**JEE Main Session 1 29 ^{th} July 2022 Shift 1**

**PHYSICS**

**Section-A**

**IMPORTANT INSTRUCTIONS:**

(1) The test is of **3 hours** duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of **Physics**, **Chemistry** and **Mathematics** having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) **Section-A:** This section contains 20 multiple choice questions which have only one correct answer. each question carries **4 marks** for correct answer and **−1** mark for wrong answer.

(ii) **Section-B:** This section contains 10 questions. In Section-B, attempt any **five questions out of 10.** The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and **−1 mark** for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. Given below are two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).

**Assertion (A):** Time period of oscillation of a liquid drop depends on surf ace tension (S), if density of the liquid is ρ and radius of the drop is r, then is dimensionally correct, where K is dimensionless.

**Reason (R):**Using dimensional analysis we get R.H.S. having different dimension than that of time period.

In the light of above statements, choose the correct answer from the options given below.

(A) Both (A) and (R) are true and (R) is the correct explanation of (A)

(B) Both (A) and (R) are true but (R) is not the correct explanation of (A)

(C) (A) is true but (R) is false

(D) (A) is false but (R) is true

2. A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height h. Find the ratio of the times in which it is at height h/3 while going up and coming down respectively.

3. If is:

(A) 4

(B) Zero

(C) 8

(D) 16

4. A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v. Which of the following curve represents the correct relation between the normal reaction on the block by the wall (N) and speed of the block (v)?

5. A ball is projected with kinetic energy E, at an angle of 60° to the horizontal. The kinetic energy of this hall at the highest point of its flight will become

(A) Zero

(B) E/2

(C) E/4

(D) E

6. Two bodies of mass 1 kg and 3 kg have position vectors The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

7. Given below are two statements: One is labelled as Assertion (A) and the other· is labelled as Reason (R).

**Assertion (A): **Clothes containing oil or grease stains cannot be cleaned by water wash.

**Reason (R):** Because the angle of contact between the oil/ grease and water is obtuse.

In the light of the above statements, choose the correct answer from the option given below.

(A) Both (A) and (R) are true and (R) is the correct explanation of (A)

(B) Both (A) and (R) are true but (R) is not the correct explanation of (A)

(C) (A) is true but (R) is false

(D) (A) is false but (R) is true

8. If the length of a wire is made double and radius is halved of its respective values. Then, the Young’s modulus of the material of the wire will :

(A) Remain same

(B) Become 8 times its initial value

(C) Become 1/4th of its initial value

(D) Become 4 times its initial value

9. The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination α, is given by:

10. A spherically symmetric charge distribution is considered with charge density varying as

Where, r(r < R) is the distance from the centre O (as shown in figure) The electric field at point P will be :

11. Given below are two statements.

**Statement I:** Electric potential is constant within and at the surface of each conductor.

**Statement II:** Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point.

In the light of the above statements, choose the most appropriate answer from the options given below.

(A) Both statement I and statement II are correct

(B) Both statement I and statement II are incorrect

(C) Statement I is correct but statement II is incorrect

(D) Statement I is incorrect but statement II is correct

12. Two metallic wires of identical dimensions are connected in series. If σ_{1} and σ_{2} are the conductivities of these wires, respectively, the effective conductivity of the combination is :

13. An alternating emf E = 440 sin100πt is applied to a circuit containing an inductance of If an a.c. ammeter is connected in the circuit, its reading will be:

(A) 4.4 A

(B) 1.55 A

(C) 2.2 A

(D) 3.11 A

14. A coil of inductance 1 H and resistance 100 Ω is connected to a battery of 6 V. Determine approximately :

(a) The time elapsed before the current acquires half of its steady – state value.

(b) The energy stored in the magnetic field associated with the coil at an instant 15 ms after the circuit is switched on.

(Given ln2 = 0.693, e^{–3/2} = 0.25)

(A) t = 10 ms; U = 2 mJ

(B) t = 10 ms; U = 1 mJ

(C) t = 7 ms; U = 1 mJ

(D) t = 7 ms; U = 2 mJ

15. Match List-I with List-II:

Choose the correct answer from the options given below :

(A) (a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)

(B) (a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)

(C) (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)

(D) (a)-(iii), (b)-(i), (c)-(ii), (d)-(iv)

16. The kinetic energy of emitted electron is E when the light incident on the metal has wavelength λ. To double the kinetic energy, the incident light must have wavelength:

17. Find the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) second permitted energy level to the first level, and (ii) the highest permitted energy level to the first permitted level.

(A) 3 : 4

(B) 4 : 3

(C) 1 : 4

(D) 4 : 1

18. Find the modulation index of an AM wave having 8 V variation where maximum amplitude of the AM wave is 9 V.

(A) 0.8

(B) 0.5

(C) 0.2

(D) 0.1

19. A travelling microscope has 20 divisions per cm on the main scale while its vernier scale has total 50 divisions and 25 vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope?

(A) 0.001 cm

(B) 0.002 mm

(C) 0.002 cm

(D) 0.005 cm

20. In an experiment to find out the diameter of the wire using a screw gauge, the following observations were noted :

(A) Screw moves 0.5 mm on main scale in one complete rotation

(B) Total divisions on circular scale = 50

(C) Main scale reading is 2.5 mm

(D) 45th division of circular scale is in the pitch line

(E) Instrument has 0.03 mm negative error

Then the diameter of wire is :

(A) 2.92 mm

(B) 2.54mm

(C) 2.98mm

(D) 3.45mm

**SECTION-B**

21. An object is projected in the air with initial velocity u at an angle θ. The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be ______ degree.

22. If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth αh(h << R_{e}) from the earth surface. The value of α will be ________.

(Use R_{e }= 6400 km)

23. The pressure P_{1} and density d_{1} of diatomic gas (γ = 7/5) changes suddenly to P_{2} (>P_{1}) and d_{2} respectively during an adiabatic process. The temperature of the gas increases and becomes _____ times of its initial temperature.

(Given d_{2}/d_{1} =32)

24. One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is then the value of α will be _______. (Assume that the given diatomic gas has no vibrational mode).

25. The current I flowing through the given circuit will be ________A.

26. A closely wounded circular coil of radius 5 cm produces a magnetic field of 37.68 × 10^{–4} T at its center. The current through the coil is _______ A.

[Given, number of turns in the coil is 100 and π =3.14]

27. Two light beams of intensities 4I and 9I interfere on a screen. The phase difference between these beams on the screen at point A is zero and at point B is π. The difference of resultant intensities, at the point A and B, will be _______I.

28. A wire of length 314 cm carrying current of 14 A is bent to form a circle. The magnetic moment of the coil is _______ A–m^{2}. [Given π =3.14]

29. The X–Y plane be taken as the boundary between two transparent media M_{1} and M_{2}. M_{1} in Z ≥ 0 has a refractive index of √2 and M_{2} with Z < 0 has a refractive index of √3. A ray of light travelling in M_{1} along the direction given by the vector is incident on the plane of separation. The value of difference between the angle of incident in M_{1} and the angle of refraction in M_{2} will be ______ degree.

30. If the potential barrier across a p–n junction is 0.6 V. Then the electric field intensity, in the depletion region having the width of 6 × 10^{–6} m, will be ______× 10^{5} N/C.

**CHEMISTRY**

**SECTION-A**

1. Which of the following pair of molecules contain odd electron molecule and an expanded octet molecule?

(A) BCl_{3} and SF_{6}

(B) NO and H_{2}SO_{4}

(C) SF_{6} and H_{2}SO_{4}

(D) BCl_{3} and NO

2.

Consider the above reaction, the limiting reagent of the reaction and number of moles of NH_{3} formed respectively are:

(A) H_{2}, 1.42 moles

(B) H_{2}, 0.71 moles

(C) N_{2}, 1.42 moles

(D) N_{2}, 0.71 moles

3. 100 mL of 5% (w/v) solution of NaCl in water was prepared in 250 mL beaker. Albumin from the egg was poured into NaCl solution and stirred well. This resulted in a/ an :

(A) Lyophilic sol

(B) Lyophobic sol

(C) Emulsion

(D) Precipitate

4. The first ionization enthalpy of Na, Mg and Si, respectively, are: 496, 737 and 786 kJ mo1^{−}^{1}. The first ionization enthalpy (kJ mol^{−}^{1}) of Al is:

(A) 487

(B) 768

(C) 577

(D) 856

5. In metallurgy the term “gangue” is used for:

(A) Contamination of undesired earthy materials.

(B) Contamination of metals, other than desired metal

(C) Minerals which are naturally occurring in pure form

(D) Magnetic impurities in an ore.

6. The reaction of zinc with excess of aqueous alkali, evolves hydrogen gas and gives :

(A) Zn(OH)_{2}

(B) ZnO

(C) [Zn(OH)_{4}]^{2}^{−}

(D) [ZnO_{2}]^{2}^{−}

7. Lithium nitrate and sodium nitrate, when heated separately, respectively, give :

(A) LiNO_{2} and NaNO_{2}

(B) Li_{2}O and Na_{2}O

(C) Li_{2}O and NaNO_{2}

(D) LiNO_{2} and Na_{2}O

8. Number of lone pairs of electrons in the central atom of SCl_{2}, O_{3}, ClF_{3} and SF_{6}, respectively, are :

(A) 0, 1, 2 and 2

(B) 2, 1, 2 and 0

(C) 1, 2, 2 and 0

(D) 2, 1, 2 and 0

9. In following pairs, the one in which both transition metal ions are colourless is :

(A) Sc^{3+}, Zn^{2+}

(B) Ti^{4+}, Cu^{2+}

(C) V^{2+}, Ti^{3+}

(D) Zn^{2+}, Mn^{2+}

10. In neutral or faintly alkaline medium, KMnO_{4} being a powerful oxidant can oxidize, thiosulphate almost quantitatively, to sulphate. In this reaction overall change in oxidation state of manganese will be :

(A) 5

(B) 1

(C) 0

(D) 3

11. Which among the following pairs has only herbicides ?

(A) Aldrin and Dieldrin

(B) Sodium chlorate and Aldrin

(C) Sodium arsinate and Dieldrin

(D) Sodium chlorate and sodium arsinite.

12. Which among the following is the strongest Bronsted base ?

13. Which among the following pairs of the structures will give different products on ozonolysis? (Consider the double bonds in the structures are rigid and not delocalized.)

14.

Considering the above reactions, the compound ‘A’ and compound ‘B’ respectively are :

15.

Consider the above reaction sequence, the Product ‘C’ is :

16.

Consider the above reaction, the compound ‘A’ is :

17.

Which among the following represent reagent ‘A’?

18. Consider the following reaction sequence :

19. Which of the following compounds is an example of hypnotic drug ?

(A) Seldane

(B) Amytal

(C) Aspartame

(D) Prontosil

20. A compound ‘X’ is acidic and it is soluble in NaOH solution, but insoluble in NaHCO_{3} Compound ‘X’ also gives violet colour with neutral FeCI_{3} solution. The compound ‘X’ is :

**SECTION-B**

21. Resistance of a conductivity cell (cell constant 129 m^{−}^{l}) filled with 74.5 ppm solution of KCl is 100Ω (labelled as solution 1). When the same cell is filled with KCl solution of 149 ppm, the resistance is 50Ω (labelled as solution 2). The ratio of molar conductivity of solution 1 and solution 2 is i.e. The value of x is ______. (Nearest integer)

Given, molar mass of KCl is 74.5 g mol^{−}^{l}

22. Ionic radii of cation A^{+} and anion B^{−} are 102 and 181 pm respectively. These ions are allowed to crystallize into an ionic solid. This crystal has cubic close packing for B^{−}. A^{+} is present in all octahedral voids. The edge length of the unit cell of the crystal AB is _____ pm. (Nearest Integer)

23. The minimum uncertainty in the speed of an electron in an one dimensional region of length 2aO (Where a_{0} = Bohr radius 52.9 pm) is _____km s^{−}^{1}. (Given : Mass of electron = 9.l × 10^{−}^{31} kg, Planck’s constant h = 6.63 × 10^{−}^{34}Js)

24. When 600 mL of 0.2 M HNO_{3} is mixed with 400 mL of 0.1M NaOH solution in a flask, the rise in temperature of the flask is _______ × 10^{−}^{2}° (Enthalpy of neutralisation = 57 kJ mo1^{−}^{1} and Specific heat of water = 4.2 JK^{−}^{1} g^{−}^{1})

(Neglect heat capacity of flask)

25. If O_{2} gas is bubbled through water at 303 K, the number of millimoles of O_{2} gas that dissolve in 1 litre of water is_______. (Nearest Integer)

(Given : Henry’s Law constant for O_{2} at 303 K is 46.82 k bar and partial pressure of O_{2} = 0.920 bar)

(Assume solubility of O_{2} in water is too small, nearly negligible)

26. If the solubility product of PbS is 8 × 10^{−}^{28}, then the solubility of PbS in pure water at 298 K is x × 10^{−}^{l6}mol L^{−}^{1}. The value of x is ________. (Nearest Integer)

[Given √2 = 1.41]

27. The reaction between X and Y is first order with respect to X and zero order with respect to Y.

Examine the data of table and calculate ratio of numerical values of M and L. (Nearest Integer)

28. In a linear tetrapeptide (Constituted with different amino acids), (number of amino acids) – (number of peptide bonds) is______.

29. In bromination of Propyne, with Bromine 1, 1, 2, 2-tetrabromopropane is obtained in 27% yield. The amount of 1, 1, 2, 2 tetrabromopropane obtained from 1 g of Bromine in this reaction is ______ × 10^{−}^{1} (Nearest integer)

(Molar Mass : Bromine = 80 g/mol)

30. [Fe(CN)_{6}]^{3}^{−} should be an inner orbital complex. Ignoring the pairing energy, the value of crystal field stabilization energy for this complex is (–) _________ ∆_{o}. (Nearest integer)

**MATHEMATICS**

**SECTION-A**

1. Let R be a relation from the set {1, 2, 3, ….., 60} to itself such that R = {(a, b) : b = pq, where p, q ≥ 3 are prime numbers}. Then, the number of elements in R is :

(A) 600

(B) 660

(C) 540

(D) 720

2. If z = 2 + 3i, then is equal to :

(A) 244

(B) 224

(C) 245

(D) 265

3. Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then

(A) the system of linear equations AX = 0 has a unique solution

(B) the system of linear equations AX = 0 has infinitely many solutions

(C) B is an invertible matrix

(D) adj(A) is an invertible matrix

4. If then the maximum value of a is:

(A) 198

(B) 202

(C) 212

(D) 218

5. If where α, β, γ ∈ R, then which of the following is NOT correct?

(A) α^{2} + β^{2} + γ^{2} = 6

(B) αβ + βγ + γα + 1 = 0

(C) αβ^{2} + βγ^{2} + γα^{2} + 3 = 0

(D) α^{2} – β^{2} + γ^{2} = 4

6. The integral is equal to

(A) tan^{−}^{1} (2)

(B)

(C)

(D) 1/2

7. Let the solution curve y = y(x) of the differential equation pass through the point (0, π/2). Then, is equal to

(A) π/4

(B) 3π/4

(C) π/2

(D) 3π/2

8. Let a line L pass through the point intersection of the lines bx + 10y – 8 = 0 and If the line L also passes through the point (1, 1) and touches the circle 17(x^{2} + y^{2}) = 16, then the eccentricity of the ellipse .

9. If the foot of the perpendicular from the point A(–1, 4, 3) on the plane P : 2x + my + nz = 4, is (-2, 7/2, 3/2), then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, –1, –4, is equal to

(A) 1

(B) √26

(C) 2√2

(D) √14

10. Let Let be a vector satisfying If are non-parallel, then the value of λ is

(A) –5

(B) 5

(C) 1

(D) –1

11. The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is If the distance of the point B from the tower is 15 units, then cot α is equal to :

(A) 6/5

(B) 9/5

(C) 4/3

(D) 7/3

12. The statement (p ∧ q) ⇒ (p ∧ r) is equivalent to :

(A) q ⇒ (p ∧ r)

(B) p ⇒ (p ∧ r)

(C) (p ∧ r) ⇒ (p ∧ q)

(D) (p ∧ q) ⇒ r

13. Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k_{1}, k_{2}), then k_{1} + k_{2} is equal to :

(A) 2

(B) 4/7

(C) 2/7

(D) 4

14. Let be two unit vectors such that the angle between them is π/4. If θ is the angle between the vectors then the value of 164 cos^{2}θ is equal to :

(A) 90 + 27√2

(B) 45 + 18√2

(C) 90 + 3√2

(D) 54 + 90√2

15. If then f(e^{3}) + f(e^{–3}) is equal to :

(A) 9

(B) 9/2

(C)

(D)

16. The area of the region is equal to

17. Let the focal chord of the parabola P :y^{2} = 4x along the line L : y = mx + c, m > 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H : x^{2} – y^{2} = 4. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is

(A) 2√6

(B) 2√14

(C) 4√6

(D) 4√14

18. The number of points, where the function f: ℝ → ℝ, f(x) = |x – 1|cos|x – 2|sin|x – 1| + (x – 3)|x^{2} – 5x + 4|, is NOT differentiable, is

(A) 1

(B) 2

(C) 3

(D) 4

19. Let S = {1, 2, 3, …, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is

20. Let Then which of the following statements are true?

P :x = 0 is a point of local minima of f

Q: x = √2 is a point of inflection of f

R : fʹ is increasing for x > √2

(A) Only P and Q

(B) Only P and R

(C) Only Q and R

(D) All P, Q and R

**SECTION-B**

21. Let S = {θ ∈ (0, 2π) : 7 cos^{2}θ – 3 sin^{2}θ – 2 cos^{2}2θ = 2}. Then, the sum of roots of all the equations x^{2} – 2 (tan^{2}θ + cot^{2}θ) x + 6 sin^{2}θ = 0, θ ∈ S, is _______.

22. Let the mean and the variance of 20 observations x_{1}, x_{2}, …., x_{20} be 15 and 9, respectively. For a ∈ R, if the mean of (x_{1} + α)^{2}, (x_{2} + α)^{2}, …., (x_{2}0 + α)^{2} is 178, then the square of the maximum value of α is equal to ___________.

23. Let a line with direction ratios a, – 4a, –7 be perpendicular to the lines with direction ratios 3, – 1, 2b and b, a, – 2. If the point of intersection of the line and the plane x – y + z = 0 is (α, β, γ), then α + β + γ is equal to ______.

24. Let a_{1}, a_{2}, a_{3}, …. be an A.P. If then 4a_{2} is equal to ________.

25. Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of in the increasing powers of If the sixth term from the beginning is then α is equal to ___________.

26. The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _________.

27. Let p and p + 2 be prime numbers and let

Then the sum of the maximum values of α and β, such that p^{α} and (p + 2)^{β} divide Δ, is _______.

28. If then 34 k is equal to ________.

29. Let S = {4, 6, 9} and T = {9, 10, 11, …,1000}. If A = {a_{1} + a_{2} + … +a_{k} :k∈N, a_{1}, a_{2}, a_{3}, …, a_{k}∈S}, then the sum of all the elements in the set T – A is equal to ________.

30. Let the mirror image of a circle c_{1} : x^{2} + y^{2} – 2x – 6y + α = 0 in line y = x + 1 be c2 : 5x^{2} + 5y^{2} + 10gx + 10fy + 38 = 0. If r is the radius of circle c_{2}, then α + 6r_{2} is equal to _________.

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