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JEE Main 18 March 2021 Shift 2 Question Paper with Answers

18 March 2021 · March session · 85 questions

85 of the 90 questions from the JEE Main 18 March 2021 Shift 2 paper, each with its correct answer and tagged to the chapter it tests. Free to read, no account needed.

5 questions are held back from this paper while we re-check the transcription or the answer key. We would rather show you nothing than show you an answer we are not confident is right.

Physics
27
Chemistry
29
Mathematics
29

Physics — JEE Main 18 March 2021 Shift 2

Q1·PhysicsSingle correct
An object of mass m1m_1m1​ collides with another object of mass m2m_2m2​, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses m2:m1m_2 : m_1m2​:m1​ is :
  1. (A)3 : 1
  2. (B)2 : 1
  3. (C)1 : 2
  4. (D)1 : 1

Correct answer: (A)

Step-by-step solution →
Q2·PhysicsSingle correct
For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where γ is the ratio of specific heats):
  1. (A)−γdVV-\gamma\frac{dV}{V}−γVdV​
  2. (B)−γVdV-\gamma\frac{V}{dV}−γdVV​
  3. (C)−1γdVV-\frac{1}{\gamma}\frac{dV}{V}−γ1​VdV​
  4. (D)dVV\frac{dV}{V}VdV​

Correct answer: (A)

Step-by-step solution →
Q3·Physics·Magnetic Field of CurrentSingle correct
A proton and an α-particle, having kinetic energies KpK_pKp​ and KαK_\alphaKα​, respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of α-particle is 2 : 1. The ratio of Kp:KαK_p : K_\alphaKp​:Kα​ is :
  1. (A)1 : 8
  2. (B)8 : 1
  3. (C)1 : 4
  4. (D)4 : 1

Correct answer: (D)

Step-by-step solution →
Q4·PhysicsSingle correct
A plane electromagnetic wave propagating along y-direction can have the following pair of electric field (E⃗)\left(\vec{E}\right)(E) and magnetic field (B⃗)\left(\vec{B}\right)(B) components.
  1. (A)EyE_yEy​, ByB_yBy​ or EzE_zEz​, BzB_zBz​
  2. (B)EyE_yEy​, BxB_xBx​ or ExE_xEx​, ByB_yBy​
  3. (C)ExE_xEx​, BzB_zBz​ or EzE_zEz​, BxB_xBx​
  4. (D)ExE_xEx​, ByB_yBy​ or EyE_yEy​, BxB_xBx​

Correct answer: (C)

Step-by-step solution →
Q5·PhysicsSingle correct
Consider a uniform wire of mass M and length L. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :
  1. (A)14ML2π2\frac{1}{4}\frac{ML^{2}}{\pi^{2}}41​π2ML2​
  2. (B)25ML2π2\frac{2}{5}\frac{ML^{2}}{\pi^{2}}52​π2ML2​
  3. (C)ML2π2\frac{ML^{2}}{\pi^{2}}π2ML2​
  4. (D)12ML2π2\frac{1}{2}\frac{ML^{2}}{\pi^{2}}21​π2ML2​

Correct answer: (C)

Step-by-step solution →
Q6·PhysicsSingle correct
The velocity-displacement graph of a particle is shown in the figure. The acceleration-displacement graph of the same particle is represented by :
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (C)

Step-by-step solution →
Q7·PhysicsSingle correct
The correct relation between α (ratio of collector current to emitter current) and β (ratio of collector current to base current) of a transistor is :
  1. (A)β=α1+α\beta=\frac{\alpha}{1+\alpha}β=1+αα​
  2. (B)α=β1−α\alpha=\frac{\beta}{1-\alpha}α=1−αβ​
  3. (C)β=11−α\beta=\frac{1}{1-\alpha}β=1−α1​
  4. (D)α=β1+β\alpha=\frac{\beta}{1+\beta}α=1+ββ​

Correct answer: (D)

Step-by-step solution →
Q8·PhysicsSingle correct
Three rays of light, namely red (R), green (G) and blue (B) are incident on the face PQ of a right angled prism PQR as shown in figure. The refractive indices of the material of the prism for red, green and blue wavelength are 1.27, 1.42 and 1.49 respectively. The colour of the ray(s) emerging out of the face PR is :
  1. (A)green
  2. (B)red
  3. (C)blue and green
  4. (D)blue

Correct answer: (B)

Step-by-step solution →
Q9·PhysicsSingle correct
If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately : (Take : g = 10 ms−2^{-2}−2, the radius of earth, R = 6400 × 103^{3}3 m, Take π = 3.14)
  1. (A)60 minutes
  2. (B)does not change
  3. (C)1200 minutes
  4. (D)84 minutes

Correct answer: (D)

Step-by-step solution →
Q10·PhysicsSingle correct
The decay of a proton to neutron is :
  1. (A)not possible as proton mass is less than the neutron mass
  2. (B)possible only inside the nucleus
  3. (C)not possible but neutron to proton conversion is possible
  4. (D)always possible as it is associated only with β+^{+}+ decay

Correct answer: (B)

Step-by-step solution →
Q11·PhysicsSingle correct
In a series LCR circuit, the inductive reactance (XLX_LXL​) is 10 Ω and the capacitive reactance (XCX_CXC​) is 4 Ω. The resistance (R) in the circuit is 6 Ω. The power factor of the circuit is :
  1. (A)12\frac{1}{2}21​
  2. (B)122\frac{1}{2\sqrt{2}}22​1​
  3. (C)12\frac{1}{\sqrt{2}}2​1​
  4. (D)32\frac{\sqrt{3}}{2}23​​

Correct answer: (C)

Step-by-step solution →
Q12·PhysicsSingle correct
The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is L⃗\vec{L}L. The magnitude of the areal velocity of the planet is :
  1. (A)4LM\frac{4L}{M}M4L​
  2. (B)LM\frac{L}{M}ML​
  3. (C)2LM\frac{2L}{M}M2L​
  4. (D)L2M\frac{L}{2M}2ML​

Correct answer: (D)

Step-by-step solution →
Q13·PhysicsSingle correct
The function of time representing a simple harmonic motion with a period of πω\frac{\pi}{\omega}ωπ​ is :
  1. (A)sin(ωt) + cos (ωt)
  2. (B)cos(ωt) + cos (2ωt) + cos (3ωt)
  3. (C)sin⁡2(ωt)\sin^{2}(\omega t)sin2(ωt)
  4. (D)3cos⁡(π4−2ωt)3\cos\left(\frac{\pi}{4}-2\omega t\right)3cos(4π​−2ωt)

Correct answer: (D)

Step-by-step solution →
Q14·PhysicsSingle correct
A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is : [The coefficient of static friction, μs\mu_sμs​, is 0.4]
  1. (A)72mg\frac{7}{2}mg27​mg
  2. (B)5 mg
  3. (C)mg5\frac{mg}{5}5mg​
  4. (D)0

Correct answer: (C)

Step-by-step solution →
Q15·PhysicsSingle correct
The time taken for the magnetic energy to reach 25% of its maximum value, when a solenoid of resistance R, inductance L is connected to a battery, is :
  1. (A)LRℓn5\frac{L}{R}\ell n5RL​ℓn5
  2. (B)infinite
  3. (C)LRℓn2\frac{L}{R}\ell n2RL​ℓn2
  4. (D)LRℓn10\frac{L}{R}\ell n10RL​ℓn10

Correct answer: (C)

Step-by-step solution →
Q16·PhysicsSingle correct
A particle of mass m moves in a circular orbit under the central potential field, U(r)=−CrU(r)=\frac{-C}{r}U(r)=r−C​, where C is a positive constant. The correct radius – velocity graph of the particle's motion is :
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (A)

Step-by-step solution →
Q17·PhysicsSingle correct
An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is S1S_1S1​ and that of the other part is S2S_2S2​. Given that S1>S2S_1 > S_2S1​>S2​. If the piston is removed then the total entropy of the system will be :
  1. (A)S1×S2S_1 \times S_2S1​×S2​
  2. (B)S1−S2S_1 - S_2S1​−S2​
  3. (C)S1S2\frac{S_1}{S_2}S2​S1​​
  4. (D)S1+S2S_1 + S_2S1​+S2​

Correct answer: (D)

Step-by-step solution →
Q18·PhysicsSingle correct
The speed of electrons in a scanning electron microscope is 1 × 107^{7}7 ms−1^{-1}−1. If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of:
  1. (A)1837
  2. (B)11837\frac{1}{1837}18371​
  3. (C)1837\sqrt{1837}1837​
  4. (D)11837\frac{1}{\sqrt{1837}}1837​1​

Correct answer: (A)

Step-by-step solution →
Q19·PhysicsNumerical
The projectile motion of a particle of mass 5 g is shown in the figure. The initial velocity of the particle is 525\sqrt{2}52​ ms−1^{-1}−1 and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points A and B is x × 10−2^{-2}−2 kgms−1^{-1}−1. The value of x, to the nearest integer, is _______.

Correct answer: 5

Step-by-step solution →
Q20·PhysicsNumerical
A ball of mass 4 kg, moving with a velocity of 10 ms−1^{-1}−1, collides with a spring of length 8 m and force constant 100 Nm−1^{-1}−1. The length of the compressed spring is x m. The value of x, to the nearest integer, is_____.

Correct answer: 6

Step-by-step solution →
Q21·PhysicsNumerical
The typical output characteristics curve for a transistor working in the common-emitter configuration is shown in the figure. The estimated current gain from the figure is

Correct answer: 200

Step-by-step solution →
Q22·PhysicsNumerical
Consider a water tank as shown in the figure. It's cross-sectional area is 0.4 m2^{2}2. The tank has an opening B near the bottom whose cross-section area is 1 cm2^{2}2. A load of 24 kg is applied on the water at the top when the height of the water level is 40 cm above the bottom, the velocity of water coming out the opening B is v ms−1^{-1}−1. The value of v, to the nearest integer, is___. [Take value of g to be 10 ms−2^{-2}−2]

Correct answer: 3

Step-by-step solution →
Q23·PhysicsNumerical
A TV transmission tower antenna is at a height of 20 m. Suppose that the receiving antenna is at. (i) ground level (ii) a height of 5 m. The increase in antenna range in case (ii) relative to case (i) is n%. The value of n, to the nearest integer, is .

Correct answer: 50

Step-by-step solution →
Q24·PhysicsNumerical
An infinite number of point charges, each carrying 1 μC charge, are placed along the y-axis at y = 1 m, 2 m, 4 m, 8 m.................. The total force on a 1 C point charge, placed at the origin, is x × 103^{3}3 N. The value of x, to the nearest integer, is________. [Take 14πϵ0\frac{1}{4\pi \epsilon_{0}}4πϵ0​1​ = 9 × 109^{9}9 Nm2^{2}2/C2^{2}2]

Correct answer: 12

Step-by-step solution →
Q25·PhysicsNumerical
Consider a 72 cm long wire AB as shown in the figure. The galvanometer jockey is placed at P on AB at a distance x cm from A. The galvanometer shows zero deflection. The value of x, to the nearest integer, is

Correct answer: 48

Step-by-step solution →
Q26·PhysicsNumerical
Two wires of same length and thickness having specific resistances 6Ω cm and 3Ω cm respectively are connected in parallel. The effective resistivity is ρ Ω cm. The value of ρ, to the nearest integer, is_____.

Correct answer: 4

Step-by-step solution →
Q27·PhysicsNumerical
A galaxy is moving away from the earth at a speed of 286 kms−1^{-1}−1. The shift in the wavelength of a red line at 630 nm is x × 10−10^{-10}−10 m. The value of x, to the nearest integer, is______. [Take the value of speed of light c, as 3 × 108^{8}8 ms−1^{-1}−1]

Correct answer: 6

Step-by-step solution →

Chemistry — JEE Main 18 March 2021 Shift 2

Q28·ChemistrySingle correct
The oxidation states of nitrogen in NO, NO2_{2}2​, N2_{2}2​O and NO3−_{3}^{-}3−​ are in the order of :
  1. (A)NO3−_{3}^{-}3−​ > NO2_{2}2​ > NO > N2_{2}2​O
  2. (B)NO2_{2}2​ > NO3−_{3}^{-}3−​ > NO > N2_{2}2​O
  3. (C)N2_{2}2​O > NO2_{2}2​ > NO > NO3−_{3}^{-}3−​
  4. (D)NO > NO2_{2}2​ > N2_{2}2​O > NO3−_{3}^{-}3−​

Correct answer: (A)

Step-by-step solution →
Q29·ChemistrySingle correct
In basic medium, H2_{2}2​O2_{2}2​ exhibits which of the following reactions ? (A) Mn2+^{2+}2+ → Mn4+^{4+}4+ (B) I2_{2}2​ → I−^{-}− (C) PbS → PbSO4_{4}4​ Choose the most appropriate answer from the options given below :
  1. (A)(A), (C) only
  2. (B)(A) only
  3. (C)(B) only
  4. (D)(A), (B) only

Correct answer: (D)

Step-by-step solution →
Q30·ChemistrySingle correct
In the reaction of hypobromite with amide, the carbonyl carbon is lost as :
  1. (A)CO32−_{3}^{2-}32−​
  2. (B)HCO3−_{3}^{-}3−​
  3. (C)CO2_{2}2​
  4. (D)CO

Correct answer: (A)

Step-by-step solution →
Q31·ChemistrySingle correct
The oxide that shows magnetic property is :
  1. (A)SiO2_{2}2​
  2. (B)Mn3_{3}3​O4_{4}4​
  3. (C)Na2_{2}2​O
  4. (D)MgO

Correct answer: (B)

Step-by-step solution →
Q32·ChemistrySingle correct
Main Products formed during a reaction of 1-methoxy naphthalene with hydroiodic acid are:
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (B)

Step-by-step solution →
Q33·ChemistrySingle correct
Deficiency of vitamin K causes :
  1. (A)Increase in blood clotting time
  2. (B)Increase in fragility of RBC's
  3. (C)Cheilosis
  4. (D)Decrease in blood clotting time

Correct answer: (A)

Step-by-step solution →
Q34·ChemistrySingle correct
An organic compound "A" on treatment with benzene sulphonyl chloride gives compound B. B is soluble in dil. NaOH solution. Compound A is :
  1. (A)C6_{6}6​H5_{5}5​–N–(CH3_{3}3​)2_{2}2​
  2. (B)C6_{6}6​H5_{5}5​–NHCH2_{2}2​CH3_{3}3​
  3. (C)C6_{6}6​H5_{5}5​–CH2_{2}2​ NHCH3_{3}3​
  4. (D)C6_{6}6​H5_{5}5​–CH(CH3_{3}3​)–NH2_{2}2​

Correct answer: (D)

Step-by-step solution →
Q35·ChemistrySingle correct
The first ionization energy of magnesium is smaller as compared to that of elements X and Y, but higher than that of Z. the elements X, Y and Z, respectively, are :
  1. (A)chlorine, lithium and sodium
  2. (B)argon, lithium and sodium
  3. (C)argon, chlorine and sodium
  4. (D)neon, sodium and chlorine

Correct answer: (C)

Step-by-step solution →
Q36·ChemistrySingle correct
The secondary valency and the number of hydrogen bonded water molecule(s) in CuSO4_{4}4​·5H2_{2}2​O, respectively, are :
  1. (A)6 and 4
  2. (B)4 and 1
  3. (C)6 and 5
  4. (D)5 and 1

Correct answer: (B)

Step-by-step solution →
Q37·ChemistrySingle correct
Given below are two statements : Statement I : Bohr's theory accounts for the stability and line spectrum of Li+^{+}+ ion. Statement II : Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field. In the light of the above statements, choose the most appropriate answer from the options given below :
  1. (A)Both statement I and statement II are true.
  2. (B)Statement I is false but statement II is true.
  3. (C)Both statement I and statement II are false.
  4. (D)Statement I is true but statement II is false.

Correct answer: (B)

Step-by-step solution →
Q38·ChemistrySingle correct
Consider the given reaction, percentage yield of :
  1. (A)C > A > B
  2. (B)B > C > A
  3. (C)A > C > B
  4. (D)C > B > A

Correct answer: (D)

Step-by-step solution →
Q39·ChemistrySingle correct
The charges on the colloidal CdS sol and TiO2_{2}2​ sol are, respectively :
  1. (A)positive and positive
  2. (B)positive and negative
  3. (C)negative and negative
  4. (D)negative and positive

Correct answer: (D)

Step-by-step solution →
Q40·ChemistrySingle correct
Match List - I with List - II : List - I (Class of Chemicals) List - II (Example) (a) Antifertility drug (i) Meprobamate (b) Antibiotic (ii) Alitame (c) Tranquilizer (iii) Norethindrone (d) Artificial Sweetener (iv) Salvarsan
  1. (A)(a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)
  2. (B)(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
  3. (C)(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)
  4. (D)(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)

Correct answer: (C)

Step-by-step solution →
Q41·ChemistrySingle correct
Consider the above reaction, the product 'X' and 'Y' respectively are :
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (C)

Step-by-step solution →
Q42·ChemistrySingle correct
Given below are two statements : Statement I : C2_{2}2​H5_{5}5​OH and AgCN both can generate nucleophile. Statement II : KCN and AgCN both will generate nitrile nucleophile with all reaction conditions. Choose the most appropriate option :
  1. (A)Statement I is true but statement II is false
  2. (B)Both statement I and statement II are true
  3. (C)Statement I is false but statement II is true
  4. (D)Both statement I and statement II are false

Correct answer: (A)

Step-by-step solution →
Q43·ChemistrySingle correct
Given below are two statements : Statement I : Non-biodegradable wastes are generated by the thermal power plants. Statement II : Bio-degradable detergents leads to eutrophication. In the light of the above statements, choose the most appropriate answer from the option given below :
  1. (A)Both statement I and statement II are false
  2. (B)Statement I is true but statement II is false
  3. (C)Statement I is false but statement II is true
  4. (D)Both statement I and statement II are true.

Correct answer: (D)

Step-by-step solution →
Q44·ChemistrySingle correct
Match list-I (metal) with list-II (refining process) : Choose the most appropriate answer from the option given below :
List-IList-II
a.Mercuryi.Vapour phase refining
b.Copperii.Distillation refining
c.Siliconiii.Electrolytic refining
d.Nickeliv.Zone refining
  1. (A)a-i, b-iv, c-ii, d-iii
  2. (B)a-ii, b-iii, c-i, d-iv
  3. (C)a-ii, b-iii, c-iv, d-i
  4. (D)a-ii, b-iv, c-iii, d-i

Correct answer: (C)

Step-by-step solution →
Q45·ChemistrySingle correct
In the following molecules, Hybridisation of carbon a, b and c respectively are :
  1. (A)sp3^{3}3, sp, sp
  2. (B)sp3^{3}3, sp2^{2}2, sp
  3. (C)sp3^{3}3, sp2^{2}2, sp2^{2}2
  4. (D)sp3^{3}3, sp, sp2^{2}2

Correct answer: (C)

Step-by-step solution →
Q46·ChemistrySingle correct
A hard substance melts at high temperature and is an insulator in both solid and in molten state. This solid is most likely to be a / an :
  1. (A)Ionic solid
  2. (B)Molecular solid
  3. (C)Metallic solid
  4. (D)Covalent solid

Correct answer: (D)

Step-by-step solution →
Q47·ChemistryNumerical
A reaction has a half life of 1 min. The time required for 99.9% completion of the reaction is ______ min. (Round off to the Nearest integer) [Use : ln 2= 0.69, ln 10 = 2.3]

Correct answer: 10

Step-by-step solution →
Q48·ChemistryNumerical
The molar conductivities at infinite dilution of barium chloride, sulphuric arid and hydrochloric acid are 280, 860 and 426 Scm2^{2}2 mol−1^{-1}−1 respectively. The molar conductivity at infinite dilution of barium sulphate is ________ S cm2^{2}2 mol−1^{-1}−1 (Round off to the Nearest Integer).

Correct answer: 288

Step-by-step solution →
Q49·ChemistryNumerical
The number of species below that have two lone pairs of electrons in their central atom is ____ (Round off to the Nearest integer) SF4_{4}4​, BF4−_{4}^{-}4−​ , ClF3_{3}3​, AsF3_{3}3​, PCl5_{5}5​, BrF5_{5}5​, XeF4_{4}4​, SF6_{6}6​

Correct answer: 2

Step-by-step solution →
Q50·ChemistryNumerical
A xenon compound 'A' upon partial hydrolysis gives XeO2_{2}2​F2_{2}2​. The number of lone pair of electrons present in compound A is _______ (Round off to the Nearest integer)

Correct answer: 19

Step-by-step solution →
Q51·ChemistryNumerical
The gas phase reaction 2A(g) ⇌ A2_{2}2​(g) at 400 K has ΔG° = + 25.2 kJ mol−1^{-1}−1. The equilibrium constant KC_{C}C​ for this reaction is ______ × 10−2^{-2}−2. (Round off to the Nearest integer) [Use : R = 8.3 J mol−1^{-1}−1K−1^{-1}−1 , ln 10 = 2.3 log10_{10}10​ 2 = 0.30, 1 atm = 1 bar] [antilog (−0.3) = 0.501]

Correct answer: 2

Step-by-step solution →
Q52·ChemistryNumerical
In Tollen's test for aldehyde, the overall number of electron(s) transferred to the Tollen's reagent formula [Ag(NH3_{3}3​)2_{2}2​]+^{+}+ per aldehyde group to form silver mirror is ________ .(Round off to the Nearest integer)

Correct answer: 2

Step-by-step solution →
Q53·ChemistryNumerical
The solubility of CdSO4_{4}4​ in water is 8.0 × 10−4^{-4}−4 mol L−1^{-1}−1. Its solubility in 0.01 M H2_{2}2​SO4_{4}4​ solution is ______ × 10−6^{-6}−6 mol L−1^{-1}−1. (Round off to the Nearest integer) (Assume that solubility is much less than 0.01 M)

Correct answer: 64

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Q54·ChemistryNumerical
A solute a dimerizes in water. The boiling point of a 2 molar solution of A is 100.52°C. The percentage association of A is. ____ . (Round off to the Nearest integer) [Use : Kb_{b}b​ for water = 0.52 K kg mol−1^{-1}−1 Boiling point of water = 100°C]

Correct answer: 100

Step-by-step solution →
Q55·ChemistryNumerical
10.0 ml of Na2_{2}2​CO3_{3}3​ solution is titrated against 0.2 M HCl solution. The following titre values were obtained in 5 readings. 4.8 ml, 4.9 ml, 5.0 ml, 5.0 ml and 5.0 ml Based on these readings, and convention of titrimetric estimation of concentration of Na2_{2}2​CO3_{3}3​ solution is ________ mM. (Round off to the Nearest integer)

Correct answer: 50

Step-by-step solution →
Q56·ChemistryNumerical
Consider the above reaction where 6.1 g of benzoic acid is used to get 7.8 g of m-bromo benzoic acid. The percentage yield of the product is _____ . (Round off to the Nearest integer) [Given : Atomic masses : C = 12.0u, H : 1.0u, O : 16.0u, Br = 80.0 u]

Correct answer: 78

Step-by-step solution →

Mathematics — JEE Main 18 March 2021 Shift 2

Q57·MathematicsSingle correct
Let y = y(x) be the solution of the differential equation dydx=(y+1)((y+1)ex2/2−x)\frac{dy}{dx} = (y+1)\left((y+1)e^{x^{2}/2} - x\right)dxdy​=(y+1)((y+1)ex2/2−x), 0 < x < 2.1, with y(2) = 0. Then the value of dydx\frac{dy}{dx}dxdy​ at x = 1 is equal to :
  1. (A)−e3/2(e2+1)2\frac{-e^{3/2}}{\left(e^{2}+1\right)^{2}}(e2+1)2−e3/2​
  2. (B)−2e2(1+e2)2-\frac{2e^{2}}{\left(1+e^{2}\right)^{2}}−(1+e2)22e2​
  3. (C)e5/2(1+e2)2\frac{e^{5/2}}{\left(1+e^{2}\right)^{2}}(1+e2)2e5/2​
  4. (D)5e1/2(e2+1)2\frac{5e^{1/2}}{\left(e^{2}+1\right)^{2}}(e2+1)25e1/2​

Correct answer: (A)

Step-by-step solution →
Q58·MathematicsSingle correct
In a triangle ABC, if ∣BC⃗∣=8\left|\vec{BC}\right| = 8​BC​=8, ∣CA⃗∣=7\left|\vec{CA}\right| = 7​CA​=7, ∣AB⃗∣=10\left|\vec{AB}\right| = 10​AB​=10, then the projection of the vector AB⃗\vec{AB}AB on AC⃗\vec{AC}AC is equal to :
  1. (A)254\frac{25}{4}425​
  2. (B)8514\frac{85}{14}1485​
  3. (C)12720\frac{127}{20}20127​
  4. (D)11516\frac{115}{16}16115​

Correct answer: (B)

Step-by-step solution →
Q59·MathematicsSingle correct
Let the system of linear equations 4x + λy + 2z = 0 2x − y + z = 0 μx + 2y + 3z = 0, λ, μ ∈ R. has a non-trivial solution. Then which of the following is true ?
  1. (A)μ = 6, λ ∈ R
  2. (B)λ = 2, μ ∈ R
  3. (C)λ = 3, μ ∈ R
  4. (D)μ = −6, λ ∈ R

Correct answer: (A)

Step-by-step solution →
Q60·MathematicsSingle correct
Let f : R − {3} → R − {1} be defined by f(x)=x−2x−3f(x) = \frac{x-2}{x-3}f(x)=x−3x−2​. Let g : R → R be given as g(x) = 2x − 3. Then, the sum of all the values of x for which f−1(x)+g−1(x)=132f^{-1}(x) + g^{-1}(x) = \frac{13}{2}f−1(x)+g−1(x)=213​ is equal to
  1. (A)7
  2. (B)2
  3. (C)5
  4. (D)3

Correct answer: (C)

Step-by-step solution →
Q61·MathematicsSingle correct
Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :
  1. (A)92\frac{9}{\sqrt{2}}2​9​
  2. (B)727\sqrt{2}72​
  3. (C)222\sqrt{2}22​
  4. (D)323\sqrt{2}32​

Correct answer: (A)

Step-by-step solution →
Q62·MathematicsSingle correct
Consider a hyperbola H : x2−2y2=4x^{2} - 2y^{2} = 4x2−2y2=4. Let the tangent at a point P(4,6)P\left(4, \sqrt{6}\right)P(4,6​) meet the x-axis at Q and latus rectum at R(x1,y1)R(x_{1}, y_{1})R(x1​,y1​), x1>0x_{1} > 0x1​>0. If F is a focus of H which is nearer to the point P, then the area of ΔQFR is equal to
  1. (A)464\sqrt{6}46​
  2. (B)6−1\sqrt{6}-16​−1
  3. (C)76−2\frac{7}{\sqrt{6}}-26​7​−2
  4. (D)46−14\sqrt{6}-146​−1

Correct answer: (C)

Step-by-step solution →
Q63·MathematicsSingle correct
If P and Q are two statements, then which of the following compound statement is a tautology ?
  1. (A)((P ⇒ Q) ∧ ~ Q) ⇒ Q
  2. (B)((P ⇒ Q) ∧ ~ Q) ⇒ ~ P
  3. (C)((P ⇒ Q) ∧ ~ Q) ⇒ P
  4. (D)((P ⇒ Q) ∧ ~ Q) ⇒ (P ∧ Q)

Correct answer: (B)

Step-by-step solution →
Q64·MathematicsSingle correct
Let g(x)=∫0xf(t) dtg(x) = \int_{0}^{x} f(t)\,dtg(x)=∫0x​f(t)dt, where f is continuous function in [0, 3] such that 13≤f(t)≤1\frac{1}{3} \le f(t) \le 131​≤f(t)≤1 for all t ∈ [0, 1] and 0≤f(t)≤120 \le f(t) \le \frac{1}{2}0≤f(t)≤21​ for all t ∈ (1, 3]. The largest possible interval in which g(3) lies is :
  1. (A)[−1,−12]\left[-1, -\frac{1}{2}\right][−1,−21​]
  2. (B)[−32,−1]\left[-\frac{3}{2}, -1\right][−23​,−1]
  3. (C)[13,2]\left[\frac{1}{3}, 2\right][31​,2]
  4. (D)[1, 3]

Correct answer: (C)

Step-by-step solution →
Q65·MathematicsSingle correct
Let S1S_{1}S1​ be the sum of first 2n terms of an arithmetic progression. Let S2S_{2}S2​ be the sum of first 4n terms of the same arithmetic progression. If (S2−S1)(S_{2} - S_{1})(S2​−S1​) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to:
  1. (A)1000
  2. (B)7000
  3. (C)5000
  4. (D)3000

Correct answer: (D)

Step-by-step solution →
Q66·MathematicsSingle correct
Let a complex number be w = 1 − 3i\sqrt{3}i3​i. Let another complex number z be such that |zw| = 1 and arg(z) − arg(w) = π2\frac{\pi}{2}2π​. Then the area of the triangle with vertices origin, z and w is equal to :
  1. (A)4
  2. (B)12\frac{1}{2}21​
  3. (C)14\frac{1}{4}41​
  4. (D)2

Correct answer: (B)

Step-by-step solution →
Q67·MathematicsSingle correct
Let in a series of 2n observations, half of them are equal to a and remaining half are equal to −a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a2+b2a^{2} + b^{2}a2+b2 is equal to :
  1. (A)425
  2. (B)650
  3. (C)250
  4. (D)925

Correct answer: (A)

Step-by-step solution →
Q68·MathematicsSingle correct
Let S1:x2+y2=9S_{1} : x^{2} + y^{2} = 9S1​:x2+y2=9 and S2:(x−2)2+y2=1S_{2} : (x-2)^{2} + y^{2} = 1S2​:(x−2)2+y2=1. Then the locus of center of a variable circle S which touches S1S_{1}S1​ internally and S2S_{2}S2​ externally always passes through the points :
  1. (A)(0,±3)\left(0, \pm\sqrt{3}\right)(0,±3​)
  2. (B)(12,±52)\left(\frac{1}{2}, \pm\frac{\sqrt{5}}{2}\right)(21​,±25​​)
  3. (C)(2,±32)\left(2, \pm\frac{3}{2}\right)(2,±23​)
  4. (D)(1,±2)\left(1, \pm 2\right)(1,±2)

Correct answer: (C)

Step-by-step solution →
Q69·MathematicsSingle correct
Let a⃗\vec{a}a and b⃗\vec{b}b be two non-zero vectors perpendicular to each other and ∣a⃗∣=∣b⃗∣\left|\vec{a}\right| = \left|\vec{b}\right|∣a∣=​b​. If ∣a⃗×b⃗∣=∣a⃗∣\left|\vec{a} \times \vec{b}\right| = \left|\vec{a}\right|​a×b​=∣a∣, then the angle between the vectors (a⃗+b⃗+(a⃗×b⃗))\left(\vec{a} + \vec{b} + \left(\vec{a} \times \vec{b}\right)\right)(a+b+(a×b)) and a⃗\vec{a}a is equal to :
  1. (A)sin⁡−1(13)\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)sin−1(3​1​)
  2. (B)cos⁡−1(13)\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)cos−1(3​1​)
  3. (C)cos⁡−1(12)\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)cos−1(2​1​)
  4. (D)sin⁡−1(16)\sin^{-1}\left(\frac{1}{\sqrt{6}}\right)sin−1(6​1​)

Correct answer: (B)

Step-by-step solution →
Q70·MathematicsSingle correct
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
  1. (A)32625\frac{32}{625}62532​
  2. (B)80243\frac{80}{243}24380​
  3. (C)40243\frac{40}{243}24340​
  4. (D)128625\frac{128}{625}625128​

Correct answer: (A)

Step-by-step solution →
Q71·MathematicsSingle correct
Let a tangent be drawn to the ellipse x227+y2=1\frac{x^{2}}{27} + y^{2} = 127x2​+y2=1 at (33cos⁡θ,sin⁡θ)\left(3\sqrt{3}\cos\theta, \sin\theta\right)(33​cosθ,sinθ) where θ∈(0,π2)\theta \in \left(0, \frac{\pi}{2}\right)θ∈(0,2π​). Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
  1. (A)π8\frac{\pi}{8}8π​
  2. (B)π4\frac{\pi}{4}4π​
  3. (C)π6\frac{\pi}{6}6π​
  4. (D)π3\frac{\pi}{3}3π​

Correct answer: (C)

Step-by-step solution →
Q72·MathematicsSingle correct
Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP−1=BPAP^{-1} = BPAP−1=B". Then which of the following is true ?
  1. (A)R is symmetric, transitive but not reflexive,
  2. (B)R is reflexive, symmetric but not transitive
  3. (C)R is an equivalence relation
  4. (D)R is reflexive, transitive but not symmetric

Correct answer: (C)

Step-by-step solution →
Q73·MathematicsSingle correct
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be π3\frac{\pi}{3}3π​. If the radius of the circumcircle ot ΔABC is 2, then the height of the pole is equal to :
  1. (A)233\frac{2\sqrt{3}}{3}323​​
  2. (B)232\sqrt{3}23​
  3. (C)3\sqrt{3}3​
  4. (D)13\frac{1}{\sqrt{3}}3​1​

Correct answer: (B)

Step-by-step solution →
Q74·MathematicsSingle correct
If 15sin⁡4α+10cos⁡4α=615\sin^{4}\alpha + 10\cos^{4}\alpha = 615sin4α+10cos4α=6, for some α ∈ R, then the value of 27sec⁡6α+8 cosec6α27\sec^{6}\alpha + 8\,\mathrm{cosec}^{6}\alpha27sec6α+8cosec6α is equal to :
  1. (A)350
  2. (B)500
  3. (C)400
  4. (D)250

Correct answer: (D)

Step-by-step solution →
Q75·MathematicsSingle correct
The area bounded by the curve 4y2=x2(4−x)(x−2)4y^{2} = x^{2}(4-x)(x-2)4y2=x2(4−x)(x−2) is equal to :
  1. (A)π8\frac{\pi}{8}8π​
  2. (B)3π8\frac{3\pi}{8}83π​
  3. (C)3π2\frac{3\pi}{2}23π​
  4. (D)π16\frac{\pi}{16}16π​

Correct answer: (C)

Step-by-step solution →
Q76·MathematicsSingle correct
Let f : R → R be a function defined as f(x)={sin⁡(a+1)x+sin⁡2x2x, if x<0b, if x=0x+bx3−xbx5/2, if x>0f(x) = \begin{cases} \dfrac{\sin(a+1)x + \sin 2x}{2x} & ,\ \text{if } x < 0 \\ b & ,\ \text{if } x = 0 \\ \dfrac{\sqrt{x + bx^{3}} - \sqrt{x}}{bx^{5/2}} & ,\ \text{if } x > 0 \end{cases}f(x)=⎩⎨⎧​2xsin(a+1)x+sin2x​bbx5/2x+bx3​−x​​​, if x<0, if x=0, if x>0​ If f is continuous at x = 0, then the value of a + b is equal to :
  1. (A)−52-\frac{5}{2}−25​
  2. (B)−2
  3. (C)−3
  4. (D)−32-\frac{3}{2}−23​

Correct answer: (D)

Step-by-step solution →
Q77·MathematicsNumerical
If f(x)f(x)f(x) and g(x) are two polynomials such that the polynomial P(x)=f(x3)+xg(x3)P(x) = f(x^{3}) + xg(x^{3})P(x)=f(x3)+xg(x3) is divisible by x2+x+1x^{2} + x + 1x2+x+1, then P(1) is equal to ________ .

Correct answer: 0

Step-by-step solution →
Q78·MathematicsNumerical
Let I be an identity matrix of order 2 × 2 and P=[2−15−3]P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}P=[25​−1−3​]. Then the value of n ∈ N for which Pn=5I−8PP^{n} = 5I - 8PPn=5I−8P is equal to ________ .

Correct answer: 6

Step-by-step solution →
Q79·MathematicsNumerical
If ∑r=110r!(r3+6r2+2r+5)=α(11!)\sum_{r=1}^{10} r!\left(r^{3} + 6r^{2} + 2r + 5\right) = \alpha(11!)∑r=110​r!(r3+6r2+2r+5)=α(11!), then the value of α is equal to ________ .

Correct answer: 160

Step-by-step solution →
Q80·MathematicsNumerical
The term independent of x in the expansion of [x+1x2/3−x1/3+1−x−1x−x1/2]10\left[\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right]^{10}[x2/3−x1/3+1x+1​−x−x1/2x−1​]10, x ≠ 1, is equal to ________ .

Correct answer: 210

Step-by-step solution →
Q81·MathematicsNumerical
Let P(x) be a real polynomial of degree 3 which vanishes at x = −3. Let P(x) have local minima at x = 1, local maxima at x = −1 and ∫−11P(x) dx=18\int_{-1}^{1} P(x)\,dx = 18∫−11​P(x)dx=18, then the sum of all the coefficients of the polynomial P(x) is equal to ________ .

Correct answer: 8

Step-by-step solution →
Q82·MathematicsNumerical
Let the mirror image of the point (1, 3, a) with respect to the plane r⃗⋅(2i^−j^+k^)−b=0\vec{r} \cdot \left(2\hat{i} - \hat{j} + \hat{k}\right) - b = 0r⋅(2i^−j^​+k^)−b=0 be (−3, 5, 2). Then the value of |a + b| is equal to ________ .

Correct answer: 1

Step-by-step solution →
Q83·MathematicsNumerical
Let fff : R → R satisfy the equation f(x+y)=f(x)⋅f(y)f(x + y) = f(x) \cdot f(y)f(x+y)=f(x)⋅f(y) for all x, y ∈ R and f(x) ≠ 0 for any x ∈ R. If the function fff is differentiable at x = 0 and f′(0)=3f'(0) = 3f′(0)=3, then lim⁡h→01h(f(h)−1)\lim_{h \to 0} \frac{1}{h}\left(f(h) - 1\right)limh→0​h1​(f(h)−1) is equal to ________ .

Correct answer: 3

Step-by-step solution →
Q84·MathematicsNumerical
Let P be a plane containing the line x−13=y+64=z+52\frac{x-1}{3} = \frac{y+6}{4} = \frac{z+5}{2}3x−1​=4y+6​=2z+5​ and parallel to the line x−34=y−2−3=z+57\frac{x-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}4x−3​=−3y−2​=7z+5​. If the point (1, −1, α) lies on the plane P, then the value of |5α| is equal to ________ .

Correct answer: 38

Step-by-step solution →
Q85·MathematicsNumerical
Let y = y(x) be the solution of the differential equation xdy−ydx=(x2−y2) dxxdy - ydx = \sqrt{\left(x^{2} - y^{2}\right)}\,dxxdy−ydx=(x2−y2)​dx , x ≥ 1, with y(1) = 0. If the area bounded by the line x = 1, x=eπx = e^{\pi}x=eπ, y = 0 and y = y(x) is αe2π+β\alpha e^{2\pi} + \betaαe2π+β, then the value of 10(α + β) is equal to ________ .

Correct answer: 4

Step-by-step solution →

Chapters tested in this paper

Open any chapter to practise every past-year question on that topic across all papers, and see how often it has actually appeared.

  • Properties of Solids and Liquids 172/186
  • Three Dimensional Geometry 176/186
  • Matrices and Determinants 180/186
  • Coordination Compounds 176/186
  • Sets, Relations and Functions 165/186
  • Current Electricity 160/186
  • Sequence and Series 164/186
  • p-Block Elements 164/186
  • Definite Integration 168/186
  • Rotational Motion 172/186
  • Redox Reactions and Electrochemistry 177/186
  • Geometrical Optics 172/186
  • Kinematics 156/186
  • Vector Algebra 173/186
  • Differential Equations 167/186
  • Probability 176/186
  • Magnetic Field of Current 147/186
  • Binomial Theorem and Its Simple Applications 158/186
  • Electromagnetic Waves 144/186
  • Chemical Bonding and Molecular Structure 151/186
  • Application of Derivatives 139/186
  • Limits and Continuity 149/186
  • d- and f-Block Elements 126/186
  • Thermodynamics 154/186
  • Aldehydes and Ketones 135/186
  • Equilibrium 163/186
  • Solutions 158/186
  • Laws of Motion 130/186
  • Chemical Thermodynamics 165/186
  • Complex Numbers 165/186
  • Trigonometric Functions 144/186
  • Biomolecules 162/186
  • Chemical Kinetics 169/186
  • Gravitation 152/186
  • Electric Field and Coulomb's Law 133/186
  • Atomic Structure 161/186
  • Electronic Devices 145/186
  • Circles 142/186
  • Dual Nature of Matter and Radiation 155/186
  • Amines 133/186
  • Work, Energy and Power 132/186
  • Some Basic Concepts in Chemistry 129/186
  • Electromagnetic Induction 120/186
  • Wave Optics 130/186
  • Area Under Curves 139/186
  • Alcohols and Ethers 106/186
  • Oscillations 117/186
  • Alternating Currents 108/186
  • Nuclei 116/186
  • Organic Compounds Containing Halogens 109/186
  • Straight Lines 114/186
  • Classification of Elements and Periodicity in Properties 113/186
  • Statistics 118/186
  • Ellipse 103/186
  • Isolation of Metals 106/186
  • Differentiability 91/186
  • Surface Chemistry 98/186
  • Environmental Chemistry 83/186
  • Hydrogen 81/186
  • Hyperbola 77/186
  • Solid State 63/186
  • Chemistry in Everyday Life 60/186
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