JEE Main 31 January 2023 Shift 2 question paper with solutions

Question 1

Physics · Motion in a Straight Line · Single correct

A body is moving with constant speed, in a circle of radius 10 m. The body completes one revolution in 4 s. At the end of 3rd second, the displacement of body (in m) from its starting point is:

1. 30
2. 15$\pi$
3. 5$\pi$
4. 10$\sqrt{2}$

Answer: (d)

Question 2

Physics · Physical World, Units and Measurements · Single correct

Match List-I with List-II. Choose the correct answer from the options given below:

1. A-I, B-IV, C-III, D-II
2. A-III, B-I, C-IV, D-II
3. A-II, B-III, C-IV, D-I
4. A-IV, B-II, C-I, D-III

Answer: (b)

Question 3

Physics · Laws of Motion · Single correct

A stone of mass 1 kg is tied to end of a massless string of length 1 m. If the breaking tension of the string is 400 N, then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is:

1. 20 ms$^{-1}$
2. 40 ms$^{-1}$
3. 400 ms$^{-1}$
4. 10 ms$^{-1}$

Answer: (a)

Question 4

Physics · Motion in a Plane · Numerical

Two bodies are projected from ground with same speeds 40 $\mathrm{ms^{-1}}$ at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of $60^\circ$, with horizontal then sum of the maximum heights, attained by the two projectiles, is ______ m. (Given $g=10\mathrm{ms^{-2}}$)

Answer: 80

Question 5

Physics · Laws of Motion · Single correct

A body of mass 10 kg is moving with an initial speed of 20 m/s. The body stops after 5 s due to friction between body and the floor. The value of the coefficient of friction is: (Take acceleration due to gravity $g = 10 \, \mathrm{m/s^2}$)

1. 0.2
2. 0.3
3. 0.5
4. 0.4

Answer: (d)

Question 6

Physics · System of Particles and Rotational Motion · Numerical

A ball is dropped from a height of 20 m. If the coefficient of restitution for the collision between ball and floor is 0.5, after hitting the floor, the ball rebounds to a height of _______ m.

Answer: 5

Question 7

Physics · System of Particles and Rotational Motion · Numerical

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of $\frac{x}{6}$. The value of $x$ is.

Answer: 5

Question 8

Physics · Gravitation · Single correct

A body weight $W$, is projected vertically upwards from earth's surface to reach a height above the earth which is equal to nine times the radius of earth. The weight of the body at that height will be:

1. $\frac{W}{91}$
2. $\frac{W}{100}$
3. $\frac{W}{9}$
4. $\frac{W}{3}$

Answer: (b)

Question 9

Physics · Mechanical Properties of Solids · Single correct

Under the same load, wire A having length 5.0 m and cross section $2.5 \times 10^{-5} \, \mathrm{m}^2$ stretches uniformly by the same amount as another wire B of length 6.0 m and a cross section of $3.0 \times 10^{-5} \, \mathrm{m}^2$ stretches. The ratio of the Young's modulus of wire A to that of wire B will be:

1. 1 : 4
2. 1 : 1
3. 1 : 10
4. 1 : 2

Answer: (b)

Question 10

Physics · Thermal Properties of Matter · Numerical

A water heater of power $2000 \, \mathrm{W}$ is used to heat water. The specific heat capacity of water is $4200 \, \mathrm{J} \, \mathrm{kg}^{-1} \, \mathrm{K}^{-1}$. The efficiency of heater is $70\%$. Time required to heat $2 \, \mathrm{kg}$ of water from $10^{\circ}\mathrm{C}$ to $60^{\circ}\mathrm{C}$ is ________ $\mathrm{s}$. (Assume that the specific heat capacity of water remains constant over the temperature range of the water).

Answer: 300

Question 11

Physics · Thermodynamics · Single correct

Heat energy of 735 J is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but do not oscillate. The increase in the internal energy of the gas will be:

1. 525 J
2. 441 J
3. 572 J
4. 735 J

Answer: (a)

Question 12

Physics · Thermodynamics · Single correct

A hypothetical gas expands adiabatically such that its volume changes from 08 litres to 27 litres. If the ratio of final pressure of the gas to initial pressure of the gas is $\frac{16}{81}$. Then the ratio of $\frac{C_P}{C_V}$ will be.

1. $\frac{4}{3}$
2. $\frac{3}{1}$
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Answer: (a)

Question 13

Physics · Mechanical Properties of Solids · Single correct

For a solid rod, the Young's modulus of elasticity is $3.2 \times 10^{11} \, \mathrm{Nm^{-2}}$ and density is $8 \times 10^{3} \, \mathrm{kg \, m^{-3}}$. The velocity of longitudinal wave in the rod will be.

1. $145.75 \times 10^{3} \, \mathrm{ms^{-1}}$
2. $3.65 \times 10^{3} \, \mathrm{ms^{-1}}$
3. $18.96 \times 10^{3} \, \mathrm{ms^{-1}}$
4. $6.32 \times 10^{3} \, \mathrm{ms^{-1}}$

Answer: (d)

Question 14

Physics · Waves · Numerical

The displacement equations of two interfering waves are given by $$y_1 = 10 \sin \left( \omega t + \frac{\pi}{3} \right) \, \mathrm{cm},$$ $$y_2 = 5 \left[ \sin (\omega t) + \sqrt{3} \cos \omega t \right] \, \mathrm{cm}$$ respectively. The amplitude of the resultant wave is _______ $\mathrm{cm}$.

Answer: 20

Question 15

Physics · Electric Charges and Fields · Single correct

Considering a group of positive charges, which of the following statements is correct?

1. Net potential of the system cannot be zero at a point but net electric field can be zero at that point.
2. Net potential of the system at a point can be zero but net electric field can't be zero at that point.
3. Both the net potential and the net field can be zero at a point.
4. Both the net potential and the net electric field cannot be zero at a point.

Answer: (a)

Question 16

Physics · Electrostatic Potential and Capacitance · Numerical

Two parallel plate capacitors $C_1$ and $C_2$ each having capacitance of $10 \, \mu \mathrm{F}$ are individually charged by a $100 \, \mathrm{V}$ D.C. source. Capacitor $C_1$ is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor $C_2$ is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor $C_1$ is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be V. (Assuming Dielectric constant = 10)

Answer: 55

Question 17

Physics · Current Electricity · Single correct

The number of turns of the coil of a moving coil galvanometer is increased in order to increase current sensitivity by 50$\%$. The percentage change in voltage sensitivity of the galvanometer will be:

1. 100$\%$
2. 50$\%$
3. 75$\%$
4. 0$\%$

Answer: (d)

Question 18

Physics · Current Electricity · Single correct

The $H$ amount of thermal energy is developed by a resistor in $10 \, \mathrm{s}$ when a current of $4 \, \mathrm{A}$ is passed through it. If the current is increased to $16 \, \mathrm{A}$, the thermal energy developed by the resistor in $10 \, \mathrm{s}$ will be:

1. $H$
2. $16H$
3. $\frac{H}{4}$
4. $4H$

Answer: (b)

Question 19

Physics · Current Electricity · Numerical

For the given circuit, in the steady state, $|V_B - V_D|$ = $V$.

Answer: 1

Question 20

Physics · Moving Charges and Magnetism · Single correct

A long conducting wire having a current I flowing through it, is bent into a circular coil of N turns. Then it is bent into a circular coil of n turns. The magnetic field is calculated at the centre of coils in both the cases. The ratio of the magnetic field in first case to that of second case is:

1. N : n
2. n^2 : N^2
3. N^2 : n^2
4. n : N

Answer: (c)

Question 21

Physics · Alternating Current · Single correct

An alternating voltage source $V = 260 \sin (628t)$ is connected across a pure inductor of $5 \, \mathrm{mH}$. Inductive reactance in the circuit is:

1. $3.14\Omega$
2. $6.28\Omega$
3. $0.5\Omega$
4. $0.318\Omega$

Answer: (a)

Question 22

Physics · Alternating Current · Numerical

A series LCR circuit consists of $R=80\,\Omega$, $X_L=100\,\Omega$, and $X_C=40\,\Omega$. The input voltage is $2500 \cos(100\, \pi t)$ V. The amplitude of current, in the circuit, is

Answer: 25

Question 23

Physics · Electromagnetic Waves · Single correct

Match List-I with List-II. Choose the correct answer from the option given below:

1. A-II, B-IV, C-III, D-I
2. A-IV, B-I, C-II, D-III
3. A-IV, B-III, C-I, D-II
4. A-III, B-II, C-I, D-IV

Answer: (c)

Question 24

Physics · Ray Optics and Optical Instruments · Single correct

A microscope is focused on an object at the bottom of a bucket. If liquid with refractive index $\frac{5}{3}$ is poured inside the bucket, then microscope have to be raised by 30 cm to focus the object again. The height of the liquid in the bucket is :

1. 75 cm
2. 50 cm
3. 18 cm
4. 12 cm

Answer: (a)

Question 25

Physics · Wave Optics · Numerical

Two light waves of wavelengths 800 and 600 nm are used in Young's double slit experiment to obtain interference fringes on a screen placed 7 m away from plane of slits. If the two slits are separated by 0.35 mm, then shortest distance from the central bright maximum to the point where the bright fringes of the two wavelength coincide will be

Answer: 48

Question 26

Physics · Dual Nature of Radiation and Matter · Subjective

If the two metals A and B are exposed to radiation of wavelength 350 nm. The work functions of metals A and B are 4.8 eV and 2.2 eV. Then choose the correct option (1) Metal B will not emit photo-electrons (2) Both metals A and B will emit photo-electrons (3) Both metals A and B will not emit photo-electrons (4) Metal A will not emit photo-electrons

1. Metal B will not emit photo-electrons
2. Both metals A and B will emit photo-electrons
3. Both metals A and B will not emit photo-electrons
4. Metal A will not emit photo-electrons

Answer: (d)

Question 27

Physics · Atoms · Single correct

The radius of electron's second stationary orbit in Bohr's atom is $R$. The radius of 3rd orbit will be

1. $\frac{R}{3}$
2. $2.25R$
3. $3R$
4. $9R$

Answer: (b)

Question 28

Physics · Atoms · Numerical

If the binding energy of ground state electron in a hydrogen atom is 13.6 $\mathrm{eV}$, then, the energy required to remove the electron from the second excited state of $\mathrm{Li}^{2+}$ will be: $x \times 10^{-1}$ $\mathrm{eV}$. The value of $x$ is

Answer: 136

Question 29

Physics · Semiconductor Electronics: Materials, Devices and Simple Circuits · Single correct

Given below are two statements: Statement I: In a typical transistor, all three regions emitter, base and collector have same doping level. Statement II: In a transistor, collector is the thickest and base is the thinnest segment. In the light of the above statements, choose the most appropriate answer from the options given below.

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

Answer: (c)

Question 30

Physics · Communication Systems · Single correct

Given below are two statements Statement I: For transmitting a signal, size of antenna $(l)$ should be comparable to wavelength of signal (at least $l = \frac{\lambda}{4}$ in dimension). Statement II: In amplitude modulation, amplitude of carrier wave remains constant (unchanged). In the light of the above statements, choose the most appropriate answer from the options given below.

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

Answer: (d)

Question 31

Chemistry · Some Basic Concepts of Chemistry · Single correct

When a hydrocarbon A undergoes complete combustion it requires 11 equivalents of oxygen and produces 4 equivalents of water. What is the molecular formula of A?

1. C_9H_8
2. C_{11}H_4
3. C_5H_8
4. C_{11}H_8

Answer: (a)

Question 32

Chemistry · Some Basic Concepts of Chemistry · Numerical

Assume carbon burns according to following equation : $$2\mathrm{C}_{(s)} + \mathrm{O}_2{(g)} \rightarrow 2\mathrm{CO}_{(g)}$$ When 12 g carbon is burnt in 48 g of oxygen, the volume of carbon monoxide produced is ______ $\times 10^{-1}$ L at STP [nearest integer] [Given : Assume CO as ideal gas, Mass of C is $12 \, \mathrm{g \, mol^{-1}}$, Mass of O is $16 \, \mathrm{g \, mol^{-1}}$ and molar volume of an ideal gas at STP is $22.7 \, \mathrm{L \, mol^{-1}}$]

Answer: 227

Question 33

Chemistry · Structure of Atom · Single correct

Arrange the following orbitals in decreasing order of energy? $(A)$ $n = 3, l = 0, m = 0$ $(B)$ $n = 4, l = 0, m = 0$ $(C)$ $n = 3, l = 1, m = 0$ $(D)$ $n = 3, l = 2, m = 1$ The correct option for the order is:

1. B > D > C > A
2. D > B > C > A
3. A > C > B > D
4. D > B > A > C

Answer: (b)

Question 34

Chemistry · Classification of Elements and Periodicity in Properties · Single correct

Given below are two statements: one is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$ Assertion $(A)$: The first ionization enthalpy of 3d series elements is more than that of group 2 metals Reason $(R)$: In 3d series of elements successive filling of d-orbitals takes place. In the light of the above statements, choose the correct answer from the options given below:

1. Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$
2. Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$
3. $(A)$ is false but $(R)$ is true
4. $(A)$ is false but $(R)$ is true

Answer: (a)

Question 35

Chemistry · Chemical Bonding and Molecular Structure · Numerical

Amongst the following, the number of species having the linear shape is ____. $\mathrm{XeF_2},\quad \mathrm{I_3^+},\quad \mathrm{C_3O_2},\quad \mathrm{I_3^-},\quad \mathrm{CO_2},\quad \mathrm{SO_2},\quad \mathrm{BeCl},\quad \text{and}\quad \mathrm{BCl_2^-}$

Answer: 5

Question 36

Chemistry · Thermodynamics · Numerical

Enthalpies of formation of $\mathrm{CCl_4(g)}$, $\mathrm{H_2O(g)}$, $\mathrm{CO_2(g)}$ and $\mathrm{HCl(g)}$ are $-105$, $-242$, $-394$ and $-92 \, \mathrm{kJ \, mol^{-1}}$ respectively. The magnitude of enthalpy of the reaction given below is ______ $\mathrm{kJ \, mol^{-1}}$ (nearest integer) $\mathrm{CCl_4(g) + 2H_2O(g) \rightarrow CO_2(g) + 4HCl(g)}$

Answer: 173

Question 37

Chemistry · Equilibrium · Single correct

Incorrect statement for the use of indicators in acid-base titration is :

1. Methyl orange may be used for a weak acid vs weak base titration.
2. Methyl orange is a suitable indicator for a strong acid vs weak base titration
3. Phenolphthalein is a suitable indicator for a weak acid vs strong base titration
4. Phenolphthalein may be used for a strong acid vs strong base titration.

Answer: (a)

Question 38

Chemistry · Equilibrium · Numerical

At 298 K, the solubility of silver chloride in water is $1.434 \times 10^{-3} \, \mathrm{g \, L^{-1}}$. The value of $-\log K_{sp}$ for silver chloride is . (Given mass of Ag is $107.9\,\mathrm{g}\ \mathrm{mol^{-1}}$ and mass of Cl is $35.5\,\mathrm{g}\ \mathrm{mol^{-1}}$)

Answer: 10

Question 39

Chemistry · Hydrogen · Single correct

Given below are two statements : Statement I : $\mathrm{H_2O_2}$ is used in the synthesis of Cephalosporin Statement II : $\mathrm{H_2O_2}$ is used for the restoration of aerobic conditions to sewage wastes. In the light of the above statements, choose the most appropriate answer from the options given below :

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

Answer: (a)

Question 40

Chemistry · The s-Block Elements · Single correct

The element playing significant role in neuromuscular function and interneuronal transmission is :

1. Be
2. Ca
3. Li
4. Mg

Answer: (b)

Question 41

Chemistry · The s-Block Elements · Numerical

The number of alkali metal(s), from Li, K, Cs, Rb having ionization enthalpy greater than 400 \, $\mathrm{kJ \, mol^{-1}}$ and forming stable super oxide is .

Answer: 2

Question 42

Chemistry · The p-Block Elements (Group-13 and 14) · Single correct

The Lewis acid character of boron tri halides follows the order:

1. BBr_3 > BI_3 > BCl_3 > BF_3
2. BCl_3 > BF_3 > BBr_3 > BI_3
3. BF_3 > BCl_3 > BBr_3 > BI_3
4. BI_3 > BBr_3 > BCl_3 > BF_3

Answer: (d)

Question 43

Chemistry · Organic Chemistry — Some Basic Principles & Techniques · Single correct

In Dumas method for the estimation of $N_2$, the sample is heated with copper oxide and the gas evolved is passed over:

1. Ni
2. Copper gauze
3. Pd
4. Copper oxide

Answer: (b)

Question 44

Chemistry · Hydrocarbons · Single correct

A hydrocarbon 'X' with formula $C_6H_8$ uses two moles of $H_2$ on catalytic hydrogenation of its one mole. On ozonolysis, 'X' yields two moles of methane dicarbaldehyde. The hydrocarbon 'X' is:

1. hexa-1, 3, 5-triene
2. 1-methylcyclopenta-1, 4-diene
3. cyclohexa-1, 3-diene
4. cyclohexa-1, 4-diene

Answer: (d)

Question 45

Chemistry · Environmental Chemistry · Single correct

The normal rain water is slightly acidic and its pH value is 5.6 because of which one of the following?

1. $\mathrm{CO_2 + H_2O \rightarrow H_2CO_3}$
2. $\mathrm{4NO_2 + O_2 + 2H_2O \rightarrow 4HNO_3}$
3. $\mathrm{2SO_2 + O_2 + 2H_2O \rightarrow 2H_2SO_4}$
4. $\mathrm{N_2O_5 + H_2O \rightarrow 2HNO_3}$

Answer: (a)

Question 46

Chemistry · Redox Reactions · Numerical

A sample of a metal oxide has formula $\mathrm{M}_{0.83}\mathrm{O}_{1.00}$. The metal $\mathrm{M}$ can exist in two oxidation states $+2$ and $+3$. In the sample of $\mathrm{M}_{0.83}\mathrm{O}_{1.00}$, the percentage of metal ions existing in $+2$ oxidation state is $\%$ (nearest integer)

Answer: 59

Question 47

Chemistry · Solutions · Single correct

Evaluate the following statements for their correctness. (A) The elevation in boiling point temperature of water will be same for 0.1 M NaCl and 0.1 M urea. (B) Azeotropic mixtures boil without change in their composition (C) Osmosis always takes place from hypertonic to hypotonic solution (D) The density of 32$\%$ $\mathrm{H_2SO_4}$ solution having molarity 4.09 $\mathrm{M}$ is approximately 1.26 $\mathrm{g \, mL^{-1}}$ (E) A negatively charged sol is obtained when $\mathrm{KI}$ solution is added to silver nitrate solution. Choose the correct answer from the options given below :

1. B, D, and E only
2. A, B, and D only
3. A and C only
4. B and D only

Answer: (d)

Question 48

Chemistry · Electrochemistry · Numerical

The resistivity of a 0.8 M solution of an electrolyte is $5 \times 10^{-3} \, \Omega \mathrm{cm}$. Its molar conductivity is _______ $\times 10^{4} \, \Omega^{-1} \mathrm{cm}^{2} \mathrm{mol}^{-1}$. (Nearest integer)

Answer: 25

Question 49

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

The rate constant for a first order reaction is $20 \, \mathrm{min}^{-1}$. The time required for the initial concentration of the reactant to reduce to its $\frac{1}{32}$ level is ______ $\times 10^{-2} \, \mathrm{min}$. (Nearest integer) (Given: $\ln 10 = 2.303$ $\log 2 = 0.3010$)

Answer: 17

Question 50

Chemistry · Surface Chemistry · Single correct

Match List-I with List-II Choose the correct answer from the options given below:

1. A – II, B – III, C – I, D – IV
2. A – III, B – IV, C – I, D - II
3. A – IV, B – II, C – III, D - I
4. A – II, B – I, C – IV, D – III

Answer: (d)

Question 51

Chemistry · General Principles and Processes of Isolation of Elements · Single correct

Which one of the following statements is incorrect?

1. Boron and Indium can be purified by zone refining method.
2. Van-Arkel method is used to purify tungsten.
3. Cast iron is obtained by melting pig iron with scrap iron and coke using hot air blast.
4. The malleable iron is prepared from cast iron by oxidising impurities in a reverberatory furnace.

Answer: (b)

Question 52

Chemistry · Classification of Elements and Periodicity in Properties · Single correct

Which of the following elements have half-filled f-orbitals in their ground state ? (Given : atomic number Sm = 62; Eu = 63; Tb = 65; Gd = 64, Pm = 61 )

1. B and D only
2. A and E only
3. A and B only
4. C and D only

Answer: (a)

Question 53

Chemistry · Co-ordination Compounds · Numerical

If the CFSE of $[\mathrm{Ti}(\mathrm{H}_2\mathrm{O})_6]^{3+}$ is $-96.0 \, \mathrm{kJ/mol}$, this complex will absorb maximum at wavelength ___ nm. (nearest integer) Assume Planck's constant $(h) = 6.4 \times 10^{-34} \, \mathrm{Js}$, Speed of light $(c) = 3.0 \times 10^8 \, \mathrm{m/s}$ and Avogadro's constant $(N_A) = 6 \times 10^{23} \, / \mathrm{mol}$.

Answer: 480

Question 54

Chemistry · Organic Chemistry — Some Basic Principles & Techniques · Single correct

In the following halogenated organic compounds the one with maximum number of chlorine atoms in its structure is :

1. Chloral
2. Gammaxene
3. Chloropicrin
4. Freon -12

Answer: (b)

Question 55

Chemistry · Amines · Single correct

Cyclohexylamine when treated with nitrous acid yields (P). On treating (P) with PCC results in (Q). When (Q) is heated with dil. NaOH we get ($R$) The final product ($R$) is :

Answer: (b)

Question 56

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Numerical

The number of molecules which gives haloform test among the following molecules is $\underline{\hspace{1cm}}$.

Answer: 3

Question 57

Chemistry · Organic Chemistry — Some Basic Principles & Techniques · Single correct

An organic compound $[A](C_4H_{11}N)$, shows optical activity and gives $N_2$ gas on treatment with $HNO_2$. The compound $[A]$ reacts with $PhSO_2Cl$ producing a compound which is soluble in $KOH$. The structure of $A$ is:

Answer: (d)

Question 58

Chemistry · Chemistry in Everyday Life · Single correct

Which of the following compounds are not used as disinfectants? $(A)$ Chloroxylenol $(B)$ Bithional $(C)$ Veronal $(D)$ Prontosil $(E)$ Terpineol Choose the correct answer from the options given below:

1. A, B, E
2. A, B
3. B, D, E
4. C, D

Answer: (d)

Question 59

Chemistry · Analytical Chemistry · Single correct

Given below are two statements : Statement I : Upon heating a borax bead dipped in cupric sulphate in a luminous flame, the colour of the bead becomes green. Statement II : The green colour observed is due to the formation of copper(I) metaborate. In the light of the above statements, choose the most appropriate answer from the options given below :

1. Both Statement I and Statement II are true
2. Statement I is true but Statement II is false
3. Both Statement I and Statement II are false
4. Statement I is false but Statement II is true

Answer: (c)

Question 60

Chemistry · Biomolecules · Single correct

Compound A, $C_5H_{10}O_5$, given a tetraacetate with $Ac_2O$ and oxidation of A with $Br_2-H_2O$ gives an acid, $C_5H_{10}O_6$. Reduction of A with HI gives isopentane. The possible structure of A is :

Answer: (a)

Question 61

Maths · Complex Numbers and Quadratic Equations · Single correct

The equation $$e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0, x \in \mathbb{R}$$ has:

1. two solutions and both are negative
2. no solution
3. four solutions two of which are negative
4. two solutions and only one of them is negative

Answer: (a)

Question 62

Maths · Complex Numbers and Quadratic Equations · Single correct

The complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ is equal to:

1. $\sqrt{2} \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)$
2. $\cos \frac{\pi}{12} - i \sin \frac{\pi}{12}$
3. $\sqrt{2} \left( \cos \frac{\pi}{12} + i \sin \frac{\pi}{12} \right)$
4. $\sqrt{2} i \left( \cos \frac{5\pi}{12} - i \sin \frac{5\pi}{12} \right)$

Answer: (a)

Question 63

Maths · Sequences and Series · Single correct

Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7 = 3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n! - 4a_{n+2}$ is equal to:

1. 24
2. $\frac{33}{4}$
3. $\frac{381}{4}$
4. 9

Answer: (a)

Question 64

Maths · Sequences and Series · Fill in the blank

The sum $1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$ is___________.

Answer: 6952

Question 65

Maths · Binomial Theorem · Numerical

The Coefficient of $x^{-6}$, in the expansion of $$\left(\frac{4x}{5} + \frac{5}{2x^2}\right)^9$$, is

Answer: 5040

Question 66

Maths · Binomial Theorem · Numerical

If the constant term in the binomial expansion of $$\left( \frac{x^{\frac{5}{2}}}{2} - \frac{4}{x^\ell} \right)^9$$ is $-84$ and the Coefficient of $x^{-3\ell}$ is $2^\alpha \beta$, where $\beta < 0$ is an odd number, Then $|\alpha \ell - \beta|$ is equal to

Answer: 98

Question 67

Maths · Permutations and Combinations · Numerical

If $$\frac{{^{2n+1}P_{n-1}}}{{^{2n-1}P_n}} = 11 : 21$$, then $n^2 + n + 15$ is equal to:

Answer: 45

Question 68

Maths · Conic Sections · Single correct

The set of all values of $a$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$ is equal to:

1. $(8, \infty)$
2. $(4, \infty)$
3. $(0, 4]$
4. $(2, 12]$

Answer: (a)

Question 69

Maths · Conic Sections · Numerical

Let S be the set of all $a \in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P \,(b, c)$, $b, c \in \mathbb{N}$, on the parabola $y^2 = 2ax$ and the lines $x = b$, $y = 0$ is $16 \, unit^2$, then $\sum_{a \in S} a$ is equal to _____.

Answer: 146

Question 70

Maths · Conic Sections · Single correct

Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is ______________.

1. 2
2. 3
3. $\frac{5}{2}$
4. $\frac{3}{2}$

Answer: (a)

Question 71

Maths · Limits and Derivatives · Single correct

$$\lim_{x \to \infty} \frac{\left( \sqrt{3x+1} + \sqrt{3x-1} \right)^6 + \left( \sqrt{3x+1} - \sqrt{3x-1} \right)^6}{\left( x + \sqrt{x^2-1} \right)^6 + \left( x - \sqrt{x^2-1} \right)^6} x^3$$

1. is equal to 9
2. is equal to 27
3. does not exist
4. is equal to $\frac{27}{2}$

Answer: (b)

Question 72

Maths · Mathematical Reasoning · Single correct

The number of values of $r \in \{p, q, \sim p, \sim q\}$ for which $((p \land q) \Rightarrow (r \lor q)) \land ((p \land r) \Rightarrow q)$ is a tautology, is:

1. 3
2. 2
3. 1
4. 4

Answer: (b)

Question 73

Maths · Statistics · Single correct

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha (> 0)$, and the mean and standard deviation of marks of class B of $n$ students be respectively 55 and $30 - \alpha$. If the mean and variance of the marks of the combined class of $100 + n$ students are respectively 50 and 350, then the sum of variances of classes A and B is:

1. 500
2. 650
3. 450
4. 900

Answer: (a)

Question 74

Maths · Relations and Functions · Single correct

Among the relations $$S = \{(a, b) : a, b \in \mathbb{R} - \{0\}, 2 + \frac{a}{b} > 0\}$$ And $$T = \{(a, b) : a, b \in \mathbb{R}, a^2 - b^2 \in \mathbb{Z}\},$$

1. S is transitive but T is not
2. T is symmetric but S is not
3. Neither S nor T is transitive
4. Both S and T are symmetric

Answer: (b)

Question 75

Maths · Matrices · Numerical

Let $A = [a_{ij}]$, $a_{ij} \in \mathbb{Z} \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2,13)$ is

Answer: 196

Question 76

Maths · Matrices · Numerical

Let A be a $n \times n$ matrix such that $|A|=2$. If the determinant of the matrix $Adj \left( 2 \cdot Adj(2A^{-1}) \right)$ is $2^{84}$, then $n$ is equal to .

Answer: 5

Question 77

Maths · Determinants · Single correct

If a point $P(\alpha, \beta, \gamma)$ satisfying $(\alpha$ $\beta$ $\gamma$)$$\begin{vmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{vmatrix} = (0\ 0\ 0)$$ lies on the plane $2x + 4y + 3z = 5$, then $6\alpha + 9\beta + 7\gamma$ is equal to:

1. 1
2. $\frac{11}{5}$
3. $\frac{5}{4}$
4. 11

Answer: (d)

Question 78

Maths · Inverse Trigonometric Functions · Single correct

Let (a, b) $\subset$ (0, 2$\pi$) be the largest interval for which $\sin^{-1}(\sin \theta) - \cos^{-1}(\sin \theta) > 0$, $\theta \in (0, 2\pi)$, holds. If $\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$ and $\alpha - \beta = b - a$, then $\alpha$ is equal to :

1. $\frac{\pi}{48}$
2. $\frac{\pi}{16}$
3. $\frac{\pi}{8}$
4. $\frac{\pi}{12}$

Answer: (d)

Question 79

Maths · Relations and Functions · Single correct

Let $f : \mathbb{R} - \{2,6\} \rightarrow \mathbb{R}$ be real valued function defined as $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$. Then range of $f$ is

1. $\left(-\infty, -\frac{21}{4}\right] \cup [0, \infty)$
2. $\left(-\infty, -\frac{21}{4}\right) \cup (0, \infty)$
3. $\left(-\infty, -\frac{21}{4}\right] \cup \left[\frac{21}{4}, \infty\right)$
4. $\left(-\infty, -\frac{21}{4}\right] \cup [1, \infty)$

Answer: (a)

Question 80

Maths · Relations and Functions · Single correct

The absolute minimum value, of the function $f(x) = |x^2 - x + 1| + \left\lfloor x^2 - x + 1 \right\rfloor$, where $[t]$ denotes the greatest integer function, in the interval $[-1, 2]$, is :

1. $\frac{3}{4}$
2. $\frac{3}{2}$
3. $\frac{1}{4}$
4. $\frac{5}{4}$

Answer: (a)

Question 81

Maths · Integrals · Single correct

Let $\alpha > 0$. If $\int_{0}^{\alpha} \frac{x}{\sqrt{x+\alpha} - \sqrt{x}} \, dx = \frac{16 + 20\sqrt{2}}{15}$, then $\alpha$ is equal to:

1. 2
2. 4
3. $\sqrt{2}$
4. 2$\sqrt{2}$

Answer: (a)

Question 82

Maths · Differential Equations · Single correct

If $\phi(x) = -\frac{1}{\sqrt{x}} \int_{\frac{\pi}{4}}^{x} (4\sqrt{2} \sin t - 3\phi'(t)) \, dt, \quad x > 0,$ then $\phi'\left(\frac{\pi}{4}\right)$ is equal to :

1. \frac{8}{\sqrt{\pi}}
2. \frac{4}{6 + \sqrt{\pi}}
3. \frac{8}{6 + \sqrt{\pi}}
4. \frac{4}{6 - \sqrt{\pi}}

Answer: (c)

Question 83

Maths · Applications of Integrals · Numerical

Let the area of the region $\{(x, y) : |2x - 1| \leq y \leq x^2 - x, 0 \leq x \leq 1\}$ be $A$. Then $(6A + 11)^2$ is equal to ______.

Answer: 125

Question 84

Maths · Differential Equations · Single correct

Let $y = y(x)$ be the solution of the differential equation $$(3y^2 - 5x^2) y \, dx + 2x(x^2 - y^2) \, dy = 0$$ such that $y(1) = 1$. then $$\left| (y(2))^3 - 12y(2) \right|$$ is equal to:

1. 32$\sqrt{2}$
2. 64
3. 16$\sqrt{2}$
4. 32

Answer: (a)

Question 85

Maths · Vector Algebra · Single correct

Let : $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\vec{c} = 5\hat{i} - 3\hat{j} + 3\hat{k}$ be there vectors. If $\vec{r}$ is a vector such that, $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$. Then $25|\vec{r}|^2$ is equal to

1. 449
2. 336
3. 339
4. 560

Answer: (c)

Question 86

Maths · Vector Algebra · Numerical

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\left| \vec{a} \right| = \sqrt{31}$, $4 \left| \vec{b} \right| = \left| \vec{c} \right| = 2$ and $2 (\vec{a} \times \vec{b}) = 3 (\vec{c} \times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$, then $\left( \frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}} \right)^2$ is equal to ________.

Answer: 3

Question 87

Maths · Three Dimensional Geometry · Single correct

Let the plane $P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0$ be parallel to the line $L: \frac{x+2}{2} = \frac{y-3}{3} = \frac{z+4}{5}$. If the intercept of $P$ on the y-axis is 1, then the distance between $P$ and $L$ is:

1. $\sqrt{14}$
2. $\frac{6}{\sqrt{14}}$
3. $\sqrt{\frac{2}{7}}$
4. $\sqrt{\frac{7}{2}}$

Answer: (a)

Question 88

Maths · Three Dimensional Geometry · Single correct

Let P be the plane, passing through the point $(1, -1, -5)$ and perpendicular to the line joining the points $(4, 1, -3)$ and $(2, 4, 3)$. Then the distance of P from the point $(3, -2, 2)$ is

1. 6
2. 4
3. 5
4. 7

Answer: (c)

Question 89

Maths · Vector Algebra · Single correct

The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A$, $B$, $C$ is $(2, a, 4)$, $a \in \mathbb{N}$. If the volume of the tetrahedron $OABC$ is $144 \, unit^3$, then which of the following points is NOT on $P$?

1. (2, 2, 4)
2. (0, 4, 4)
3. (3, 0, 4)
4. (0, 6, 3)

Answer: (c)

Question 90

Maths · Probability · Numerical

Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space $[0, 60]$ is less than or equal to $a$. If $P(A) = \frac{11}{36}$, then $a$ is equal to ________.

Answer: 10