32}{3}\)

Let ABC be a triangle such that \(\vec{BC}\)=\(\vec a\),\(\vec{CA} \)=\(\vec b\),\(\vec AB\)=\(\vec c\),|\(\vec a\)|=6\(\sqrt2\),|\(\vec b\)|=2\(\sqrt3\) and \(\vec b\)⋅\(\vec c\)=12 Consider the statements:
(S₁):|(\(\vec a\)×\(\vec b\))+(\(\vec c\)×\(\vec d\))|−|\(\vec c\)|=6(2\(\sqrt2\)−1)
(S₂):\(\angle\)ACB=cos⁻¹⁡(\(\sqrt{\frac{2}{3}}\))
Then

• (1) Both (S1) and (S2) are true
• (2) Only (S1) is true
• (3) Only (S2) is true
• (4) Both (S1) and (S2) are false

• (1) \(\frac{17}{36}\)
• (2) \(\frac{4}{9}\)
• (3) \(\frac{1}{2}\)
• (4) \(\frac{19}{36}\)

Let P be the plane containing the straight line\(\frac{ x−3}{9}\)=\(\frac{y+4}{-1}\)=\(\frac{z-7}{-5}\)and perpendicular to the plane containing the straight lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\) and \(\frac{x}{3}=\frac{y}{7}=\frac{z}{8}\).If d is the distance P from the point (2, –5, 11), then d² is equal to :

• (1) \(\frac{147}{2}\)
• (2) 96
• (3) \(\frac{32}{3}\)
• (4) 54

• (1) \(\frac{33}{2^{32}}\)
• (2) \(\frac{33}{2^{29}}\)
• (3) \(\frac{33}{2^{28}}\)
• (4) \(\frac{33}{2^{27}}\)

• (1) 4
• (2) 6
• (3) 8
• (4) 12

• (1) 5(2\(\sqrt3\)+3)m
• (2) 5(\(\sqrt3\)+3)m
• (3) 10(\(\sqrt3\)+1)m
• (4) 10(2\(\sqrt3\)+1)m

Which of the following statements is a tautology?

• (1) \(((∼p)∨q)⇒p\)
• (2) \(p⇒((∼p)∨q)\)
• (3) \(((∼p)∨q)⇒q\)
• (4) \(q⇒((∼p)∨q)\)

Let A=\(\begin{pmatrix} 2 &-1  &-1 \\   1&0  &-1 \\   1&-1  &0  \end{pmatrix}\) and B=A–I. If ω=\(\frac{\sqrt3 i-1}{2}\), then the number of elements in the set {n∈{1,2,⋯,100}:\(A^n+(ωB)^n\)=A+B} is equal to _______.

If the maximum value of the term independent of t in the expansion of (t²x\(^{\frac{1}{5}}\)+\(\frac{(1−x)^{\frac{1}{10}}}{t}\)),x≥0 is K, then 8 K is equal to__________.

Let a, b be two non-zero real numbers. If p and r are the roots of the equation x² – 8ax + 2a = 0 and q and s are the roots of the equation x² + 12bx + 6b = 0, such that \(\frac{1}{p}\),\(\frac{1}{q}\),\(\frac{1}{r}\),\(\frac{1}{s}\) are in A.P., then a^–1 – b^–1 is equal to _________.

Let f(x)=\(\left\{\begin{matrix}  |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\   |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.\) where [α] denotes the greatest integer less than or equal to α.Then the number of points in R where f is not differentiable is

Let a₁ = b₁ = 1, a_n = a_{n – 1} + 2 and b_n = a_a + b_{n – 1} for every natural number n ≥ 2. Then\(\sum_{n=1}^{15}\) \(a_n⋅b_n\) is equal to ______.

If \(\lim_{n\rightarrow \infty}\) \(\frac{(n+1)^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+....+(nk+n)]=33.\lim_{n\rightarrow \infty}\frac{1}{n^{k+1}}.[1^k+2^k+3^k+....+n^k]\)
then the integral value of k is equal to _______.

The sum of diameters of the circles that touch (i) the parabola \(75x^2 \)= \(64(5y – 3)\) at the point (\(\frac{8}{5}\), \(\frac{6}{5}\)) and (ii) the y-axis is equal to _______.
 

The line of the shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of cos⁻¹⁡\(\sqrt{\frac{2}{27}}\) with the plane P: ax – y – z = 0, (a> 0). If the image of the point (1, 1, –5) in the plane P is (α, β, γ), then α + β – γ is equal to ________.

If momentum \([ P ]\), area \([ A ]\) and time \([ T ]\) are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

• (1) \(\left[ P A ^{-1} T ^0\right]\)
• (2) \(\left[ P A T ^{-1}\right]\)
• (3) \(\left[ PA ^{-1} T \right]\)
• (4) \(\left[ P A ^{-1} T ^{-1}\right]\)

Which of the following physical quantities have the same dimensions?

• (1) Electric displacement (\(\vec D\)) and surface charge density
• (2) Displacement current and electric field
• (3) Current density and surface charge density
• (4) Electric potential and energy

A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is 60 m, then the magnitude of displacement would be
(Given cos135°= –0.7)

• (1) 42 m
• (2) 47 m
• (3) 19 m
• (4) 40 m

A body of mass 0.5 kg travels on straight line path with velocity v = (3x² + 4) m/s. The network done by the force during its displacement from x = 0 to x = 2 m is

• (1) 64 J
• (2) 60 J
• (3) 120 J
• (4) 128 J

A solid cylinder and a solid sphere, having same mass M and radius R, roll down the same inclined plane from top without slipping They start from rest The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be :

• (1) \(\sqrt{\frac{5}{3}}\)
• (2) \(\sqrt{\frac{4}{5}}\)
• (3) \(\sqrt{\frac{3}{5}}\)
• (4) \(\sqrt{\frac{14}{15}}\)

Three identical particles A, B and C of mass 100 kg each are placed in a straight line with AB = BC = 13 m. The gravitational force on a fourth particle P of the same mass is F, when placed at a distance 13 m from the particle B on the perpendicular bisector of the line AC. The value of F will be approximately

• (1) 21G
• (2) 100G
• (3) 59G
• (4) 42G

A certain amount of gas of volume V at 27°C temperature and pressure 2 × 10⁷ Nm^–2 expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use, γ = 1.5)

• (1) \(3.536\times10^5 \)Pa
• (2) \(3.536\times10^6\) Pa
• (3) \(1.25\times10^6\) Pa
• (4) \(1.25\times10^5 \)Pa

Following statements are given:
(A) The average kinetic energy of a gas molecule decreases when the temperature is reduced.
(B) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature.
(C) The average kinetic energy of a gas molecule decreases with increase in volume.
(D) Pressure of a gas increases with increase in temperature at constant pressure.
(E) The volume of gas decreases with increase in temperature.
Choose the correct answer from the options given below:

• (1) (A) & (D) only
• (2) (A), (B) & (D) only
• (3) (B) & (D) only
• (4) (A), (B) & (E) only

In figure (A), mass ‘2 m‘ is fixed on mass ‘m‘, which is attached to two springs of spring constant k. In figure (B), mass ‘m‘ is attached to two springs of spring constant ‘k‘ and ‘2k‘. If mass ‘m‘ in (A) and in (B) are displaced by distance’ x’ horizontally and then released, then time period T1 and T2 corresponding to (A) and (B) respectively follow the relation.

• (1) \(\frac{T_1}{T_2}=\frac{3}{\sqrt2}\)
• (2) \(\frac{T_1}{T_2}=\sqrt{\frac{3}{2}}\)
• (3) \(\frac{T_1}{T_2}=\sqrt{\frac{2}{3}}\)
• (4) \(\frac{T_1}{T_2}=\frac{\sqrt2}{3}\)

A condenser of 2μF capacitance is charged steadily from 0 to 5 C.Which of the following graph represents correctly the variation of potential difference (V) across it’s plates with respect to the charge (Q) on the condenser?

• (1)
• (2)
• (3)
• (4)

Two charged particles, having same kinetic energy, are allowed to pass through a uniform magnetic field perpendicular to the direction of motion If the ratio of radii of their circular paths is \(6: 5\) and their respective masses ratio is \(9: 4\) Then, the ratio of their charges will be :

• (1) \(8: 5\)
• (2) \(5: 4\)
• (3) \(5: 3\)
• (4) \(8: 7\)

To increase the resonant frequency in series LCR circuit-

• (1) Source frequency should be increased
• (2) Another resistance should be added in series with the first resistance.
• (3) Another capacitor should be added in series with the first capacitor.
• (4) The source frequency should be decreased.

A small square loop of wire of side l is placed inside a large square loop of wire L(L>>l). As shown in figure, both loops are coplanar and their centers coincide at point O. The mutual inductance of the system is:

• (1) \(\frac{2\sqrt2 \mu_0L^2}{\pi l}\)
• (2) \(\frac{\mu_0L^2}{2\sqrt2 \pi L}\)
• (3) \(\frac{2\sqrt 2 \mu_0 l^2}{\pi L}\)
• (4) \(\frac{\mu_0L^2}{2\sqrt2\pi l}\)

The rms value of the conduction current in a parallel plate capacitor is 6.9 μA. The capacity of this capacitor, if it is connected to 230 V ac supply with an angular frequency of 600 rad/s, will be :

• (1) 5 pF
• (2) 50 pF
• (3) 100 pF
• (4) 200 pF

Which of the following statement is correct ?

• (1) In primary rainbow, observer sees red colour on the top and violet on the bottom
• (2) In primary rainbow, observer sees violet colour on the top and red on the bottom
• (3) In primary rainbow, light wave suffers total internal reflection twice before coming out of water drops
• (4) Primary rainbow is less bright than secondary rainbow.

Time taken by light to travel in two different materials A and B of refractive indices μ_A and μ_B of same thickness is t₁ and t₂ respectively. If t₂ – t₁ = 5 × 10^–10 s and the ratio of μA to μB is 1 : 2. Then, the thickness of material, in meter is: (Given v_A and v_B are velocities of light in A and B materials, respectively.)

• (1) 5 × 10^-10 v_A m
• (2) 5 × 10^-10 m
• (3) 1.5 × 10^-10 m
• (4) 5 × 10^-10 v_B m

A metal exposed to light of wavelength \(800\, nm\) and emits photoelectrons with a certain kinetic energy The maximum kinetic energy of photo-electron doubles when light of wavelength \(500\, nm\) is used The work function of the metal is (Take \(hc =1230 \,eV - nm )\)

• (1) \(1.537 \,eV\)
• (2) \(2.46 \,eV\)
• (3) \(0.615\, eV\)
• (4) \(1.23 \,eV\)

The momentum of an electron revolving in n^th orbit is given by: (Symbols have their usual meanings)

• (1) \(\frac{nh}{2πr}\)
• (2) \(\frac{nh}{2r}\)
• (3) \(\frac{nh}{2\pi}\)
• (4) \(\frac{2πr}{nh}\)

The magnetic moment of an electron (e) revolving in an orbit around nucleus with an orbital angular momentum is given by:

• (1) \(\vec{\mu L}\)=\(\frac{\vec{eL}}{2m}\)
• (2) \(\vec{\mu L}\)=\(-\frac{\vec{eL}}{2m}\)
• (3) \(\vec{\mu L}\)=−\(\frac{\vec{eL}}{m}\)
• (4) \(\vec{\mu L}\)=\(\frac{\vec{2eL}}{m}\)

In the circuit, the logical value of A = 1 or B = 1 when potential at A or B is 5 V and the logical value of A = 0 or B = 0 when potential at A or B is 0 V. The truth table of the given circuit will be:

 

• (1) A B Y
  0 0 0
  1 0 0
  0 1 0
  1 1 1
• (2) A B Y 
  0 0 0 
  1 0 1 
  0 1 1 
  1 1 1
• (3) A B Y 
  0 0 0 
  1 0 0 
  0 1 0 
  1 1 0
• (4) A B Y 
  0 0 1 
  1 0 1 
  0 1 1 
  1 1 0

A car is moving with speed of 150 km/h and after applying the break it will move 27 m before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling ___ m distance.

Four forces are acting at a point \(P\) in equilibrium as shown in figure.The ratio of force \(F_1\) to \(F_2\) is \(1: x\) where \(x=\) _____

A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force F, its length increases by 5 cm. Another wire of the same material of length 4L and radius 4r is pulled by a force 4F under same conditions. The increase in length of this wire is ___ cm.

A unit scale is to be prepared whose length does not change with temperature and remains \(20\, cm\), using a bimetallic strip made of brass and iron each of different length The length of both components would change in such a way that difference between their lengths remains constant If length of brass is \(40 \,cm\) and length of iron will be ________\(cm\)\((\alpha_{iron}=1.2\times10^{-5}K^{-1} \,\,\,and \,\,\,\alpha_{brass}=1.8\times10^{-5}K^{-1}).\)

The volume charge density of a sphere of radius 6 m is 2μC cm^-3. The number of lines of force per unit surface area coming out from the surface of the sphere is _______ × 10¹⁰ NC^-1.

In the given figure, the value of V₀ will be ______ V.

Eight copper wire of length l and diameter d are joined in parallel to form a single composite conductor of resistance R. If a single copper wire of length 2l have the same resistance (R) then its diameter will be ________ d.

The energy band gap of semiconducting material to produce violet (wavelength = \(4000 \mathring{A}\) ) LED is ______\(eV\) (Round off to the nearest integer)

An observer is riding on a bicycle and moving towards a hill at \(18 \,kmh ^{-1}\) He hears a sound from a source at some distance behind him directly as well as after its reflection from the hill If the original frequency of the sound as emitted by source is \(640\, Hz\) and velocity of the sound in air is \(320\, m / s\), the beat frequency between the two sounds heard by observer will be _______Hz

The required height of a TV tower which can cover the population of \(603\) lakh is \(h\) If the average population density is \(100\) per square \(km\) and the radius of earth is \(6400\, km\), then the value of \(h\) will be ______\(m\)

\(SO _2 Cl _2\) on reaction with excess of water results into acidic mixture \(SO _2 Cl _2+2 H _2 O \rightarrow H _2 SO _4+2 HCl\) 16 moles of \(NaOH\) is required for the complete neutralisation of the resultant acidic mixture The number of moles of \(SO _2 Cl _2\) used is :

• (1) 16
• (2) 8
• (3) 4
• (4) 2

• (1) \(n=3,l=2,m_l=0,s=+\frac{1}{2}\)
• (2) \(n=3,l=2,m_l=-2,s=+\frac{1}{2}\)
• (3) \(n=3,l=3,m_l=-3,s=-\frac{1}{2}\)
• (4) \(n=3,l=0,m_l=0,s=-\frac{1}{2}\)

• (1) 0.8
• (2) 1.1
• (3) 1.9
• (4) 2.4

20 mL of 0.1 M NH₄OH is mixed with 40 mL of 0.05 M HCl.The pH of the mixture is nearest to:(Given: K_b(NH₄OH) = 1 × 10^–5 ,log 2 = 0.30,log 3 = 0.48, log 5 = 0.69, log 7 = 0.84, log 11 =1.04)

• (1) 3.2
• (2) 4.2
• (3) 5.2
• (4) 6.2

• (1) Unnilbium
• (2) Unnilunium
• (3) Unnilquadium
• (4) Unniltrium

A compound A on reaction with X and Y produces the same major product but different by product a and b Oxidation of  a gives a substance produced by ants X and Y respectively are :

• (1) \(KMnO _4 / H ^{+}\)and dil. \(KMnO _4, 273 \,K\)
• (2) \(KMnO _4\), (dilute), \(273 K\) and \(KMnO _4 / H ^{+}\)
• (3) \(KMnO _4 / H ^{+}\)and \(O _3, H _2 O / Zn\)
• (4) \(O _3, H _2 O / Zn\) and \(KMnO _4 / H ^{+}\)

Most stable product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

• (1)
• (2)
• (3)
• (4)

• (1)
• (2)
• (3)
• (4)

• (1) Primary
• (2) Secondary
• (3) Tertiary
• (4) Quaternary

Given below are two statements:
Statement I: On heating with KHSO₄, glycerol is dehydrated and acrolein is formed.
Statement II: Acrolein has fruity odour and can be used to test glycerol’s presence.
Choose the correct option

• (1) Both Statement I and Statement II are correct.
• (2) Both Statement I and Statement II are incorrect.
• (3) Statement I is correct but Statement II is incorrect.
• (4) Statement I is incorrect but Statement II is correct.

Match List-I with List-II.

Choose the correct answer from the options given below :

• (1) (A) – (II), (B) – (IV), (C) – (I), (D) – (III)
• (2) (A) – (II), (B) – (I), (C) – (IV), (D) – (III)
• (3) (A) – (III), (B) – (IV), (C) – (I), (D) – (II)
• (4) (A) – (III), (B) – (I), (C) – (IV), (D) – (II)

The compound(s) that is(are) removed as slag during the extraction of copper is 
(A) CaO 
(B) FeO 
(C) Al₂O₃ 
(D) ZnO 
(E) NiO 
Choose the correct answer from the options given below :

• (1) (C), (D) only
• (2) (A), (B), (E) only
• (3) (A), (B) only
• (4) (B) only

• (1) Mn²⁺
• (2) Mn⁴⁺
• (3) Mn³⁺
• (4) Mn⁶⁺

• (1) Li < K < Na < Rb < Cs
• (2) Li < Na < K < Rb < Cs
• (3) Cs < Rb < K < Na < Li
• (4) Li < Na < K < Cs < Rb

The geometry around boron in the product 'B' formed from the following reaction is \( BF _3+ NaH \xrightarrow{450 K } A + NaF\) \( A + NMe _3 \rightarrow B\)

• (1) trigonal planar
• (2) tetrahedral
• (3) pyramidal
• (4) square planar

The interhalogen compound formed from the reaction of bromine with excess of fluorine is a :

• (1) hypohalite
• (2) halate
• (3) perhalate
• (4) halite

• (1) NO
• (2) NO₂
• (3) SO₂
• (4) HCHO

• (1)
• (2)
• (3)
• (4)

• (1) Agonists
• (2) Antagonists
• (3) Allosterists
• (4) Anti histaminists

Among the following species
N₂,N₂⁺,N₂⁻,N₂²⁻,O₂,O₂⁺,O₂⁻,O₂²⁻
the number of species showing diamagnetism is ________.

The enthalpy of combustion of propane, graphite and dihydrogen at 298 K are –2220.0 kJ mol^–1, –393.5 kJ mol^–1 and –285.8 kJ mol^–1, respectively.The magnitude of enthalpy of formation of propane (C₃H₈) is _____ kJ mol^–1. (Nearest integer)

The cell potential for \(\text{Zn}|\text{Zn}^{2+}\text{(aq)}||\text{Sn}^{x+}|\text{Sn}\) is 0.801 V at 298 K.The reaction quotient for the above reaction is 10^–2. The number of electrons involved in the given electrochemical cell reaction is _____. (Given: \(Eº_Zn^{2+}\)|Zn=-0.763V,Eºsn^x+|sn=+0.008V and 2.303RT/F=0.06V)

While estimating the nitrogen present in an organic compound by Kjeldahl’s method, the ammonia evolved from 0.25g of the compound neutralized 2.5mL of 2M H₂SO₄. The percentage of nitrogen present in organic compound is ________

 In the given reaction,

(where Et is –C₂H₅)
The number of chiral carbon(s) in product A is ______.

The pressure of a moist gas at 27°C is 4 atm. The volume of the container is doubled at the same temperature. The new pressure of the moist gas is ______ ×10^–1 atm. (Nearest integer)(Given: The vapor pressure of water at 27°C is 0.4 atm.)