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JEE Main 9 April 2024 Shift 1 Question Paper with Answers

9 April 2024 · April session · 90 questions

The complete JEE Main 9 April 2024 Shift 1 paper — all 90 questions with the correct answer for each, tagged to the chapter it tests. Free to read, no account needed.

Physics
30
Chemistry
30
Mathematics
30

Physics — JEE Main 9 April 2024 Shift 1

Q1·PhysicsSingle correct
A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as:
  1. (A)λe>λα>λp\lambda_e > \lambda_\alpha > \lambda_pλe​>λα​>λp​
  2. (B)λα<λp<λe\lambda_\alpha < \lambda_p < \lambda_eλα​<λp​<λe​
  3. (C)λp<λe<λα\lambda_p < \lambda_e < \lambda_\alphaλp​<λe​<λα​
  4. (D)λp>λe>λα\lambda_p > \lambda_e > \lambda_\alphaλp​>λe​>λα​

Correct answer: (B)

Step-by-step solution →
Q2·PhysicsSingle correct
A particle moving in a straight line covers half the distance with speed 6 m/s. The other half is covered in two equal time intervals with speeds 9 m/s and 15 m/s respectively. The average speed of the particle during the motion is :
  1. (A)8.8 m/s
  2. (B)10 m/s
  3. (C)9.2 m/s
  4. (D)8 m/s

Correct answer: (D)

Step-by-step solution →
Q3·PhysicsSingle correct
A plane EM wave is propagating along x direction. It has a wavelength of 4 mm. If electric field is in y direction with the maximum magnitude of 60 Vm−1^{-1}−1, the equation for magnetic field is:
  1. (A)Bz=60sin⁡[π2(x−3×108t)]k^B_z = 60\sin\left[\dfrac{\pi}{2}\left(x - 3\times10^8 t\right)\right]\hat{k}Bz​=60sin[2π​(x−3×108t)]k^ T
  2. (B)Bz=2×10−7sin⁡[π2×103(x−3×108t)]k^B_z = 2\times10^{-7}\sin\left[\dfrac{\pi}{2}\times10^3\left(x - 3\times10^8 t\right)\right]\hat{k}Bz​=2×10−7sin[2π​×103(x−3×108t)]k^ T
  3. (C)Bx=60sin⁡[π2(x−3×108t)]i^B_x = 60\sin\left[\dfrac{\pi}{2}\left(x - 3\times10^8 t\right)\right]\hat{i}Bx​=60sin[2π​(x−3×108t)]i^ T
  4. (D)Bz=2×10−7sin⁡[π2(x−3×108t)]k^B_z = 2\times10^{-7}\sin\left[\dfrac{\pi}{2}\left(x - 3\times10^8 t\right)\right]\hat{k}Bz​=2×10−7sin[2π​(x−3×108t)]k^ T

Correct answer: (B)

Step-by-step solution →
Q4·PhysicsSingle correct
Given below are two statements: Statement (I): When an object is placed at the centre of curvature of a concave lens, image is formed at the centre of curvature of the lens on the other side. Statement (II): Concave lens always forms a virtual and erect image. In the light of the above statements, choose the correct answer from the options given below:
  1. (A)Statement I is false but Statement II is true.
  2. (B)Both Statement I and Statement II are false.
  3. (C)Statement I is true but Statement II is false.
  4. (D)Both Statement I and Statement II are true.

Correct answer: (A)

Step-by-step solution →
Q5·PhysicsSingle correct
A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is 1.42 eV. The wavelength of light emitted from the LED is:
  1. (A)650 nm
  2. (B)1243 nm
  3. (C)875 nm
  4. (D)1400 nm

Correct answer: (C)

Step-by-step solution →
Q6·PhysicsSingle correct
A sphere of relative density σ\sigmaσ and diameter D has concentric cavity of diameter d. The ratio of Dd\dfrac{D}{d}dD​, if it just floats on water in a tank is:
  1. (A)(σσ−1)13\left(\dfrac{\sigma}{\sigma-1}\right)^{\frac{1}{3}}(σ−1σ​)31​
  2. (B)(σ+1σ−1)13\left(\dfrac{\sigma+1}{\sigma-1}\right)^{\frac{1}{3}}(σ−1σ+1​)31​
  3. (C)(σ−1σ)13\left(\dfrac{\sigma-1}{\sigma}\right)^{\frac{1}{3}}(σσ−1​)31​
  4. (D)(σ−2σ+2)13\left(\dfrac{\sigma-2}{\sigma+2}\right)^{\frac{1}{3}}(σ+2σ−2​)31​

Correct answer: (A)

Step-by-step solution →
Q7·PhysicsSingle correct
A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is A3\dfrac{A}{3}3A​ and the height is d, the capacitance of the arrangement is:
  1. (A)11ε0A18d\dfrac{11\varepsilon_0 A}{18 d}18d11ε0​A​
  2. (B)13ε0A17d\dfrac{13\varepsilon_0 A}{17 d}17d13ε0​A​
  3. (C)11ε0A20d\dfrac{11\varepsilon_0 A}{20 d}20d11ε0​A​
  4. (D)18ε0A11d\dfrac{18\varepsilon_0 A}{11 d}11d18ε0​A​

Correct answer: (A)

Step-by-step solution →
Q8·PhysicsSingle correct
A light unstretchable string passing over a smooth light pulley connects two blocks of masses m1m_1m1​ and m2m_2m2​. If the acceleration of the system is g8\dfrac{g}{8}8g​, then the ratio of the masses m2m1\dfrac{m_2}{m_1}m1​m2​​ is:
  1. (A)9 : 7
  2. (B)4 : 3
  3. (C)5 : 3
  4. (D)8 : 1

Correct answer: (A)

Step-by-step solution →
Q9·PhysicsSingle correct
The dimensional formula of latent heat is:
  1. (A)[M0LT−2][M^0 L T^{-2}][M0LT−2]
  2. (B)[MLT−2][M L T^{-2}][MLT−2]
  3. (C)[M0L2T−2][M^0 L^2 T^{-2}][M0L2T−2]
  4. (D)[ML2T−2][M L^2 T^{-2}][ML2T−2]

Correct answer: (C)

Step-by-step solution →
Q10·PhysicsSingle correct
The volume of an ideal gas (γ=1.5\gamma = 1.5γ=1.5) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is:
  1. (A)45\dfrac{4}{5}54​
  2. (B)1625\dfrac{16}{25}2516​
  3. (C)855\dfrac{8}{5\sqrt{5}}55​8​
  4. (D)25\dfrac{2}{\sqrt{5}}5​2​

Correct answer: (C)

Step-by-step solution →
Q11·PhysicsSingle correct
The energy equivalent of 1g of substance is:
  1. (A)11.2×102411.2 \times 10^{24}11.2×1024 MeV
  2. (B)5.6×10125.6 \times 10^{12}5.6×1012 MeV
  3. (C)5.6 eV
  4. (D)5.6×10265.6 \times 10^{26}5.6×1026 MeV

Correct answer: (D)

Step-by-step solution →
Q12·PhysicsSingle correct
An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is xGMem21Rex\dfrac{GM_e m}{21 R_e}x21Re​GMe​m​. The value of x is (take Re=6370R_e = 6370Re​=6370 km):
  1. (A)11
  2. (B)9
  3. (C)12
  4. (D)10

Correct answer: (A)

Step-by-step solution →
Q13·PhysicsSingle correct
Given below are two statements: Statement (I): When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account. Statement (II): Ampere's circuital law does not depend on Biot-Savart's law. In the light of the above statements, choose the correct answer from the options 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: (B)

Step-by-step solution →
Q14·PhysicsSingle correct
A particle of mass m moves on a straight line with its velocity increasing with distance according to the equation v=αxv = \alpha\sqrt{x}v=αx​, where α\alphaα is a constant. The total work done by all the forces applied on the particle during its displacement from x=0x = 0x=0 to x=dx = dx=d, will be:
  1. (A)m2α2d\dfrac{m}{2\alpha^2 d}2α2dm​
  2. (B)md2α2\dfrac{md}{2\alpha^2}2α2md​
  3. (C)mα2d2\dfrac{m\alpha^2 d}{2}2mα2d​
  4. (D)2mα2d2m\alpha^2 d2mα2d

Correct answer: (C)

Step-by-step solution →
Q15·PhysicsSingle correct
A galvanometer has a coil of resistance 200 Ω\OmegaΩ with a full scale deflection at 20 μ\muμA. The value of resistance to be added to use it as an ammeter of range (0–20) mA is:
  1. (A)0.40 Ω\OmegaΩ
  2. (B)0.20 Ω\OmegaΩ
  3. (C)0.50 Ω\OmegaΩ
  4. (D)0.10 Ω\OmegaΩ

Correct answer: (B)

Step-by-step solution →
Q16·PhysicsSingle correct
A heavy iron bar, of weight WWW is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle θ\thetaθ with the horizontal. The weight experienced by the person is:
  1. (A)W2\frac{W}{2}2W​
  2. (B)WWW
  3. (C)Wcos⁡θW\cos\thetaWcosθ
  4. (D)Wsin⁡θW\sin\thetaWsinθ

Correct answer: (A)

Step-by-step solution →
Q17·PhysicsSingle correct
One main scale division of a vernier caliper is equal to mmm units. If nthn^{th}nth division of main scale coincides with (n+1)th(n+1)^{th}(n+1)th division of vernier scale, the least count of the vernier caliper is:
  1. (A)n(n+1)\frac{n}{(n+1)}(n+1)n​
  2. (B)m(n+1)\frac{m}{(n+1)}(n+1)m​
  3. (C)1(n+1)\frac{1}{(n+1)}(n+1)1​
  4. (D)mn(n+1)\frac{m}{n(n+1)}n(n+1)m​

Correct answer: (B)

Step-by-step solution →
Q18·PhysicsSingle correct
A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb:
  1. (A)increases
  2. (B)remains same
  3. (C)becomes zero
  4. (D)decreases

Correct answer: (A)

Step-by-step solution →
Q19·PhysicsSingle correct
The equivalent resistance between A and B is:
  1. (A)18 Ω18\ \Omega18 Ω
  2. (B)25 Ω25\ \Omega25 Ω
  3. (C)27 Ω27\ \Omega27 Ω
  4. (D)19 Ω19\ \Omega19 Ω

Correct answer: (D)

Step-by-step solution →
Q20·PhysicsSingle correct
A sample of 1 mole gas at temperature TTT is adiabatically expanded to double its volume. If adiabatic constant for the gas is γ=32\gamma=\frac{3}{2}γ=23​, then the work done by the gas in the process is:
  1. (A)RT[2−2]RT\left[2-\sqrt{2}\right]RT[2−2​]
  2. (B)RT[2−2]\frac{R}{T}\left[2-\sqrt{2}\right]TR​[2−2​]
  3. (C)RT[2+2]RT\left[2+\sqrt{2}\right]RT[2+2​]
  4. (D)TR[2+2]\frac{T}{R}\left[2+\sqrt{2}\right]RT​[2+2​]

Correct answer: (A)

Step-by-step solution →
Q21·PhysicsNumerical
If a⃗\vec{a}a and b⃗\vec{b}b makes an angle cos⁡−1(59)\cos^{-1}\left(\frac{5}{9}\right)cos−1(95​) with each other, then ∣a⃗+b⃗∣=2 ∣a⃗−b⃗∣|\vec{a}+\vec{b}|=\sqrt{2}\,|\vec{a}-\vec{b}|∣a+b∣=2​∣a−b∣ for ∣a⃗∣=n ∣b⃗∣|\vec{a}|=n\,|\vec{b}|∣a∣=n∣b∣. The integer value of nnn is ______.

Correct answer: 3

Step-by-step solution →
Q22·PhysicsNumerical
At the centre of a half ring of radius R=10R=10R=10 cm and linear charge density 4n C m−14n\ \mathrm{C\ m^{-1}}4n C m−1, the potential is x π Vx\,\pi\ \mathrm{V}xπ V. The value of xxx is ______.

Correct answer: 36

Step-by-step solution →
Q23·PhysicsNumerical
A star has 100% helium composition. It starts to convert three 4He^{4}\mathrm{He}4He into one 12C^{12}\mathrm{C}12C via triple alpha process as 4He+4He+4He→12C+Q^{4}\mathrm{He}+{}^{4}\mathrm{He}+{}^{4}\mathrm{He}\to{}^{12}\mathrm{C}+Q4He+4He+4He→12C+Q. The mass of the star is 2.0×10322.0\times10^{32}2.0×1032 kg and it generates energy at the rate of 5.808×10305.808\times10^{30}5.808×1030 W. The rate of converting these 4He^{4}\mathrm{He}4He to 12C^{12}\mathrm{C}12C is n×1042 s−1n\times10^{42}\ \mathrm{s^{-1}}n×1042 s−1, where nnn is ______. [Take, mass of 4He=4.0026^{4}\mathrm{He}=4.00264He=4.0026 u, mass of 12C=12^{12}\mathrm{C}=1212C=12 u]

Correct answer: 5

Step-by-step solution →
Q24·PhysicsNumerical
In a Young's double slit experiment, the intensity at a point is (14)th\left(\frac{1}{4}\right)^{th}(41​)th of the maximum intensity, the minimum distance of the point from the central maximum is ______ μ\muμm. (Given: λ=600\lambda=600λ=600 nm, d=1.0d=1.0d=1.0 mm, D=1.0D=1.0D=1.0 m)

Correct answer: 200

Step-by-step solution →
Q25·PhysicsNumerical
A string is wrapped around the rim of a wheel of moment of inertia 0.40 kgm20.40\ \mathrm{kgm^{2}}0.40 kgm2 and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is xxx rad/s, where xxx is ______.

Correct answer: 100

Step-by-step solution →
Q26·PhysicsNumerical
A square loop of edge length 2 m carrying current of 2 A is placed with its edges parallel to the x-y axis. A magnetic field is passing through the x-y plane and expressed as B⃗=B0(1+4x)k^\vec{B}=B_{0}(1+4x)\hat{k}B=B0​(1+4x)k^, where B0=5B_{0}=5B0​=5 T. The net magnetic force experienced by the loop is ______ N.

Correct answer: 160

Step-by-step solution →
Q27·PhysicsNumerical
Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire. Young's modulus of the material of wire is 1×1011 N m−21\times10^{11}\ \mathrm{N\ m^{-2}}1×1011 N m−2. Original length of the wire is 2 m and the area of cross section is 2 cm22\ \mathrm{cm^{2}}2 cm2. The wire will extend in length by ______ μ\muμm.

Correct answer: 20

Step-by-step solution →
Q28·PhysicsNumerical
When a coil is connected across a 20 V dc supply, it draws a current of 5 A. When it is connected across 20 V, 50 Hz ac supply, it draws a current of 4 A. The self inductance of the coil is ______ mH. (Take π=3\pi=3π=3)

Correct answer: 10

Step-by-step solution →
Q29·PhysicsNumerical
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of 4 m, 2 ms−12\ \mathrm{ms^{-1}}2 ms−1 and 16 ms−216\ \mathrm{ms^{-2}}16 ms−2 at a certain instant. The amplitude of the motion is x\sqrt{x}x​ m where xxx is ______.

Correct answer: 17

Step-by-step solution →
Q30·PhysicsNumerical
The current flowing through the 1 Ω1\ \Omega1 Ω resistor is n10\frac{n}{10}10n​ A. The value of nnn is ______.

Correct answer: 25

Step-by-step solution →

Chemistry — JEE Main 9 April 2024 Shift 1

Q31·ChemistrySingle correct
The molar conductivity for electrolytes A and B are plotted against C1/2C^{1/2}C1/2 as shown below. Electrolytes A and B respectively are :
  1. (A)A: Weak electrolyte, B: weak electrolyte
  2. (B)A: Strong electrolyte, B: strong electrolyte
  3. (C)A: Weak electrolyte, B: strong electrolyte
  4. (D)A: Strong electrolyte, B: weak electrolyte

Correct answer: (C)

Step-by-step solution →
Q32·ChemistrySingle correct
Methods used for purification of organic compounds are based on :
  1. (A)neither on nature of compound nor on the impurity present.
  2. (B)nature of compound only.
  3. (C)nature of compound and presence of impurity.
  4. (D)presence of impurity only.

Correct answer: (C)

Step-by-step solution →
Q33·ChemistrySingle correct
In the following sequence of reaction, the major products B and C respectively are :
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (A)

Step-by-step solution →
Q34·ChemistrySingle correct
Correct order of basic strength of Pyrrole, Pyridine and Piperidine is :
  1. (A)Piperidine > Pyridine > Pyrrole
  2. (B)Pyrrole > Pyridine > Piperidine
  3. (C)Pyridine > Piperidine > Pyrrole
  4. (D)Pyrrole > Piperidine > Pyridine

Correct answer: (A)

Step-by-step solution →
Q35·ChemistrySingle correct
In which one of the following pairs the central atoms exhibit sp2sp^2sp2 hybridization ?
  1. (A)BF3BF_3BF3​ and NO2−NO_2^-NO2−​
  2. (B)NH2−NH_2^-NH2−​ and H2OH_2OH2​O
  3. (C)H2OH_2OH2​O and NO2NO_2NO2​
  4. (D)NH2−NH_2^-NH2−​ and BF3BF_3BF3​

Correct answer: (A)

Step-by-step solution →
Q36·ChemistrySingle correct
The F−F^-F− ions make the enamel on teeth much harder by converting hydroxyapatite (the enamel on the surface of teeth) into much harder fluoroapatite having the formula.
  1. (A)[3(Ca3(PO4)2).CaF2][3(Ca_3(PO_4)_2).CaF_2][3(Ca3​(PO4​)2​).CaF2​]
  2. (B)[3(Ca2(PO4)2).Ca(OH)2][3(Ca_2(PO_4)_2).Ca(OH)_2][3(Ca2​(PO4​)2​).Ca(OH)2​]
  3. (C)[3(Ca3(PO4)3).CaF2][3(Ca_3(PO_4)_3).CaF_2][3(Ca3​(PO4​)3​).CaF2​]
  4. (D)[3(Ca3(PO4)2).Ca(OH)2][3(Ca_3(PO_4)_2).Ca(OH)_2][3(Ca3​(PO4​)2​).Ca(OH)2​]

Correct answer: (A)

Step-by-step solution →
Q37·Chemistry·Electronic Effects and StabilitySingle correct
Relative stability of the contributing structures is : (I) CH2=CH−C(=O)−HCH_2=CH-C(=O)-HCH2​=CH−C(=O)−H ; (II) CH2−CH=C(O−)−HCH_2-CH=C(O^-)-HCH2​−CH=C(O−)−H (with + charge on terminal CH2CH_2CH2​) ; (III) CH2−CH=C(O+)−HCH_2-CH=C(O^+)-HCH2​−CH=C(O+)−H (with carbanion lone pair on terminal CH2CH_2CH2​)
  1. (A)(I) > (III) > (II)
  2. (B)(I) > (II) > (III)
  3. (C)(II) > (I) > (III)
  4. (D)(III) > (II) > (I)

Correct answer: (B)

Step-by-step solution →
Q38·ChemistrySingle correct
Given below are two statements : Statement (I) : The oxidation state of an element in a particular compound is the charge acquired by its atom on the basis of electron gain enthalpy consideration from other atoms in the molecule. Statement (II) : pπp\pipπ-pπp\pipπ bond formation is more prevalent in second period elements over other periods. 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 incorrect
  2. (B)Statement I is correct but Statement II is incorrect
  3. (C)Both Statement I and Statement II are correct
  4. (D)Statement I is incorrect but Statement II is correct

Correct answer: (D)

Step-by-step solution →
Q39·ChemistrySingle correct
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R) : Assertion (A) : SN2S_N2SN​2 reaction of C6H5CH2BrC_6H_5CH_2BrC6​H5​CH2​Br occurs more readily than the SN2S_N2SN​2 reaction of CH3CH2BrCH_3CH_2BrCH3​CH2​Br. Reason (R) : The partially bonded unhybridized p-orbital that develops in the trigonal bipyramidal transition state is stabilized by conjugation with the phenyl ring. In the light of the above statements, choose the most appropriate answer from the options given below :
  1. (A)(A) is not correct but (R) is correct
  2. (B)Both (A) and (R) are correct but (R) is not the correct explanation of (A)
  3. (C)Both (A) and (R) are correct and (R) is the correct explanation of (A)
  4. (D)(A) is correct but (R) is not correct

Correct answer: (C)

Step-by-step solution →
Q40·ChemistrySingle correct
For the given compounds, the correct order of increasing pKapK_apKa​ value :
  1. (A)(E) < (D) < (C) < (B) < (A)
  2. (B)(D) < (E) < (C) < (B) < (A)
  3. (C)(E) < (D) < (B) < (A) < (C)
  4. (D)(B) < (D) < (A) < (C) < (E)

Correct answer: (D)

Step-by-step solution →
Q41·ChemistrySingle correct
Given below are two statements : one is labelled as Assertion (A) : and the other is labelled as Reason (R). Assertion (A) : Both rhombic and monoclinic sulphur exist as S8S_8S8​ while oxygen exists as O2O_2O2​. Reason (R) : Oxygen forms pπp\pipπ-pπp\pipπ multiple bonds with itself and other elements having small size and high electronegativity like C, N, which is not possible for sulphur. In the light of the above statements, choose the most appropriate answer from the options given below :
  1. (A)Both (A) and (R) are correct and (R) is the correct explanation of (A).
  2. (B)Both (A) and (R) are correct but (R) is not the correct explanation of (A).
  3. (C)(A) is correct but (R) is not correct.
  4. (D)(A) is not correct but (R) is correct.

Correct answer: (A)

Step-by-step solution →
Q42·ChemistrySingle correct
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The total number of geometrical isomers shown by [Co(en)2Cl2]+[Co(en)_2Cl_2]^+[Co(en)2​Cl2​]+ complex ion is three. Reason (R) : [Co(en)2Cl2]+[Co(en)_2Cl_2]^+[Co(en)2​Cl2​]+ complex ion has an octahedral geometry. In the light of the above statements, choose the most appropriate answer from the options given below :
  1. (A)Both (A) and (R) are correct and (R) is the correct explanation of (A).
  2. (B)(A) is correct but (R) is not correct.
  3. (C)(A) is not correct but (R) is correct.
  4. (D)Both (A) and (R) are correct but (R) is not the correct explanation of (A).

Correct answer: (C)

Step-by-step solution →
Q43·ChemistrySingle correct
The electronic configuration of Cu(II) is 3d93d^93d9 whereas that of Cu(I) is 3d103d^{10}3d10. Which of the following is correct ?
  1. (A)Cu(II) is less stable
  2. (B)Stability of Cu(I) and Cu(II) depends on nature of copper salts
  3. (C)Cu(II) is more stable
  4. (D)Cu(I) and Cu(II) are equally stable

Correct answer: (C)

Step-by-step solution →
Q44·ChemistrySingle correct
In the following sequence of reactions, benzene reacts with succinic anhydride in presence of AlCl3AlCl_3AlCl3​ to give A, which on treatment with Zn-Hg/HCl gives B, which on treatment with Conc. H2SO4H_2SO_4H2​SO4​ gives C. What is the structure of C ?
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (A)

Step-by-step solution →
Q45·ChemistrySingle correct
Compare the energies of following sets of quantum numbers for multielectron system. (A) n=4,l=1n = 4, l = 1n=4,l=1 ; (B) n=4,l=2n = 4, l = 2n=4,l=2 ; (C) n=3,l=1n = 3, l = 1n=3,l=1 ; (D) n=3,l=2n = 3, l = 2n=3,l=2 ; (E) n=4,l=0n = 4, l = 0n=4,l=0. Choose the correct answer from the options given below :
  1. (A)(B) > (A) > (C) > (E) > (D)
  2. (B)(E) > (C) < (D) < (A) < (B)
  3. (C)(E) > (C) > (A) > (D) > (B)
  4. (D)(C) < (E) < (D) < (A) < (B)

Correct answer: (D)

Step-by-step solution →
Q46·ChemistrySingle correct
Identify major product "X" formed in the following reaction: Benzene→Anhydrous AlCl3/CuClCO,HClXBenzene \xrightarrow[\text{Anhydrous } AlCl_3/CuCl]{CO, HCl} XBenzeneCO,HClAnhydrous AlCl3​/CuCl​X (Major product)
  1. (A)(1)
  2. (B)(2)
  3. (C)(3)
  4. (D)(4)

Correct answer: (C)

Step-by-step solution →
Q47·ChemistrySingle correct
Identify the product A and product B in the following set of reactions. Propene (CH3−CH=CH2CH_3-CH=CH_2CH3​−CH=CH2​) undergoes (i) H2O,H+H_2O, H^+H2​O,H+ to give major product A, and (ii) hydroboration-oxidation with (BH3)2(BH_3)_2(BH3​)2​ then H2O,H2O2,OH−H_2O, H_2O_2, OH^-H2​O,H2​O2​,OH− to give major product B.
  1. (A)A-CH3CH2CH2−OHCH_3CH_2CH_2-OHCH3​CH2​CH2​−OH, B-CH3CH2CH2−OHCH_3CH_2CH_2-OHCH3​CH2​CH2​−OH
  2. (B)A-CH3CH2CH2−OHCH_3CH_2CH_2-OHCH3​CH2​CH2​−OH, B-CH3CH(OH)CH3CH_3CH(OH)CH_3CH3​CH(OH)CH3​
  3. (C)A-CH3CH(OH)CH3CH_3CH(OH)CH_3CH3​CH(OH)CH3​, B-CH3CH2CH2−OHCH_3CH_2CH_2-OHCH3​CH2​CH2​−OH
  4. (D)A-CH3CH2CH3CH_3CH_2CH_3CH3​CH2​CH3​, B-CH3CH2CH3CH_3CH_2CH_3CH3​CH2​CH3​

Correct answer: (C)

Step-by-step solution →
Q48·ChemistrySingle correct
On reaction of Lead Sulphide with dilute nitric acid which of the following is notnotnot formed?
  1. (A)Lead nitrate
  2. (B)Sulphur
  3. (C)Nitric oxide
  4. (D)Nitrous oxide

Correct answer: (D)

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Q49·ChemistrySingle correct
Identify the incorrectincorrectincorrect statements regarding primary standard of titrimetric analysis (A) It should be purely available in dry form. (B) It should not undergo chemical change in air. (C) It should be hygroscopic and should react with another chemical instantaneously and stoichiometrically. (D) It should be readily soluble in water. (E) KMnO4KMnO_4KMnO4​ & NaOHNaOHNaOH can be used as primary standard. Choose the correct answer from the options given below:
  1. (A)(C) and (D) only
  2. (B)(B) and (E) only
  3. (C)(A) and (B) only
  4. (D)(C) and (E) only

Correct answer: (D)

Step-by-step solution →
Q50·ChemistrySingle correct
0.05M CuSO4CuSO_4CuSO4​ when treated with 0.01M K2Cr2O7K_2Cr_2O_7K2​Cr2​O7​ gives green colour solution of Cu2Cr2O7Cu_2Cr_2O_7Cu2​Cr2​O7​. [SPM : Semi Permeable Membrane]. A container is divided by a semi-permeable membrane (SPM): Side X contains K2Cr2O7K_2Cr_2O_7K2​Cr2​O7​ and Side Y contains CuSO4CuSO_4CuSO4​. Due to osmosis:
  1. (A)Green colour formation observed on side Y.
  2. (B)Green colour formation observed on side X.
  3. (C)Molarity of K2Cr2O7K_2Cr_2O_7K2​Cr2​O7​ solution is lowered.
  4. (D)Molarity of CuSO4CuSO_4CuSO4​ solution is lowered.

Correct answer: (D)

Step-by-step solution →
Q51·ChemistryNumerical
The heat of solution of anhydrous CuSO4CuSO_4CuSO4​ and CuSO4⋅5H2OCuSO_4\cdot 5H_2OCuSO4​⋅5H2​O are −70-70−70 kJ mol−1mol^{-1}mol−1 and +12+12+12 kJ mol−1mol^{-1}mol−1 respectively. The heat of hydration of CuSO4CuSO_4CuSO4​ to CuSO4⋅5H2OCuSO_4\cdot 5H_2OCuSO4​⋅5H2​O is −x-x−x kJ. The value of x is _____.

Correct answer: 82

Step-by-step solution →
Q52·ChemistryNumerical
Given below are two statements: Statement I: The rate law for the reaction A+B→CA + B \rightarrow CA+B→C is rate (r)=k[A]2[B](r) = k[A]^2[B](r)=k[A]2[B]. When the concentration of both A and B is doubled, the reaction rate is increased "x" times. Statement II: The figure is showing "the variation in concentration against time plot" for a "y" order reaction. The value of x + y is _____.

Correct answer: 8

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Q53·Chemistry·Electronic Effects and StabilityNumerical
How many compounds among the following compounds show inductive, mesomeric as well as hyperconjugation effects?

Correct answer: 4

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Q54·ChemistryNumerical
The standard reduction potentials at 298 K for the following half cells are given below: Cr2O72−+14H++6e−→2Cr3++7H2OCr_2O_7^{2-} + 14H^+ + 6e^- \rightarrow 2Cr^{3+} + 7H_2OCr2​O72−​+14H++6e−→2Cr3++7H2​O, E∘=1.33E^\circ = 1.33E∘=1.33V Fe3+(aq)+3e−→FeFe^{3+}(aq) + 3e^- \rightarrow FeFe3+(aq)+3e−→Fe, E∘=−0.04E^\circ = -0.04E∘=−0.04V Ni2+(aq)+2e−→NiNi^{2+}(aq) + 2e^- \rightarrow NiNi2+(aq)+2e−→Ni, E∘=−0.25E^\circ = -0.25E∘=−0.25V Ag+(aq)+e−→AgAg^+(aq) + e^- \rightarrow AgAg+(aq)+e−→Ag, E∘=0.80E^\circ = 0.80E∘=0.80V Au3+(aq)+3e−→AuAu^{3+}(aq) + 3e^- \rightarrow AuAu3+(aq)+3e−→Au, E∘=1.40E^\circ = 1.40E∘=1.40V Consider the given electrochemical reactions, The number of metal(s) which will be oxidized be Cr2O72−Cr_2O_7^{2-}Cr2​O72−​, in aqueous solution is _____.

Correct answer: 3

Step-by-step solution →
Q55·ChemistryNumerical
When equal volume of 1M HClHClHCl and 1M H2SO4H_2SO_4H2​SO4​ are separately neutralised by excess volume of 1M NaOHNaOHNaOH solution. X and y kJ of heat is liberated respectively. The value of y/x is _____.

Correct answer: 2

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Q56·ChemistryNumerical
Molarity (M) of an aqueous solution containing x g of anhyd. CuSO4CuSO_4CuSO4​ in 500 mL solution at 32 ∘^\circ∘C is 2×10−12 \times 10^{-1}2×10−1 M. Its molality will be _____ ×10−3\times 10^{-3}×10−3 m. (nearest integer). [Given density of the solution = 1.25 g/mL.]

Correct answer: 164

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Q57·ChemistryNumerical
The total number of species from the following in which one unpaired electron is present, is _____. N2,O2,C2−,O2−,O22−,H2+,CN−,He2+N_2, O_2, C_2^-, O_2^-, O_2^{2-}, H_2^+, CN^-, He_2^+N2​,O2​,C2−​,O2−​,O22−​,H2+​,CN−,He2+​

Correct answer: 4

Step-by-step solution →
Q58·ChemistryNumerical
Number of ambidentate ligands among the following is _____. NO2−,SCN−,C2O42−,NH3,CN−,SO42−,H2ONO_2^-, SCN^-, C_2O_4^{2-}, NH_3, CN^-, SO_4^{2-}, H_2ONO2−​,SCN−,C2​O42−​,NH3​,CN−,SO42−​,H2​O.

Correct answer: 3

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Q59·ChemistryNumerical
Total number of essential amino acid among the given list of amino acids is _____. Arginine, Phenylalanine, Aspartic acid, Cysteine, Histidine, Valine, Proline

Correct answer: 4

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Q60·ChemistryNumerical
Number of colourless lanthanoid ions among the following is _____. Eu3+,Lu3+,Nd3+,La3+,Sm3+Eu^{3+}, Lu^{3+}, Nd^{3+}, La^{3+}, Sm^{3+}Eu3+,Lu3+,Nd3+,La3+,Sm3+

Correct answer: 2

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Mathematics — JEE Main 9 April 2024 Shift 1

Q61·MathematicsSingle correct
Let the line L intersect the lines x−2=−y=z−1x - 2 = -y = z - 1x−2=−y=z−1, 2(x+1)=2(y−1)=z+12(x + 1) = 2(y - 1) = z + 12(x+1)=2(y−1)=z+1 and be parallel to the line x−23=y−11=z−22\frac{x - 2}{3} = \frac{y - 1}{1} = \frac{z - 2}{2}3x−2​=1y−1​=2z−2​. Then which of the following points lies on L?
  1. (A)(−13,1,1)\left(-\frac{1}{3}, 1, 1\right)(−31​,1,1)
  2. (B)(−13,1,−1)\left(-\frac{1}{3}, 1, -1\right)(−31​,1,−1)
  3. (C)(−13,−1,−1)\left(-\frac{1}{3}, -1, -1\right)(−31​,−1,−1)
  4. (D)(−13,−1,1)\left(-\frac{1}{3}, -1, 1\right)(−31​,−1,1)

Correct answer: (B)

Step-by-step solution →
Q62·MathematicsSingle correct
The parabola y2=4xy^2 = 4xy2=4x divides the area of the circle x2+y2=5x^2 + y^2 = 5x2+y2=5 in two parts. The area of the smaller part is equal to:
  1. (A)23+5sin⁡−1(25)\frac{2}{3} + 5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)32​+5sin−1(5​2​)
  2. (B)13+5sin⁡−1(25)\frac{1}{3} + 5\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)31​+5sin−1(5​2​)
  3. (C)13+5sin⁡−1(25)\frac{1}{3} + \sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)31​+5​sin−1(5​2​)
  4. (D)23+5sin⁡−1(25)\frac{2}{3} + \sqrt{5}\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)32​+5​sin−1(5​2​)

Correct answer: (A)

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Q63·MathematicsSingle correct
The solution curve, of the differential equation 2ydydx+3=5dydx2y\frac{dy}{dx} + 3 = 5\frac{dy}{dx}2ydxdy​+3=5dxdy​, passing through the point (0,1)(0, 1)(0,1) is a conic, whose vertex lies on the line:
  1. (A)2x+3y=92x + 3y = 92x+3y=9
  2. (B)2x+3y=−92x + 3y = -92x+3y=−9
  3. (C)2x+3y=−62x + 3y = -62x+3y=−6
  4. (D)2x+3y=62x + 3y = 62x+3y=6

Correct answer: (A)

Step-by-step solution →
Q64·MathematicsSingle correct
A ray of light coming from the point P(1,2)P(1, 2)P(1,2) gets reflected from the point Q on the x-axis and then passes through the point R(4,3)R(4, 3)R(4,3). If the point S(h,k)S(h, k)S(h,k) is such that PQRS is a parallelogram, then hk2hk^2hk2 is equal to:
  1. (A)80
  2. (B)90
  3. (C)60
  4. (D)70

Correct answer: (D)

Step-by-step solution →
Q65·MathematicsSingle correct
Let λ,μ∈R\lambda, \mu \in \mathbb{R}λ,μ∈R. If the system of equations 3x+5y+λz=33x + 5y + \lambda z = 33x+5y+λz=3, 7x+11y−9z=27x + 11y - 9z = 27x+11y−9z=2, 97x+155y−189z=μ97x + 155y - 189z = \mu97x+155y−189z=μ has infinitely many solutions, then μ+2λ\mu + 2\lambdaμ+2λ is equal to:
  1. (A)25
  2. (B)24
  3. (C)27
  4. (D)22

Correct answer: (A)

Step-by-step solution →
Q66·MathematicsSingle correct
The coefficient of x70x^{70}x70 in x2(1+x)98+x3(1+x)97+x4(1+x)96+…+x54(1+x)46x^2(1 + x)^{98} + x^3(1 + x)^{97} + x^4(1 + x)^{96} + \ldots + x^{54}(1 + x)^{46}x2(1+x)98+x3(1+x)97+x4(1+x)96+…+x54(1+x)46 is 99Cp−46Cq{}^{99}C_p - {}^{46}C_q99Cp​−46Cq​. Then a possible value to p+qp + qp+q is:
  1. (A)55
  2. (B)61
  3. (C)68
  4. (D)83

Correct answer: (D)

Step-by-step solution →
Q67·MathematicsSingle correct
Let ∫2−tan⁡x3+tan⁡x dx=12(αx+log⁡e∣βsin⁡x+γcos⁡x∣)+C\int \frac{2 - \tan x}{3 + \tan x}\,dx = \frac{1}{2}\left(\alpha x + \log_e |\beta \sin x + \gamma \cos x|\right) + C∫3+tanx2−tanx​dx=21​(αx+loge​∣βsinx+γcosx∣)+C, where C is the constant of integration. Then α+γβ\alpha + \frac{\gamma}{\beta}α+βγ​ is equal to:
  1. (A)3
  2. (B)1
  3. (C)4
  4. (D)7

Correct answer: (C)

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Q68·MathematicsSingle correct
A variable line L passes through the point (3,5)(3, 5)(3,5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is:
  1. (A)30
  2. (B)25
  3. (C)40
  4. (D)35

Correct answer: (A)

Step-by-step solution →
Q69·MathematicsSingle correct
Let ∣cos⁡θcos⁡(60−θ)cos⁡(60+θ)∣≤18|\cos\theta \cos(60 - \theta)\cos(60 + \theta)| \le \frac{1}{8}∣cosθcos(60−θ)cos(60+θ)∣≤81​, θ∈[0,2π]\theta \in [0, 2\pi]θ∈[0,2π]. Then, the sum of all θ∈[0,2π]\theta \in [0, 2\pi]θ∈[0,2π], where cos⁡3θ\cos 3\thetacos3θ attains its maximum value, is:
  1. (A)9π9\pi9π
  2. (B)18π18\pi18π
  3. (C)6π6\pi6π
  4. (D)15π15\pi15π

Correct answer: (C)

Step-by-step solution →
Q70·MathematicsSingle correct
Let OA⃗=2a⃗\vec{OA} = 2\vec{a}OA=2a, OB⃗=6a⃗+5b⃗\vec{OB} = 6\vec{a} + 5\vec{b}OB=6a+5b and OC⃗=3b⃗\vec{OC} = 3\vec{b}OC=3b, where O is the origin. If the area of the parallelogram with adjacent sides OA⃗\vec{OA}OA and OC⃗\vec{OC}OC is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to:
  1. (A)38
  2. (B)40
  3. (C)32
  4. (D)35

Correct answer: (D)

Step-by-step solution →
Q71·MathematicsSingle correct
If the domain of the function f(x)=sin⁡−1(x−12x+3)f(x) = \sin^{-1}\left(\frac{x - 1}{2x + 3}\right)f(x)=sin−1(2x+3x−1​) is R−(α,β)\mathbb{R} - (\alpha, \beta)R−(α,β), then 12αβ12\alpha\beta12αβ is equal to:
  1. (A)36
  2. (B)24
  3. (C)40
  4. (D)32

Correct answer: (D)

Step-by-step solution →
Q72·MathematicsSingle correct
If the sum of series 11⋅(1+d)+1(1+d)(1+2d)+…+1(1+9d)(1+10d)\frac{1}{1\cdot(1 + d)} + \frac{1}{(1 + d)(1 + 2d)} + \ldots + \frac{1}{(1 + 9d)(1 + 10d)}1⋅(1+d)1​+(1+d)(1+2d)1​+…+(1+9d)(1+10d)1​ is equal to 5, then 50d50d50d is equal to:
  1. (A)20
  2. (B)5
  3. (C)15
  4. (D)10

Correct answer: (B)

Step-by-step solution →
Q73·Mathematics·DifferentiabilitySingle correct
Let f(x)=ax3+bx2+cx+41f(x) = ax^3 + bx^2 + cx + 41f(x)=ax3+bx2+cx+41 be such that f(1)=40f(1) = 40f(1)=40, f′(1)=2f'(1) = 2f′(1)=2 and f′′(1)=4f''(1) = 4f′′(1)=4. Then a2+b2+c2a^2 + b^2 + c^2a2+b2+c2 is equal to:
  1. (A)62
  2. (B)73
  3. (C)54
  4. (D)51

Correct answer: (D)

Step-by-step solution →
Q74·MathematicsSingle correct
Let a circle passing through (2,0)(2, 0)(2,0) have its centre at the point (h,k)(h, k)(h,k). Let (xc,yc)(x_c, y_c)(xc​,yc​) be the point of intersection of the lines 3x+5y=13x + 5y = 13x+5y=1 and (2+c)x+5c2y=1(2 + c)x + 5c^2 y = 1(2+c)x+5c2y=1. If h=lim⁡c→1xch = \lim_{c \to 1} x_ch=limc→1​xc​ and k=lim⁡c→1yck = \lim_{c \to 1} y_ck=limc→1​yc​, then the equation of the circle is:
  1. (A)25x2+25y2−20x+2y−60=025x^2 + 25y^2 - 20x + 2y - 60 = 025x2+25y2−20x+2y−60=0
  2. (B)5x2+5y2−4x−2y−12=05x^2 + 5y^2 - 4x - 2y - 12 = 05x2+5y2−4x−2y−12=0
  3. (C)25x2+25y2−2x+2y−60=025x^2 + 25y^2 - 2x + 2y - 60 = 025x2+25y2−2x+2y−60=0
  4. (D)5x2+5y2−4x+2y−12=05x^2 + 5y^2 - 4x + 2y - 12 = 05x2+5y2−4x+2y−12=0

Correct answer: (A)

Step-by-step solution →
Q75·MathematicsSingle correct
The shortest distance between the line x−34=y+7−11=z−15\frac{x - 3}{4} = \frac{y + 7}{-11} = \frac{z - 1}{5}4x−3​=−11y+7​=5z−1​ and x−53=y−9−6=z+21\frac{x - 5}{3} = \frac{y - 9}{-6} = \frac{z + 2}{1}3x−5​=−6y−9​=1z+2​ is:
  1. (A)187563\frac{187}{\sqrt{563}}563​187​
  2. (B)178563\frac{178}{\sqrt{563}}563​178​
  3. (C)185563\frac{185}{\sqrt{563}}563​185​
  4. (D)179563\frac{179}{\sqrt{563}}563​179​

Correct answer: (A)

Step-by-step solution →
Q76·MathematicsSingle correct
The frequency distribution of the age of students in a class of 40 students is given below. Age: 15, 16, 17, 18, 19, 20; No. of Students: 5, 8, 5, 12, xxx, yyy. If the mean deviation about the median is 1.251.251.25, then 4x+5y4x + 5y4x+5y is equal to :
  1. (A)43
  2. (B)44
  3. (C)47
  4. (D)46

Correct answer: (B)

Step-by-step solution →
Q77·MathematicsSingle correct
The solution of the differential equation (x2+y2)dx−5xy dy=0(x^2 + y^2)dx - 5xy\, dy = 0(x2+y2)dx−5xydy=0, y(1)=0y(1) = 0y(1)=0, is :
  1. (A)∣x2−4y2∣5=x2\left|x^2 - 4y^2\right|^5 = x^2​x2−4y2​5=x2
  2. (B)∣x2−2y2∣6=x\left|x^2 - 2y^2\right|^6 = x​x2−2y2​6=x
  3. (C)∣x2−4y2∣6=x\left|x^2 - 4y^2\right|^6 = x​x2−4y2​6=x
  4. (D)∣x2−2y2∣5=x2\left|x^2 - 2y^2\right|^5 = x^2​x2−2y2​5=x2

Correct answer: (A)

Step-by-step solution →
Q78·MathematicsSingle correct
Let three vectors a⃗=αi^+4j^+2k^\vec{a} = \alpha\hat{i} + 4\hat{j} + 2\hat{k}a=αi^+4j^​+2k^, b⃗=5i^+3j^+4k^\vec{b} = 5\hat{i} + 3\hat{j} + 4\hat{k}b=5i^+3j^​+4k^, c⃗=xi^+yj^+zk^\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}c=xi^+yj^​+zk^ form a triangle such that c⃗=a⃗−b⃗\vec{c} = \vec{a} - \vec{b}c=a−b and the area of the triangle is 565\sqrt{6}56​. If α\alphaα is a positive real number, then ∣c⃗∣2|\vec{c}|^2∣c∣2 is :
  1. (A)16
  2. (B)14
  3. (C)12
  4. (D)10

Correct answer: (B)

Step-by-step solution →
Q79·MathematicsSingle correct
Let α,β\alpha, \betaα,β be the roots of the equation x2+22 x−1=0x^2 + 2\sqrt{2}\,x - 1 = 0x2+22​x−1=0. The quadratic equation, whose roots are α4+β4\alpha^4 + \beta^4α4+β4 and 110(α6+β6)\frac{1}{10}(\alpha^6 + \beta^6)101​(α6+β6), is :
  1. (A)x2−190x+9466=0x^2 - 190x + 9466 = 0x2−190x+9466=0
  2. (B)x2−195x+9466=0x^2 - 195x + 9466 = 0x2−195x+9466=0
  3. (C)x2−195x+9506=0x^2 - 195x + 9506 = 0x2−195x+9506=0
  4. (D)x2−180x+9506=0x^2 - 180x + 9506 = 0x2−180x+9506=0

Correct answer: (C)

Step-by-step solution →
Q80·MathematicsSingle correct
Let f(x)=x2+9f(x) = x^2 + 9f(x)=x2+9, g(x)=xx−9g(x) = \frac{x}{x-9}g(x)=x−9x​ and a=f∘g(10)a = f\circ g(10)a=f∘g(10), b=g∘f(3)b = g\circ f(3)b=g∘f(3). If eee and lll denote the eccentricity and the length of the latus rectum of the ellipse x2a+y2b=1\frac{x^2}{a} + \frac{y^2}{b} = 1ax2​+by2​=1, then 8e2+l28e^2 + l^28e2+l2 is equal to.
  1. (A)16
  2. (B)8
  3. (C)6
  4. (D)12

Correct answer: (B)

Step-by-step solution →
Q81·MathematicsNumerical
Let aaa, bbb and ccc denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1,2,3,41, 2, 3, 41,2,3,4. If the probability that ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 has all real roots is mn\frac{m}{n}nm​, gcd⁡(m,n)=1\gcd(m, n) = 1gcd(m,n)=1, then m+nm + nm+n is equal to ________.

Correct answer: 19

Step-by-step solution →
Q82·MathematicsNumerical
The sum of the square of the modulus of the elements in the set {z=a+ib:a,b∈Z,z∈C,∣z−1∣≤1,∣z−5∣≤∣z−5i∣}\{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z-1| \le 1, |z-5| \le |z-5i|\}{z=a+ib:a,b∈Z,z∈C,∣z−1∣≤1,∣z−5∣≤∣z−5i∣} is ________.

Correct answer: 9

Step-by-step solution →
Q83·MathematicsNumerical
Let the set of all positive values of λ\lambdaλ, for which the point of local minimum of the function (1+x(λ2−x2))\left(1 + x(\lambda^2 - x^2)\right)(1+x(λ2−x2)) satisfies x2+x+2x2+5x+6<0\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0x2+5x+6x2+x+2​<0, be (α,β)(\alpha, \beta)(α,β). Then α2+β2\alpha^2 + \beta^2α2+β2 is equal to ________.

Correct answer: 39

Step-by-step solution →
Q84·MathematicsNumerical
Let lim⁡n→∞(nn4+1−2n(n2+1)n4+1+nn4+16−8n(n2+4)n4+16+⋯+nn4+n4−2n⋅n2(n2+n2)n4+n4)\lim_{n \to \infty} \left( \frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \cdots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}} \right)limn→∞​(n4+1​n​−(n2+1)n4+1​2n​+n4+16​n​−(n2+4)n4+16​8n​+⋯+n4+n4​n​−(n2+n2)n4+n4​2n⋅n2​) be πk\frac{\pi}{k}kπ​, using only the principal values of the inverse trigonometric functions. Then k2k^2k2 is equal to ________.

Correct answer: 32

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Q85·MathematicsNumerical
The remainder when 4282024428^{2024}4282024 is divided by 212121 is ________.

Correct answer: 1

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Q86·Mathematics·Limits and ContinuityNumerical
Let f:(0,π)→Rf:(0, \pi) \to \mathbb{R}f:(0,π)→R be a function given by f(x)={(87)tan⁡8xtan⁡7x,0<x<π2a−8,x=π2(1+∣cot⁡x∣)ba∣tan⁡x∣,π2<x<πf(x) = \begin{cases} \left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ (1 + |\cot x|)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi \end{cases}f(x)=⎩⎨⎧​(78​)tan7xtan8x​,a−8,(1+∣cotx∣)ab​∣tanx∣,​0<x<2π​x=2π​2π​<x<π​ Where a,b∈Za, b \in \mathbb{Z}a,b∈Z. If fff is continuous at x=π2x = \frac{\pi}{2}x=2π​, then a2+b2a^2 + b^2a2+b2 is equal to ________.

Correct answer: 81

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Q87·MathematicsNumerical
Let AAA be a non-singular matrix of order 333. If det⁡(3 adj(2 adj((det⁡A)A)))=3−13⋅2−10\det(3\,\mathrm{adj}(2\,\mathrm{adj}((\det A)A))) = 3^{-13} \cdot 2^{-10}det(3adj(2adj((detA)A)))=3−13⋅2−10 and det⁡(3 adj(2A))=2m⋅3n\det(3\,\mathrm{adj}(2A)) = 2^m \cdot 3^ndet(3adj(2A))=2m⋅3n, then ∣3m+2n∣|3m + 2n|∣3m+2n∣ is equal to ________.

Correct answer: 14

Step-by-step solution →
Q88·MathematicsNumerical
Let the centre of a circle, passing through the point (0,0)(0, 0)(0,0), (1,0)(1, 0)(1,0) and touching the circle x2+y2=9x^2 + y^2 = 9x2+y2=9, be (h,k)(h, k)(h,k). Then for all possible values of the coordinates of the centre (h,k)(h, k)(h,k), 4(h2+k2)4(h^2 + k^2)4(h2+k2) is equal to ________.

Correct answer: 9

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Q89·MathematicsNumerical
If a function fff satisfies f(m+n)=f(m)+f(n)f(m + n) = f(m) + f(n)f(m+n)=f(m)+f(n) for all m,n∈Nm, n \in \mathbb{N}m,n∈N and f(1)=1f(1) = 1f(1)=1, then the largest natural number λ\lambdaλ such that ∑k=12022f(λ+k)≤(2022)2\sum_{k=1}^{2022} f(\lambda + k) \le (2022)^2∑k=12022​f(λ+k)≤(2022)2 is equal to ________.

Correct answer: 1010

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Q90·MathematicsNumerical
Let A={2,3,6,7}A = \{2, 3, 6, 7\}A={2,3,6,7} and B={4,5,6,8}B = \{4, 5, 6, 8\}B={4,5,6,8}. Let RRR be a relation defined on A×BA \times BA×B by (a1,b1) R (a2,b2)(a_1, b_1)\, R\, (a_2, b_2)(a1​,b1​)R(a2​,b2​) is and only if a1+a2=b1+b2a_1 + a_2 = b_1 + b_2a1​+a2​=b1​+b2​. Then the number of elements in RRR is ________.

Correct answer: 25

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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
  • Solutions 158/186
  • Laws of Motion 130/186
  • Chemical Thermodynamics 165/186
  • Units and Measurements 149/186
  • Hydrocarbons 126/186
  • 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
  • Quadratic Equations 148/186
  • Some Basic Concepts in Chemistry 129/186
  • Wave Optics 130/186
  • Area Under Curves 139/186
  • Alcohols and Ethers 106/186
  • Purification and Characterisation of Organic Compounds 116/186
  • Oscillations 117/186
  • Alternating Currents 108/186
  • Nuclei 116/186
  • Organic Compounds Containing Halogens 109/186
  • Straight Lines 114/186
  • Capacitors and Dielectrics 115/186
  • Statistics 118/186
  • Differentiability 91/186
  • Electronic Effects and Stability 74/186
  • Experimental Skills 68/186
  • Indefinite Integration 66/186
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