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JEE Advanced 2017 Paper 1 Question Paper with Answers

52 questions · Physics, Chemistry & Mathematics

52 of the 54 questions from the JEE Advanced 2017 Paper 1 paper, each with its correct answer and tagged to the chapter it tests. Free to read, no account needed.

2 questions are held back while we re-check the transcription or the answer key.

Physics
17
Chemistry
17
Mathematics
18

Physics — JEE Advanced 2017 Paper 1

Q1·PhysicsMultiple correct
A block M hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at O. A transverse wave pulse (Pulse 1) of wavelength λ0\lambda_{0}λ0​ is produced at point O on the rope. The pulse takes time TOAT_{OA}TOA​ to reach point A. If the wave pulse of wavelength λ0\lambda_{0}λ0​ is produced at point A (Pulse 2) without disturbing the position of M it takes time TAOT_{AO}TAO​ to reach point O. Which of the following options is/are correct?
  1. (A)The time TAO=TOAT_{AO} = T_{OA}TAO​=TOA​
  2. (B)The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope.
  3. (C)The wavelength of Pulse 1 becomes longer when it reaches point A.
  4. (D)The velocity of any pulse along the rope is independent of its frequency and wavelength.

Correct answer: (A), (B), (D)

Step-by-step solution →
Q2·PhysicsMultiple correct
A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at x = 0, in a co-ordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is x and the velocity is v. At that instant, which of the following options is/are correct?
  1. (A)The x component of displacement of the center of mass of the block M is : −mRM+m-\frac{mR}{M+m}−M+mmR​.
  2. (B)The position of the point mass is : x=−2 mRM+mx = -\sqrt{2}\,\frac{mR}{M+m}x=−2​M+mmR​.
  3. (C)The velocity of the point mass m is : v=2gR1+mMv = \sqrt{\frac{2gR}{1+\frac{m}{M}}}v=1+Mm​2gR​​.
  4. (D)The velocity of the block M is: V=−mM2gRV = -\frac{m}{M}\sqrt{2gR}V=−Mm​2gR​.

Correct answer: (A), (C)

Step-by-step solution →
Q3·PhysicsMultiple correct
A circular insulated copper wire loop is twisted to form two loops of area A and 2A as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field B⃗\vec{B}B points into the plane of the paper. A uniform magnetic field B⃗\vec{B}B points into the plane of the paper. At t = 0, the loop starts rotating about the common diameter as axis with a constant angular velocity ω\omegaω in the magnetic field. Which of the following options is/are correct?
  1. (A)The rate of change of the flux is maximum when the plane of the loops is perpendicular to plane of the paper.
  2. (B)The net emf induced due to both the loops is proportional to cos ωt\omega tωt.
  3. (C)The emf induced in the loop is proportional to the sun of the areas of the two loops.
  4. (D)The amplitude of the maximum net emf induced due to both the loops is equal to the amplitude of maximum emf induced in the smaller loop alone.

Correct answer: (A), (D)

Step-by-step solution →
Q4·PhysicsMultiple correct
For an isosceles prism of angle A and refractive index μ\muμ, it is found that the angle of minimum deviation δm\delta_{m}δm​ = A. Which of the following options is/are correct?
  1. (A)At minimum deviation, the incident angle i1i_{1}i1​ and the refracting angle r1r_{1}r1​ at the first refracting surface are related by r1=(i1/2)r_{1} = (i_{1}/2)r1​=(i1​/2).
  2. (B)For this prism the refractive index μ\muμ and the angle of prism A are related as A=12cos⁡−1(μ/2)A = \frac{1}{2}\cos^{-1}(\mu/2)A=21​cos−1(μ/2).
  3. (C)For this prism, the emergent ray at the second surface will be tangential to the surface when the angle of incidence at the first surface is i1=sin⁡−1[sin⁡A4cos⁡2A2−1−cos⁡A]i_{1} = \sin^{-1}\left[\sin A\sqrt{4\cos^{2}\frac{A}{2}-1}-\cos A\right]i1​=sin−1[sinA4cos22A​−1​−cosA].
  4. (D)For the angle of incidence i1i_{1}i1​ = A, the ray inside the prism is parallel to the base of the prism.

Correct answer: (A), (C), (D)

Step-by-step solution →
Q5·PhysicsMultiple correct
In the circuit shown L = 1 μ\muμH, C = 1 μ\muμF and R = 1 kΩk\OmegakΩ. They are connected in series with an a.c. source V = V0V_{0}V0​ sin ωt\omega tωt as shown. Which of the following options is/are correct?
  1. (A)The frequency at which the current will be in the phase with the voltage is independent of R.
  2. (B)At ω∼0\omega \sim 0ω∼0 the current flowing through the circuit becomes nearly zero.
  3. (C)At ω>>106\omega >> 10^{6}ω>>106 rad.s−1rad.s^{-1}rad.s−1, the circuit behave like a capacitor.
  4. (D)The current will be in phase with the voltage if ω=104\omega = 10^{4}ω=104 rad. s−1s^{-1}s−1

Correct answer: (A), (B)

Step-by-step solution →
Q6·PhysicsMultiple correct
A flat plate is moving normal to its plane through a gas under the action of constant force FFF. The gas is kept at a very low pressure. The speed of the plate v is much less than the average speed u of the gas molecules. Which of the following options is/are true?
  1. (A)The resistive force experienced by the plate is proportional to v
  2. (B)The pressure difference between the leading and trailing faces of the plate is proportional to uv
  3. (C)The plate will continue to move with constant non-zero acceleration, at all times
  4. (D)At a later time the external force FFF balances the resistive force.

Correct answer: (A), (B), (D)

Step-by-step solution →
Q7·PhysicsInteger
A drop of liquid of radius R = 10−210^{-2}10−2 m having surface tension S = 0.14π\frac{0.1}{4\pi}4π0.1​ Nm−1Nm^{-1}Nm−1 divides itself into K identical drops. In this process the total change in the surface energy ΔU=10−3\Delta U = 10^{-3}ΔU=10−3 J. If K = 10α10^{\alpha}10α then the value of α\alphaα is

Correct answer: 6

Step-by-step solution →
Q8·PhysicsInteger
131I^{131}I131I is an isotope of Iodine that β\betaβ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with 131I^{131}I131I is injected into the blood of a person. The activity of the amount of 131I^{131}I131I injected was 2.4 ×\times× 10510^{5}105 Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of 115 Bq. The total volume of blood in the person's body, in liters is approximately (you may use ex≈1+xe^{x} \approx 1 + xex≈1+x for ∣x∣≪1|x| \ll 1∣x∣≪1 and ln 2 ≈\approx≈ 0.7).

Correct answer: 5

Step-by-step solution →
Q9·PhysicsInteger
An electron in a hydrogen atom undergoes a transition from an orbit with quantum numebr nin_{i}ni​ to another with quantum number nfn_{f}nf​. ViV_{i}Vi​ and VfV_{f}Vf​ are respectively the initial and final potential energies of the electon. If ViVf=6.25\frac{V_{i}}{V_{f}} = 6.25Vf​Vi​​=6.25, then the smallest possible nfn_{f}nf​ is

Correct answer: 5

Step-by-step solution →
Q10·PhysicsInteger
A monochromatic light is travelling in a medium of refractive index n = 1.6. It enters a stack of glass layers from the bottom side at an angle θ\thetaθ = 30∘^{\circ}∘. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as nmn_{m}nm​ = n −-− mΔ\DeltaΔn, where nmn_{m}nm​ is the refractive index of the mthm^{th}mth slab and Δ\DeltaΔn = 0.1 (see the figure). The ray is refracted out parallel to the interface between the (m −-−1)th^{th}th and mthm^{th}mth slabs from the right side of the stack. What is the value of m?

Correct answer: 8

Step-by-step solution →
Q11·PhysicsInteger
A stationary source emits sound of frequency f0f_{0}f0​ = 492 Hz. The sound is reflected by a large car approaching the source with a speed of 2 ms−1ms^{-1}ms−1. The reflected signal is received by the soruce and superposed with the original. What will be the beat frequency of the resulting signal in Hz? (Given that the speed of sound in air is 330 ms−1ms^{-1}ms−1 and the car reflects the sound at the frequency it has received).

Correct answer: 6

Step-by-step solution →
Q12·PhysicsSingle correct
A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity v⃗\vec{v}v. A uniform electric field E⃗\vec{E}E and magnetic field B⃗\vec{B}B are given in columns 1, 2 and 3, respectively. The quantities E0E_{0}E0​, B0B_{0}B0​ are positive in magnitude. In which case will the particle move in a straight line with constant velocity?
Column IColumn 2Column 3
(I) Electron with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(i) E⃗=E02z^\vec{E} = E_{0}^{2}\hat{z}E=E02​z^(P) B⃗=−B0x^\vec{B} = -B_{0}\hat{x}B=−B0​x^
(II) Electron with v⃗=E0B0y^\vec{v} = \frac{E_{0}}{B_{0}}\hat{y}v=B0​E0​​y^​(ii) E⃗=−E0y^\vec{E} = -E_{0}\hat{y}E=−E0​y^​(Q) B⃗=B0x^\vec{B} = B_{0}\hat{x}B=B0​x^
(III) Proton with v⃗=0\vec{v} = 0v=0(iii) E⃗=−E0x^\vec{E} = -E_{0}\hat{x}E=−E0​x^(R) B⃗=B0y^\vec{B} = B_{0}\hat{y}B=B0​y^​
(IV) Proton with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(iv) E⃗=E0x^\vec{E} = E_{0}\hat{x}E=E0​x^(S) B⃗=B0z^\vec{B} = B_{0}\hat{z}B=B0​z^
  1. (A)(II) (iii) (S)
  2. (B)(IV) (i) (S)
  3. (C)(III) (ii) (R)
  4. (D)(III) (iii) (P)

Correct answer: (A)

Step-by-step solution →
Q13·PhysicsSingle correct
A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity v⃗\vec{v}v. A uniform electric field E⃗\vec{E}E and magnetic field B⃗\vec{B}B are given in columns 1, 2 and 3, respectively. The quantities E0E_{0}E0​, B0B_{0}B0​ are positive in magnitude. In which case will the particle describe a helical path with axis along the positive z direction?
Column IColumn 2Column 3
(I) Electron with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(i) E⃗=E02z^\vec{E} = E_{0}^{2}\hat{z}E=E02​z^(P) B⃗=−B0x^\vec{B} = -B_{0}\hat{x}B=−B0​x^
(II) Electron with v⃗=E0B0y^\vec{v} = \frac{E_{0}}{B_{0}}\hat{y}v=B0​E0​​y^​(ii) E⃗=−E0y^\vec{E} = -E_{0}\hat{y}E=−E0​y^​(Q) B⃗=B0x^\vec{B} = B_{0}\hat{x}B=B0​x^
(III) Proton with v⃗=0\vec{v} = 0v=0(iii) E⃗=−E0x^\vec{E} = -E_{0}\hat{x}E=−E0​x^(R) B⃗=B0y^\vec{B} = B_{0}\hat{y}B=B0​y^​
(IV) Proton with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(iv) E⃗=E0x^\vec{E} = E_{0}\hat{x}E=E0​x^(S) B⃗=B0z^\vec{B} = B_{0}\hat{z}B=B0​z^
  1. (A)(II) (ii) (R)
  2. (B)(IV) (ii) (R)
  3. (C)(IV) (i) (S)
  4. (D)(III) (iii) (P)

Correct answer: (C)

Step-by-step solution →
Q14·PhysicsSingle correct
A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial velocity v⃗\vec{v}v. A uniform electric field E⃗\vec{E}E and magnetic field B⃗\vec{B}B are given in columns 1, 2 and 3, respectively. The quantities E0E_{0}E0​, B0B_{0}B0​ are positive in magnitude. In which case would be particle move in a straight line along the negative direction of y-axis (i.e., more along −y^-\hat{y}−y^​) ?
Column IColumn 2Column 3
(I) Electron with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(i) E⃗=E02z^\vec{E} = E_{0}^{2}\hat{z}E=E02​z^(P) B⃗=−B0x^\vec{B} = -B_{0}\hat{x}B=−B0​x^
(II) Electron with v⃗=E0B0y^\vec{v} = \frac{E_{0}}{B_{0}}\hat{y}v=B0​E0​​y^​(ii) E⃗=−E0y^\vec{E} = -E_{0}\hat{y}E=−E0​y^​(Q) B⃗=B0x^\vec{B} = B_{0}\hat{x}B=B0​x^
(III) Proton with v⃗=0\vec{v} = 0v=0(iii) E⃗=−E0x^\vec{E} = -E_{0}\hat{x}E=−E0​x^(R) B⃗=B0y^\vec{B} = B_{0}\hat{y}B=B0​y^​
(IV) Proton with v⃗=2E0B0x^\vec{v} = 2\frac{E_{0}}{B_{0}}\hat{x}v=2B0​E0​​x^(iv) E⃗=E0x^\vec{E} = E_{0}\hat{x}E=E0​x^(S) B⃗=B0z^\vec{B} = B_{0}\hat{z}B=B0​z^
  1. (A)(IV) (ii) (S)
  2. (B)(III) (ii) (P)
  3. (C)(II) (iii) (Q)
  4. (D)(III) (ii) (R)

Correct answer: (D)

Step-by-step solution →
Q15·PhysicsSingle correct
An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P−VP-VP−V diagrams in column 3 of the table. Consider only the path from state 1 to 2. WWW denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ\gammaγ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is nnn. Which of the following options is the only correct representation of a process in which ΔU=ΔQ−PΔV\Delta U = \Delta Q - P\Delta VΔU=ΔQ−PΔV ?
Column IColumn 2Column 3
(I) W1−2=1γ−1(P2V2−P1V2)W_{1-2} = \frac{1}{\gamma-1}\left(P_{2}V_{2}-P_{1}V_{2}\right)W1−2​=γ−11​(P2​V2​−P1​V2​)(i) Isothermal(P)
(II) W1→2=−PV2+PV1W_{1\rightarrow 2} = -PV_{2}+PV_{1}W1→2​=−PV2​+PV1​(ii) isochoric(Q)
(III) W1→2=0W_{1\rightarrow 2} = 0W1→2​=0(iii) Isobaric(R)
(IV) W1→2=−nRTln⁡(V2V1)W_{1\rightarrow 2} = -nRT\ln\left(\frac{V_{2}}{V_{1}}\right)W1→2​=−nRTln(V1​V2​​)(iv) Adiabatic(S)
  1. (A)(II) (iv) (R)
  2. (B)(II) (iii) (P)
  3. (C)(II) (iii) (S)
  4. (D)(III) (iii) (P)

Correct answer: (B)

Step-by-step solution →
Q16·PhysicsSingle correct
An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P−VP-VP−V diagrams in column 3 of the table. Consider only the path from state 1 to 2. WWW denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ\gammaγ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is nnn. Which one of the following options is the correct combination?
Column IColumn 2Column 3
(I) W1−2=1γ−1(P2V2−P1V2)W_{1-2} = \frac{1}{\gamma-1}\left(P_{2}V_{2}-P_{1}V_{2}\right)W1−2​=γ−11​(P2​V2​−P1​V2​)(i) Isothermal(P)
(II) W1→2=−PV2+PV1W_{1\rightarrow 2} = -PV_{2}+PV_{1}W1→2​=−PV2​+PV1​(ii) isochoric(Q)
(III) W1→2=0W_{1\rightarrow 2} = 0W1→2​=0(iii) Isobaric(R)
(IV) W1→2=−nRTln⁡(V2V1)W_{1\rightarrow 2} = -nRT\ln\left(\frac{V_{2}}{V_{1}}\right)W1→2​=−nRTln(V1​V2​​)(iv) Adiabatic(S)
  1. (A)(III) (ii) (S)
  2. (B)(II) (iv) (R)
  3. (C)(II) (iv) (P)
  4. (D)(IV) (ii) (S)

Correct answer: (A)

Step-by-step solution →
Q17·PhysicsSingle correct
An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P−VP-VP−V diagrams in column 3 of the table. Consider only the path from state 1 to 2. WWW denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here γ\gammaγ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is nnn. Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
Column IColumn 2Column 3
(I) W1−2=1γ−1(P2V2−P1V2)W_{1-2} = \frac{1}{\gamma-1}\left(P_{2}V_{2}-P_{1}V_{2}\right)W1−2​=γ−11​(P2​V2​−P1​V2​)(i) Isothermal(P)
(II) W1→2=−PV2+PV1W_{1\rightarrow 2} = -PV_{2}+PV_{1}W1→2​=−PV2​+PV1​(ii) isochoric(Q)
(III) W1→2=0W_{1\rightarrow 2} = 0W1→2​=0(iii) Isobaric(R)
(IV) W1→2=−nRTln⁡(V2V1)W_{1\rightarrow 2} = -nRT\ln\left(\frac{V_{2}}{V_{1}}\right)W1→2​=−nRTln(V1​V2​​)(iv) Adiabatic(S)
  1. (A)(III) (iv) (R)
  2. (B)(I) (ii) (Q)
  3. (C)(IV) (ii) (R)
  4. (D)(I) (iv) (Q)

Correct answer: (D)

Step-by-step solution →

Chemistry — JEE Advanced 2017 Paper 1

Q18·ChemistryMultiple correct
The correct statement(s) for the following addition reactions is(are)
  1. (A)(M and O) and (N and P) are two pairs of diastereomers
  2. (B)Bromination proceeds through trans-addition in both the reactions
  3. (C)O and P are identical molecules
  4. (D)(M and O) and (N and P) are two pairs of enantiomers

Correct answer: (A), (B)

Step-by-step solution →
Q19·ChemistryMultiple correct
Addition of excess aqueous ammonia to a pink coloured aqueous solution of MCl2.6H2OMCl_{2}.6H_{2}OMCl2​.6H2​O(X) and NH4ClNH_{4}ClNH4​Cl gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1 :3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y. Among the following options, which statement(s) is(are) correct?
  1. (A)The hybridization of the central metal ion in Y is d2sp3d^{2}sp^{3}d2sp3
  2. (B)Z is a tetrahedral complex
  3. (C)Addition of silver nitrate to Y gives only two equivalents of silver chloride
  4. (D)When X and Z are in equilibrium at 00C0^{0}C00C, the colour of the solution is pink

Correct answer: (A), (B), (D)

Step-by-step solution →
Q20·ChemistryMultiple correct
For a solution formed by mixing liquids L and M, the vapour pressure of L plotted against the mole fraction of M in solution is shown in the following figure. Here xLx_{L}xL​ and xMx_{M}xM​ represent mole fractions of L and M, respectively, in the solution. The correct statement(s) applicable to this system is(are)
  1. (A)Attractive intermolecular interactions between L-L in pure liquid L and M-M in pure liquid M are stronger than those between L-M when mixed in solution
  2. (B)The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed when xL→0x_{L} \rightarrow 0xL​→0
  3. (C)The point Z represents vapour pressure of pure liquid L and Raoult's law is obeyed when xL→1x_{L} \rightarrow 1xL​→1
  4. (D)The point Z represents vapour pressure of pure liquid M and Raoult's law is obeyed from xL=0x_{L} = 0xL​=0 to xL=1x_{L} = 1xL​=1

Correct answer: (A), (C)

Step-by-step solution →
Q21·ChemistryMultiple correct
An ideal gas is expanded from (p1,V1,T1)(p_{1}, V_{1}, T_{1})(p1​,V1​,T1​) to (p2,V2,T2)(p_{2}, V_{2}, T_{2})(p2​,V2​,T2​) under different conditions. The correct statement(s) among the following is(are)
  1. (A)The work done on the gas is maximum when it is compressed irreversibly from (p2,V2)(p_{2}, V_{2})(p2​,V2​) to (p1,V1)(p_{1}, V_{1})(p1​,V1​) against constant pressure p1p_{1}p1​
  2. (B)The work done by the gas is less when it is expanded reversibly from V1V_{1}V1​ to V2V_{2}V2​ under adiabatic conditions as compared to that when expanded reversibly from V1V_{1}V1​ to V2V_{2}V2​ under isothermal conditions
  3. (C)The change in internal energy of the gas is (i) zero, if it is expanded reversibly with T1=T2T_{1} = T_{2}T1​=T2​ , and (ii) positive, if it is expanded reversibly under adiabatic conditions with T1≠T2T_{1} \neq T_{2}T1​=T2​
  4. (D)If the expansion is carried out freely, it is simultaneously both isothermal as well as adiabatic

Correct answer: (A), (B), (D)

Step-by-step solution →
Q22·ChemistryMultiple correct
The correct statement(s) about the oxoacids, HClO4HClO_{4}HClO4​ and HClOHClOHClO, is(are)
  1. (A)HClO4HClO_{4}HClO4​ is more acidic than HClOHClOHClO because of the resonance stabilization of its anion
  2. (B)HClO4HClO_{4}HClO4​ is formed in the reaction between Cl2Cl_{2}Cl2​ and H2OH_{2}OH2​O
  3. (C)The central atom in both HClO4HClO_{4}HClO4​ and HClOHClOHClO is sp3sp^{3}sp3 hybridized
  4. (D)The conjugate base of HClO4HClO_{4}HClO4​ is weaker base than H2OH_{2}OH2​O

Correct answer: (A), (C), (D)

Step-by-step solution →
Q23·ChemistryMultiple correct
The colour of the X2X_{2}X2​ molecules of group 17 elements changes gradually from yellow to violet down the group. This is due to
  1. (A)the physical state of X2X_{2}X2​ at atom temperature changes from gas to solid down the group
  2. (B)decrease in HOMO-LUMO gap down the group
  3. (C)decrease in π∗\pi^{*}π∗-σ∗\sigma^{*}σ∗ gap down the group
  4. (D)decrease in ionization energy down the group

Correct answer: (B), (C)

Step-by-step solution →
Q24·ChemistryInteger
Among H2,He2+,Li2,Be2,B2,C2,N2,O2−H_{2}, He_{2}^{+}, Li_{2}, Be_{2}, B_{2}, C_{2}, N_{2}, O_{2}^{-}H2​,He2+​,Li2​,Be2​,B2​,C2​,N2​,O2−​, and F2F_{2}F2​ , the number of diamagnetic species is (Atomic numbers: H = 1, He = 2, Li = 3, Be = 4, B = 5, C = 6, N = 7, O = 8, F = 9)

Correct answer: 6

Step-by-step solution →
Q25·ChemistryInteger
Among the following, the number of aromatic compound(s) is

Correct answer: 5

Step-by-step solution →
Q26·ChemistryInteger
The conductance of a 0.0015 M aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized Pt electrodes. The distance between the electrodes is 120 cm with an area of cross section of 1 cm2cm^{2}cm2. The conductance of this solution was found to be 5×10−75 \times 10^{-7}5×10−7 S. The pH of the solution is 4. The value of limiting molar conductivity (Λmo)\left(\Lambda_{m}^{o}\right)(Λmo​) of this weak monobasic acid in aqueous solution is Z×102Z \times 10^{2}Z×102 S cm−1cm^{-1}cm−1 mol−1mol^{-1}mol−1. The value of Z is

Correct answer: 6

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Q27·ChemistryInteger
The sum of the number of lone pairs of electrons on each central atom in the following species is [TeBr6]2−,[BrF2]+,SNF3\left[TeBr_{6}\right]^{2-}, \left[BrF_{2}\right]^{+}, SNF_{3}[TeBr6​]2−,[BrF2​]+,SNF3​, and [XeF3]−\left[XeF_{3}\right]^{-}[XeF3​]− (Atomic numbers : N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

Correct answer: 6

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Q28·ChemistryInteger
A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of 400 pm. If the density of the substance in the crystal is 8 g cm−3cm^{-3}cm−3, then the number of atoms present in 256 g of the crystal is N×1024N \times 10^{24}N×1024. The value of NNN is

Correct answer: 2

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Q29·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. The wave function ψn,l,ml\psi_{n,l,m_{l}}ψn,l,ml​​ is a mathematical function whose value depends upon spherical polar coordinates (r,θ,ϕ)(r, \theta, \phi)(r,θ,ϕ) of the electron and characterized by the quantum numbers nnn, lll and mlm_{l}ml​. Here rrr is distance from nucleus, θ\thetaθ is colatitude and ϕ\phiϕ is azimuth. In the mathematical functions given in the Table, Z is atomic number and a0a_{0}a0​ is Bohr radius. For the given orbital in Column 1, the only CORRECT combination for any hydrogen-like species is
Column 1Column 2Column 3
(I) 1s orbital(i) ψn,l,ml∝(Za0)32e−(Zra0)\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{3}{2}} e^{-\left(\frac{Zr}{a_{0}}\right)}ψn,l,ml​​∝(a0​Z​)23​e−(a0​Zr​)(P)
(II) 2s orbital(ii) One radial node(Q) Probability density at nucleus ∝1a03\propto \frac{1}{a_{0}^{3}}∝a03​1​
(III) 2pz_{z}z​ orbital(iii) ψn,l,ml∝(Za0)52re−(Zr2a0)cos⁡θ\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{5}{2}} r e^{-\left(\frac{Zr}{2a_{0}}\right)} \cos\thetaψn,l,ml​​∝(a0​Z​)25​re−(2a0​Zr​)cosθ(R) Probability density is maximum at nucleus
(IV) 3dz2_{z}^{2}z2​ orbital(iv) xyxyxy-plane is a nodal plane(S) Energy needed to excite electron from n=2n = 2n=2 state to n=4n = 4n=4 state is 2732\frac{27}{32}3227​ times the energy needed to excite electron from n=2n = 2n=2 state to n=6n = 6n=6 state
  1. (A)(IV) (iv) (R)
  2. (B)(II) (ii) (P)
  3. (C)(III) (iii) (P)
  4. (D)(I) (ii) (S)

Correct answer: (B)

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Q30·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. The wave function ψn,l,ml\psi_{n,l,m_{l}}ψn,l,ml​​ is a mathematical function whose value depends upon spherical polar coordinates (r,θ,ϕ)(r, \theta, \phi)(r,θ,ϕ) of the electron and characterized by the quantum numbers nnn, lll and mlm_{l}ml​. Here rrr is distance from nucleus, θ\thetaθ is colatitude and ϕ\phiϕ is azimuth. In the mathematical functions given in the Table, Z is atomic number and a0a_{0}a0​ is Bohr radius. For He+He^{+}He+ ion, the only INCORRECT combination is
Column 1Column 2Column 3
(I) 1s orbital(i) ψn,l,ml∝(Za0)32e−(Zra0)\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{3}{2}} e^{-\left(\frac{Zr}{a_{0}}\right)}ψn,l,ml​​∝(a0​Z​)23​e−(a0​Zr​)(P)
(II) 2s orbital(ii) One radial node(Q) Probability density at nucleus ∝1a03\propto \frac{1}{a_{0}^{3}}∝a03​1​
(III) 2pz_{z}z​ orbital(iii) ψn,l,ml∝(Za0)52re−(Zr2a0)cos⁡θ\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{5}{2}} r e^{-\left(\frac{Zr}{2a_{0}}\right)} \cos\thetaψn,l,ml​​∝(a0​Z​)25​re−(2a0​Zr​)cosθ(R) Probability density is maximum at nucleus
(IV) 3dz2_{z}^{2}z2​ orbital(iv) xyxyxy-plane is a nodal plane(S) Energy needed to excite electron from n=2n = 2n=2 state to n=4n = 4n=4 state is 2732\frac{27}{32}3227​ times the energy needed to excite electron from n=2n = 2n=2 state to n=6n = 6n=6 state
  1. (A)(II) (ii) (Q)
  2. (B)(I) (i) (S)
  3. (C)(I) (i) (R)
  4. (D)(I) (iii) (R)

Correct answer: (D)

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Q31·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. The wave function ψn,l,ml\psi_{n,l,m_{l}}ψn,l,ml​​ is a mathematical function whose value depends upon spherical polar coordinates (r,θ,ϕ)(r, \theta, \phi)(r,θ,ϕ) of the electron and characterized by the quantum numbers nnn, lll and mlm_{l}ml​. Here rrr is distance from nucleus, θ\thetaθ is colatitude and ϕ\phiϕ is azimuth. In the mathematical functions given in the Table, Z is atomic number and a0a_{0}a0​ is Bohr radius. For hydrogen atom, the only CORRECT combination is
Column 1Column 2Column 3
(I) 1s orbital(i) ψn,l,ml∝(Za0)32e−(Zra0)\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{3}{2}} e^{-\left(\frac{Zr}{a_{0}}\right)}ψn,l,ml​​∝(a0​Z​)23​e−(a0​Zr​)(P)
(II) 2s orbital(ii) One radial node(Q) Probability density at nucleus ∝1a03\propto \frac{1}{a_{0}^{3}}∝a03​1​
(III) 2pz_{z}z​ orbital(iii) ψn,l,ml∝(Za0)52re−(Zr2a0)cos⁡θ\psi_{n,l,m_{l}} \propto \left(\frac{Z}{a_{0}}\right)^{\frac{5}{2}} r e^{-\left(\frac{Zr}{2a_{0}}\right)} \cos\thetaψn,l,ml​​∝(a0​Z​)25​re−(2a0​Zr​)cosθ(R) Probability density is maximum at nucleus
(IV) 3dz2_{z}^{2}z2​ orbital(iv) xyxyxy-plane is a nodal plane(S) Energy needed to excite electron from n=2n = 2n=2 state to n=4n = 4n=4 state is 2732\frac{27}{32}3227​ times the energy needed to excite electron from n=2n = 2n=2 state to n=6n = 6n=6 state
  1. (A)(I) (iv) (R)
  2. (B)(I) (i) (P)
  3. (C)(II) (i) (Q)
  4. (D)(I) (i) (S)

Correct answer: (D)

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Q32·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively. For the synthesis of benzoic acid, the only CORRECT combination is
Column 1Column 2Column 3
(I) Toluene(i) NaOH/Br2NaOH/Br_{2}NaOH/Br2​(P) Condensation
(II) Acetophenone(ii) Br2/hνBr_{2}/h\nuBr2​/hν(Q) Carboxylation
(III) Benzaldehyde(iii) (CH3CO)2O/CH3COOK(CH_{3}CO)_{2}O/CH_{3}COOK(CH3​CO)2​O/CH3​COOK(R) Substitution
(IV) Phenol(iv) NaOH/CO2NaOH/CO_{2}NaOH/CO2​(S) Haloform
  1. (A)(III) (iv) (R)
  2. (B)(IV)(ii) (P)
  3. (C)(I) (iv) (Q)
  4. (D)(II) (i) (S)

Correct answer: (D)

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Q33·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively. The only CORRECT combination in which the reaction proceeds through radical mechanism is
Column 1Column 2Column 3
(I) Toluene(i) NaOH/Br2NaOH/Br_{2}NaOH/Br2​(P) Condensation
(II) Acetophenone(ii) Br2/hνBr_{2}/h\nuBr2​/hν(Q) Carboxylation
(III) Benzaldehyde(iii) (CH3CO)2O/CH3COOK(CH_{3}CO)_{2}O/CH_{3}COOK(CH3​CO)2​O/CH3​COOK(R) Substitution
(IV) Phenol(iv) NaOH/CO2NaOH/CO_{2}NaOH/CO2​(S) Haloform
  1. (A)(I) (ii) (R)
  2. (B)(II) (iii) (R)
  3. (C)(III) (ii) (P)
  4. (D)(IV) (i) (Q)

Correct answer: (A)

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Q34·ChemistrySingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain starting materials, reaction conditions, and type of reactions, respectively. The only CORRECT combination that gives two different carboxylic acids is
Column 1Column 2Column 3
(I) Toluene(i) NaOH/Br2NaOH/Br_{2}NaOH/Br2​(P) Condensation
(II) Acetophenone(ii) Br2/hνBr_{2}/h\nuBr2​/hν(Q) Carboxylation
(III) Benzaldehyde(iii) (CH3CO)2O/CH3COOK(CH_{3}CO)_{2}O/CH_{3}COOK(CH3​CO)2​O/CH3​COOK(R) Substitution
(IV) Phenol(iv) NaOH/CO2NaOH/CO_{2}NaOH/CO2​(S) Haloform
  1. (A)(IV) (iii) (Q)
  2. (B)(III) (iii) (P)
  3. (C)(II) (iv) (R)
  4. (D)(I) (i) (S)

Correct answer: (B)

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Mathematics — JEE Advanced 2017 Paper 1

Q35·MathematicsMultiple correct
Which of the following is(are) NOT the square of a 3×33 \times 33×3 matrix with real entries ?
  1. (A)[10001000−1]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}​100​010​00−1​​
  2. (B)[−1000−1000−1]\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}​−100​0−10​00−1​​
  3. (C)[100010001]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}​100​010​001​​
  4. (D)[1000−1000−1]\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}​100​0−10​00−1​​

Correct answer: (A), (B)

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Q36·MathematicsMultiple correct
If a chord, which is not a tangent, of the parabola y2=16xy^{2} = 16xy2=16x has the equation 2x+y=p2x + y = p2x+y=p, and midpoint (h,k)(h, k)(h,k), then which of the following is(are) possible value(s) of ppp, hhh and kkk ?
  1. (A)p=5p = 5p=5, h=4h = 4h=4, k=−3k = -3k=−3
  2. (B)p=−1p = -1p=−1, h=1h = 1h=1, k=−3k = -3k=−3
  3. (C)p=−2p = -2p=−2, h=2h = 2h=2, k=−4k = -4k=−4
  4. (D)p=2p = 2p=2, h=3h = 3h=3, k=−4k = -4k=−4

Correct answer: (D)

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Q37·MathematicsMultiple correct
Let aaa, bbb, xxx and yyy be real numbers such that a−b=1a - b = 1a−b=1 and y≠0y \neq 0y=0. If the complex number z=x+iyz = x + iyz=x+iy satisfies Im(az+bz+1)=y\mathrm{Im}\left(\frac{az + b}{z + 1}\right) = yIm(z+1az+b​)=y, then which of the following is(are) possible value(s) of xxx ?
  1. (A)−1−1−y2-1 - \sqrt{1 - y^{2}}−1−1−y2​
  2. (B)1+1+y21 + \sqrt{1 + y^{2}}1+1+y2​
  3. (C)1−1+y21 - \sqrt{1 + y^{2}}1−1+y2​
  4. (D)−1+1−y2-1 + \sqrt{1 - y^{2}}−1+1−y2​

Correct answer: (A), (D)

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Q38·MathematicsMultiple correct
Let X and Y be two events such that P(X)=13P(X) = \frac{1}{3}P(X)=31​, P(X∣Y)=12P(X|Y) = \frac{1}{2}P(X∣Y)=21​ and P(Y∣X)=25P(Y|X) = \frac{2}{5}P(Y∣X)=52​. Then
  1. (A)P(X′∣Y)=12P(X'|Y) = \frac{1}{2}P(X′∣Y)=21​
  2. (B)P(X∩Y)=15P(X \cap Y) = \frac{1}{5}P(X∩Y)=51​
  3. (C)P(X∪Y)=25P(X \cup Y) = \frac{2}{5}P(X∪Y)=52​
  4. (D)P(Y)=415P(Y) = \frac{4}{15}P(Y)=154​

Correct answer: (A), (D)

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Q39·MathematicsMultiple correct
Let [x][x][x] be the greatest integer less than or equals to xxx. Then, at which of the following point(s) the function f(x)=xcos⁡(π(x+[x]))f(x) = x \cos(\pi(x + [x]))f(x)=xcos(π(x+[x])) is discontinuous ?
  1. (A)x=−1x = -1x=−1
  2. (B)x=0x = 0x=0
  3. (C)x=2x = 2x=2
  4. (D)x=1x = 1x=1

Correct answer: (A), (C), (D)

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Q40·MathematicsMultiple correct
If 2x−y+1=02x - y + 1 = 02x−y+1=0 is a tangent to the hyperbola x2a2−y216=1\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1a2x2​−16y2​=1, then which of the following CANNOT be sides of a right angled triangle ?
  1. (A)2a2a2a, 444, 111
  2. (B)2a2a2a, 888, 111
  3. (C)aaa, 444, 111
  4. (D)aaa, 444, 222

Correct answer: (B), (C), (D)

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Q41·MathematicsMultiple correct
Let f:R→(0,1)f : R \to (0, 1)f:R→(0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0,1)(0, 1)(0,1) ?
  1. (A)ex−∫0xf(t)sin⁡t dte^{x} - \int_{0}^{x} f(t)\sin t\, dtex−∫0x​f(t)sintdt
  2. (B)x9−f(x)x^{9} - f(x)x9−f(x)
  3. (C)f(x)+∫0π/2f(t)sin⁡t dtf(x) + \int_{0}^{\pi/2} f(t)\sin t\, dtf(x)+∫0π/2​f(t)sintdt
  4. (D)x−∫0π2−xf(t)cos⁡t dtx - \int_{0}^{\frac{\pi}{2} - x} f(t)\cos t\, dtx−∫02π​−x​f(t)costdt

Correct answer: (B), (D)

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Q42·MathematicsInteger
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side ?

Correct answer: 6

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Q43·MathematicsInteger
For how many values of ppp, the circle x2+y2+2x+4y−p=0x^{2} + y^{2} + 2x + 4y - p = 0x2+y2+2x+4y−p=0 and the coordinate axes have exactly three common points ?

Correct answer: 2

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Q44·MathematicsInteger
For a real number α\alphaα, if the system [1αα2α1αα2α1][xyz]=[1−11]\begin{bmatrix} 1 & \alpha & \alpha^{2} \\ \alpha & 1 & \alpha \\ \alpha^{2} & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}​1αα2​α1α​α2α1​​​xyz​​=​1−11​​ of linear equations, has infinitely many solutions, then 1+α+α2=1 + \alpha + \alpha^{2} =1+α+α2=

Correct answer: 1

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Q45·MathematicsInteger
Words of length 10 are formed using the letters, A, B, C, D, E, F, G, H, I, J. Let xxx be the number of such words where no letter is repeated ; and let yyy be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, y9x=\frac{y}{9x} =9xy​=

Correct answer: 5

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Q46·MathematicsInteger
Let f:R→Rf : R \to Rf:R→R be a differentiable function such that f(0)=0f(0) = 0f(0)=0, f(π2)=3f\left(\frac{\pi}{2}\right) = 3f(2π​)=3 and f′(0)=1f'(0) = 1f′(0)=1. If g(x)=∫xπ/2[f′(t)cosec⁡t−cot⁡t cosec⁡t f(t)]dtg(x) = \int_{x}^{\pi/2} \left[ f'(t)\operatorname{cosec} t - \cot t\ \operatorname{cosec} t\ f(t) \right] dtg(x)=∫xπ/2​[f′(t)cosect−cott cosect f(t)]dt for x∈(0,π2]x \in \left(0, \frac{\pi}{2}\right]x∈(0,2π​], then lim⁡x→0g(x)=\lim_{x \to 0} g(x) =limx→0​g(x)=

Correct answer: 2

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Q47·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. The tangent to a suitable conic (Column 1) at (3,12)\left(\sqrt{3}, \frac{1}{2}\right)(3​,21​) is found to be 3x+2y=4\sqrt{3}x + 2y = 43​x+2y=4, then which of the following options is the only CORRECT combination ?
Column 1Column 2Column 3
(I) x2+y2=a2x^{2} + y^{2} = a^{2}x2+y2=a2(i) my=m2x+amy = m^{2}x + amy=m2x+a(P) (am2,2am)\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)(m2a​,m2a​)
(II) x2+a2y2=a2x^{2} + a^{2}y^{2} = a^{2}x2+a2y2=a2(ii) y=mx+am2+1y = mx + a\sqrt{m^{2} + 1}y=mx+am2+1​(Q) (−mam2+1,am2+1)\left(\frac{-ma}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}\right)(m2+1​−ma​,m2+1​a​)
(III) y2=4axy^{2} = 4axy2=4ax(iii) y=mx+a2m2−1y = mx + \sqrt{a^{2}m^{2} - 1}y=mx+a2m2−1​(R) (−a2ma2m2+1,1a2m2+1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} + 1}}, \frac{1}{\sqrt{a^{2}m^{2} + 1}}\right)(a2m2+1​−a2m​,a2m2+1​1​)
(IV) x2−a2y2=a2x^{2} - a^{2}y^{2} = a^{2}x2−a2y2=a2(iv) y=mx+a2m2+1y = mx + \sqrt{a^{2}m^{2} + 1}y=mx+a2m2+1​(S) (−a2ma2m2−1,−1a2m2−1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} - 1}}, \frac{-1}{\sqrt{a^{2}m^{2} - 1}}\right)(a2m2−1​−a2m​,a2m2−1​−1​)
  1. (A)(II) (iii) (R)
  2. (B)(IV) (iv) (S)
  3. (C)(IV) (iii) (S)
  4. (D)(II) (iv) (R)

Correct answer: (D)

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Q48·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. If a tangent to a suitable conic (Column 1) is found to be y=x+8y = x + 8y=x+8 and its point of contact is (8,16)(8, 16)(8,16), then which of the following options is the only CORRECT combination ?
Column 1Column 2Column 3
(I) x2+y2=a2x^{2} + y^{2} = a^{2}x2+y2=a2(i) my=m2x+amy = m^{2}x + amy=m2x+a(P) (am2,2am)\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)(m2a​,m2a​)
(II) x2+a2y2=a2x^{2} + a^{2}y^{2} = a^{2}x2+a2y2=a2(ii) y=mx+am2+1y = mx + a\sqrt{m^{2} + 1}y=mx+am2+1​(Q) (−mam2+1,am2+1)\left(\frac{-ma}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}\right)(m2+1​−ma​,m2+1​a​)
(III) y2=4axy^{2} = 4axy2=4ax(iii) y=mx+a2m2−1y = mx + \sqrt{a^{2}m^{2} - 1}y=mx+a2m2−1​(R) (−a2ma2m2+1,1a2m2+1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} + 1}}, \frac{1}{\sqrt{a^{2}m^{2} + 1}}\right)(a2m2+1​−a2m​,a2m2+1​1​)
(IV) x2−a2y2=a2x^{2} - a^{2}y^{2} = a^{2}x2−a2y2=a2(iv) y=mx+a2m2+1y = mx + \sqrt{a^{2}m^{2} + 1}y=mx+a2m2+1​(S) (−a2ma2m2−1,−1a2m2−1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} - 1}}, \frac{-1}{\sqrt{a^{2}m^{2} - 1}}\right)(a2m2−1​−a2m​,a2m2−1​−1​)
  1. (A)(III) (i) (P)
  2. (B)(III) (ii) (Q)
  3. (C)(II) (iv) (R)
  4. (D)(I) (ii) (Q)

Correct answer: (A)

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Q49·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. For a=2a = \sqrt{2}a=2​, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1,1)(-1, 1)(−1,1), then which of the following options is the only CORRECT combination for obtaining its equation ?
Column 1Column 2Column 3
(I) x2+y2=a2x^{2} + y^{2} = a^{2}x2+y2=a2(i) my=m2x+amy = m^{2}x + amy=m2x+a(P) (am2,2am)\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)(m2a​,m2a​)
(II) x2+a2y2=a2x^{2} + a^{2}y^{2} = a^{2}x2+a2y2=a2(ii) y=mx+am2+1y = mx + a\sqrt{m^{2} + 1}y=mx+am2+1​(Q) (−mam2+1,am2+1)\left(\frac{-ma}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}\right)(m2+1​−ma​,m2+1​a​)
(III) y2=4axy^{2} = 4axy2=4ax(iii) y=mx+a2m2−1y = mx + \sqrt{a^{2}m^{2} - 1}y=mx+a2m2−1​(R) (−a2ma2m2+1,1a2m2+1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} + 1}}, \frac{1}{\sqrt{a^{2}m^{2} + 1}}\right)(a2m2+1​−a2m​,a2m2+1​1​)
(IV) x2−a2y2=a2x^{2} - a^{2}y^{2} = a^{2}x2−a2y2=a2(iv) y=mx+a2m2+1y = mx + \sqrt{a^{2}m^{2} + 1}y=mx+a2m2+1​(S) (−a2ma2m2−1,−1a2m2−1)\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} - 1}}, \frac{-1}{\sqrt{a^{2}m^{2} - 1}}\right)(a2m2−1​−a2m​,a2m2−1​−1​)
  1. (A)(II) (ii) (Q)
  2. (B)(III) (i) (P)
  3. (C)(I) (i) (P)
  4. (D)(I) (ii) (Q)

Correct answer: (D)

Step-by-step solution →
Q50·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Let f(x)=x+log⁡ex−xlog⁡exf(x) = x + \log_{e} x - x\log_{e} xf(x)=x+loge​x−xloge​x, x∈(0,∞)x \in (0, \infty)x∈(0,∞). • Column 1 contains information about zeros of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x). • Column 2 contains information about the limiting behavior of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x)f(x)f(x) and f′(x)f'(x)f′(x). Which of the following options is the only CORRECT combination ?
Column 1Column 2Column 3
(I) f(x)=0f(x) = 0f(x)=0 for some x∈(1,e2)x \in (1, e^{2})x∈(1,e2)(i) lim⁡x→∞f(x)=0\lim_{x \to \infty} f(x) = 0limx→∞​f(x)=0(P) fff is increasing in (0,1)(0, 1)(0,1)
(II) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(ii) lim⁡x→∞f(x)=−∞\lim_{x \to \infty} f(x) = -\inftylimx→∞​f(x)=−∞(Q) fff is decreasing in (e,e2)(e, e^{2})(e,e2)
(III) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(0,1)x \in (0, 1)x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞\lim_{x \to \infty} f'(x) = -\inftylimx→∞​f′(x)=−∞(R) f′f'f′ is increasing in (0,1)(0, 1)(0,1)
(IV) f′′(x)=0f''(x) = 0f′′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(iv) lim⁡x→∞f′′(x)=0\lim_{x \to \infty} f''(x) = 0limx→∞​f′′(x)=0(S) f′f'f′ is decreasing in (e,e2)(e, e^{2})(e,e2)
  1. (A)(IV) (i) (S)
  2. (B)(I) (ii) (R)
  3. (C)(III) (iv) (P)
  4. (D)(II) (iii) (S)

Correct answer: (D)

Step-by-step solution →
Q51·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Let f(x)=x+log⁡ex−xlog⁡exf(x) = x + \log_{e} x - x\log_{e} xf(x)=x+loge​x−xloge​x, x∈(0,∞)x \in (0, \infty)x∈(0,∞). • Column 1 contains information about zeros of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x). • Column 2 contains information about the limiting behavior of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x)f(x)f(x) and f′(x)f'(x)f′(x). Which of the following options is the only CORRECT combination ?
Column 1Column 2Column 3
(I) f(x)=0f(x) = 0f(x)=0 for some x∈(1,e2)x \in (1, e^{2})x∈(1,e2)(i) lim⁡x→∞f(x)=0\lim_{x \to \infty} f(x) = 0limx→∞​f(x)=0(P) fff is increasing in (0,1)(0, 1)(0,1)
(II) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(ii) lim⁡x→∞f(x)=−∞\lim_{x \to \infty} f(x) = -\inftylimx→∞​f(x)=−∞(Q) fff is decreasing in (e,e2)(e, e^{2})(e,e2)
(III) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(0,1)x \in (0, 1)x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞\lim_{x \to \infty} f'(x) = -\inftylimx→∞​f′(x)=−∞(R) f′f'f′ is increasing in (0,1)(0, 1)(0,1)
(IV) f′′(x)=0f''(x) = 0f′′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(iv) lim⁡x→∞f′′(x)=0\lim_{x \to \infty} f''(x) = 0limx→∞​f′′(x)=0(S) f′f'f′ is decreasing in (e,e2)(e, e^{2})(e,e2)
  1. (A)(III) (iii) (R)
  2. (B)(I) (i) (P)
  3. (C)(IV) (iv) (S)
  4. (D)(II) (ii) (Q)

Correct answer: (D)

Step-by-step solution →
Q52·MathematicsSingle correct
Answer by appropriately matching the information given in the three columns of the following table. Let f(x)=x+log⁡ex−xlog⁡exf(x) = x + \log_{e} x - x\log_{e} xf(x)=x+loge​x−xloge​x, x∈(0,∞)x \in (0, \infty)x∈(0,∞). • Column 1 contains information about zeros of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x). • Column 2 contains information about the limiting behavior of f(x)f(x)f(x), f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x) at infinity. • Column 3 contains information about increasing/decreasing nature of f(x)f(x)f(x) and f′(x)f'(x)f′(x). Which of the following options is the only INCORRECT combination ?
Column 1Column 2Column 3
(I) f(x)=0f(x) = 0f(x)=0 for some x∈(1,e2)x \in (1, e^{2})x∈(1,e2)(i) lim⁡x→∞f(x)=0\lim_{x \to \infty} f(x) = 0limx→∞​f(x)=0(P) fff is increasing in (0,1)(0, 1)(0,1)
(II) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(ii) lim⁡x→∞f(x)=−∞\lim_{x \to \infty} f(x) = -\inftylimx→∞​f(x)=−∞(Q) fff is decreasing in (e,e2)(e, e^{2})(e,e2)
(III) f′(x)=0f'(x) = 0f′(x)=0 for some x∈(0,1)x \in (0, 1)x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞\lim_{x \to \infty} f'(x) = -\inftylimx→∞​f′(x)=−∞(R) f′f'f′ is increasing in (0,1)(0, 1)(0,1)
(IV) f′′(x)=0f''(x) = 0f′′(x)=0 for some x∈(1,e)x \in (1, e)x∈(1,e)(iv) lim⁡x→∞f′′(x)=0\lim_{x \to \infty} f''(x) = 0limx→∞​f′′(x)=0(S) f′f'f′ is decreasing in (e,e2)(e, e^{2})(e,e2)
  1. (A)(II) (iii) (P)
  2. (B)(II) (iv) (Q)
  3. (C)(I) (iii) (P)
  4. (D)(III) (i) (R)

Correct answer: (D)

Step-by-step solution →

Chapters tested in this paper

  • Properties of Solids and Liquids 172/186
  • Matrices and Determinants 180/186
  • Coordination Compounds 176/186
  • 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
  • Probability 176/186
  • Permutations and Combinations 162/186
  • Magnetic Field of Current 147/186
  • Chemical Bonding and Molecular Structure 151/186
  • Application of Derivatives 139/186
  • Limits and Continuity 149/186
  • Thermodynamics 154/186
  • Aldehydes and Ketones 135/186
  • Solutions 158/186
  • Chemical Thermodynamics 165/186
  • Hydrocarbons 126/186
  • Complex Numbers 165/186
  • Atomic Structure 161/186
  • Circles 142/186
  • Electromagnetic Induction 120/186
  • Kinetic Theory of Gases 135/186
  • Alternating Currents 108/186
  • Nuclei 116/186
  • Waves 109/186
  • Atoms 112/186
  • Parabola 101/186
  • Ellipse 103/186
  • Hyperbola 77/186
  • Solid State 63/186
  • Isomerism 51/186
  • Aromaticity 22/186
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