JEE Main 2023 — 24 Jan Shift 1 (Previous Year Paper)

Match List I with List II. List I: A. Planck's constant (h); B. Stopping potential (Vs); C. Work function ($\phi$); D. Momentum (p). List II: I. $[M^1L^2T^{-2}]$; II. $[M^1L^1T^{-1}]$; III. $[M^1L^2T^{-1}]$; IV. $[M^1L^2T^{-3}A^{-1}]$. Choose the correct answer from the options given below:

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

Correct answer: D

A travelling wave is described by the equation $y(x,t)=[0.05\sin(8x-4t)]$ m. The velocity of the wave is: [all the quantities are in SI unit]

1. A.8 ms$^{-1}$
2. B.4 ms$^{-1}$
3. C.0.5 ms$^{-1}$✓
4. D.2 ms$^{-1}$

Correct answer: C

The maximum vertical height to which a man can throw a ball is 136 m. The maximum horizontal distance upto which he can throw the same ball is:

1. A.136 m
2. B.68 m
3. C.272 m✓
4. D.192 m

Correct answer: C

Given below are two statements: Statement I: An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable. Statement II: Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes down with increasing speed. 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 true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false✓
4. D.Statement I is false but Statement II is true

Correct answer: C

As per given figure, a weightless pulley P is attached on a double inclined frictionless surfaces. The tension in the string (massless) will be (if g = 10 m/s$^2$)

1. A.$(4\sqrt3-1)$ N
2. B.$(4\sqrt3+1)$ N
3. C.$4(\sqrt3+1)$ N✓
4. D.$4(\sqrt3-1)$ N

Correct answer: C

The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth's surface is (given, radius of earth $R_e$ = 6400 km):

1. A.4.9 N
2. B.19.6 N
3. C.8 N✓
4. D.9.8 N

Correct answer: C

A 100 m long wire having cross-sectional area $6.25\times10^{-4}$ m$^2$ and Young's modulus is $10^{10}$ Nm$^{-2}$ is subjected to a load of 250 N, then the elongation in the wire will be:

1. A.$6.25\times10^{-6}$ m
2. B.$6.25\times10^{-3}$ m
3. C.$4\times10^{-4}$ m
4. D.$4\times10^{-3}$ m✓

Correct answer: D

1 g of a liquid is converted to vapour at $3\times10^5$ Pa pressure. If 10% of the heat supplied is used for increasing the volume by 1600 cm$^3$ during this phase change, then the increase in internal energy in the process will be:

1. A.4800 J
2. B.432000 J
3. C.$4.32\times10^8$ J
4. D.4320 J✓

Correct answer: D

Given below are two statements: Statement I: The temperature of a gas is $-73^\circ$C. When the gas is heated to $527^\circ$C, the root mean square speed of the molecules is doubled. Statement II: The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules. 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 true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false✓
4. D.Statement I is false but Statement II is true

Correct answer: C

A modulating signal is a square wave, as shown in the figure. If the carrier wave is given as $c(t)=2\sin(8\pi t)$ volts, the modulation index is:

1. A.$\frac14$
2. B.1
3. C.$\frac12$✓
4. D.$\frac13$

Correct answer: C

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Photodiodes are preferably operated in reverse bias condition for light intensity measurement. Reason R: The current in the forward bias is more than the current in the reverse bias for a $p-n$ junction diode. In the light of the above statements, choose the correct answer from the options given below:

1. A.Both A and R are true and R is the correct explanation of A
2. B.Both A and R are true but R is NOT the correct explanation of A✓
3. C.A is true but R is false
4. D.A is false but R is true

Correct answer: B

Consider the following radioactive decay process ${}^{218}_{84}A\xrightarrow{\alpha}A_1\xrightarrow{\beta^-}A_2\xrightarrow{\gamma}A_3\xrightarrow{\alpha}A_4\xrightarrow{\beta^+}A_5\xrightarrow{\gamma}A_6$. The mass number and the atomic number of $A_6$ are given by:

1. A.210 and 82
2. B.210 and 80✓
3. C.210 and 84
4. D.211 and 80

Correct answer: B

From the photoelectric effect experiment, following observations are made. Identify which of these are correct. A. The stopping potential depends only on the work function of the metal. B. The saturation current increases as the intensity of incident light increases. C. The maximum kinetic energy of a photo electron depends on the intensity of the incident light. D. Photoelectric effect can be explained using wave theory of light. Choose the correct answer from the options given below:

1. A.B only✓
2. B.A, B, D only
3. C.B, C only
4. D.A, C, D only

Correct answer: A

Given below are two statements: Statement I: If the Brewster's angle for the light propagating from air to glass is $\theta_B$, then Brewster's angle for the light propagating from glass to air is $\frac{\pi}{2}-\theta_B$. Statement II: The Brewster's angle for the light propagating from glass to air is $\tan^{-1}(\mu_g)$ where $\mu_g$ is the refractive index of glass. 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 true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false✓
4. D.Statement I is false but Statement II is true

Correct answer: C

If $\vec E$ and $\vec K$ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by: ($\omega$ - angular frequency):

1. A.$\omega(\vec K\times\vec E)$
2. B.$\omega(\vec E\times\vec K)$
3. C.$\vec K\times\vec E$
4. D.$\frac{1}{\omega}(\vec K\times\vec E)$✓

Correct answer: D

A conducting circular loop of radius $\frac{10}{\sqrt\pi}$ cm is placed perpendicular to a uniform magnetic field of 0.5 T. The magnetic field is decreased to zero in 0.5 s at a steady rate. The induced emf in the circular loop at 0.25 s is:

1. A.emf = 100 mV
2. B.emf = 1 mV
3. C.emf = 10 mV✓
4. D.emf = 5 mV

Correct answer: C

Two long straight wires P and Q carrying equal current 10A each were kept parallel to each other at 5 cm distance. Magnitude of magnetic force experienced by 10 cm length of wire P is $F_1$. If distance between wires is halved and currents on them are doubled, force $F_2$ on 10 cm length of wire P will be:

1. A.$8F_1$✓
2. B.$10F_1$
3. C.$\frac{F_1}{8}$
4. D.$\frac{F_1}{10}$

Correct answer: A

A circular loop of radius $r$ is carrying current I A. The ratio of magnetic field at the center of circular loop and at a distance r from the center of the loop on its axis is:

1. A.$1:3\sqrt2$
2. B.$2\sqrt2:1$✓
3. C.$1:\sqrt2$
4. D.$3\sqrt2:2$

Correct answer: B

As shown in the figure, a network of resistors is connected to a battery of 24V with an internal resistance of 3$\Omega$. The currents through the resistors $R_4$ and $R_5$ are $I_4$ and $I_5$ respectively. The values of $I_4$ and $I_5$ are:

1. A.$I_4=\frac25$ A and $I_5=\frac85$ A✓
2. B.$I_4=\frac85$ A and $I_5=\frac25$ A
3. C.$I_4=\frac65$ A and $I_5=\frac{24}{5}$ A
4. D.$I_4=\frac{24}{5}$ A and $I_5=\frac65$ A

Correct answer: A

If two charges $q_1$ and $q_2$ are separated with distance 'd' and placed in a medium of dielectric constant K. What will be the equivalent distance between charges in air for the same electrostatic force?

1. A.$d\sqrt k$✓
2. B.$1.5d\sqrt k$
3. C.$2d\sqrt k$
4. D.$k\sqrt d$

Correct answer: A

A block of a mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $\frac{\pi}{\sqrt x}$ in SI unit. The value of $x$ is ________.

Correct answer: 5

Vectors $a\hat i+b\hat j+\hat k$ and $2\hat i-3\hat j+4\hat k$ are perpendicular to each other when $3a+2b=7$, the ratio of $a$ to $b$ is $\frac x2$. The value of $x$ is ________.

Correct answer: 1

A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of 5$^{th}$ second. The force acted on the body is ________ N.

Correct answer: 40

Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be $\sqrt x$ cm. The value of $x$ is ________.

Correct answer: 110

A hole is drilled in a metal sheet. At $27^\circ$C, the diameter of hole is 5 cm. When the sheet is heated to $177^\circ$C, the change in the diameter of hole is $d\times10^{-3}$ cm. The value of d will be ________ if coefficient of linear expansion of the metal is $1.6\times10^{-5}/^\circ$C.

Correct answer: 12

Assume that protons and neutrons have equal masses. Mass of a nucleon is $1.6\times10^{-27}$ kg and radius of nucleus is $1.5\times10^{-15}A^{1/3}$ m. The approximate ratio of the nuclear density and water density is $n\times10^{13}$. The value of $n$ is ________.

Correct answer: 11

As shown in the figure, a combination of a thin plano concave lens and a thin plano convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is 30 cm and refraction index of the material for both the lenses is 1.75. Both the lenses are placed at distance of 40 cm from each other. Due to the combination, the image of the object is formed at distance $x$ = ________ cm, from concave lens.

Correct answer: 120

In the circuit shown in the figure, the ratio of the quality factor and the band width is ________ s.

Correct answer: 10

A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm respectively. The resistance of the conductor is $n\times10^{-3}\Omega$. If the resistivity of the material is $2.4\times10^{-8}\Omega$m. The value of $n$ is ________.

Correct answer: 2

A stream of a positively charged particles having $\frac qm=2\times10^{11}\frac{C}{kg}$ and velocity $\vec v_0=3\times10^7\hat i$ m/s is deflected by an electric field $1.8\hat j$ kV/m. The electric field exists in a region of 10 cm along $x$ direction. Due to the electric field, the deflection of the charge particles in the $y$ direction is ________ mm.

Correct answer: 2

Order of Covalent bond; A. KF > KI; LiF > KF. B. KF < KI; LiF > KF. C. SnCl$_4$ > SnCl$_2$; CuCl > NaCl. D. LiF > KF; CuCl < NaCl. E. KF < KI; CuCl > NaCl. Choose the correct answer from the options given below:

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

Correct answer: D

In the depression of freezing point experiment A. Vapour pressure of the solution is less than that of pure solvent B. Vapour pressure of the solution is more than that of pure solvent C. Only solute molecules solidify at the freezing point D. Only solvent molecules solidify at the freezing point. Choose the most appropriate answer from the options given below:

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

Correct answer: D

Statement I: For colloidal particles, the values of colligative properties are of small order as compared to values shown by true solutions at same concentration. Statement II: For colloidal particles, the potential difference between the fixed layer and the diffused layer of same charges is called the electrokinetic potential or zeta potential. 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 true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false✓
4. D.Statement I is false but Statement II is true

Correct answer: C

It is observed that characteristic X-ray spectra of elements show regularity. When frequency to the power "n" i.e. $\nu^n$ of X-rays emitted is plotted against atomic number "Z", following graph is obtained. The value of "n" is

1. A.1
2. B.2
3. C.3
4. D.$\frac12$✓

Correct answer: D

Match List I with List II. List I: A. Reverberatory furnace; B. Electrolytic cell; C. Blast furnace; D. Zone Refining furnace. List II: I. Pig Iron; II. Aluminum; III. Silicon; IV. Copper. Choose the correct answer from the options given below:

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

Correct answer: A

Decreasing order of the hydrogen bonding in following forms of water is correctly represented by A. Liquid water B. Ice C. Impure water. Choose the correct answer from the options given below:

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

Correct answer: B

Reaction of BeO with ammonia and hydrogen fluoride gives A which on thermal decomposition gives BeF$_2$ and NH$_4$F. What is 'A'?

1. A.(NH$_4$)BeF$_3$
2. B.(NH$_4$)$_2$BeF$_4$✓
3. C.H$_3$NBeF$_3$
4. D.(NH$_4$)Be$_2$F$_5$

Correct answer: B

Match List I with List II. List I: A. Chlorophyll; B. Soda ash; C. Dentistry, Ornamental work; D. Used in white washing. List II: I. Na$_2$CO$_3$; II. CaSO$_4$; III. Mg$^{2+}$; IV. Ca(OH)$_2$. Choose the correct answer from the options given below:

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

Correct answer: D

Which of the Phosphorus oxoacid can create silver mirror from AgNO$_3$ solution?

1. A.H$_4$P$_2$O$_5$✓
2. B.(HPO$_3$)$_n$
3. C.H$_4$P$_2$O$_6$
4. D.H$_4$P$_2$O$_7$

Correct answer: A

The magnetic moment of a transition metal compound has been calculated to be 3.87 B.M. The metal ion is

1. A.Ti$^{2+}$
2. B.V$^{2+}$✓
3. C.Cr$^{2+}$
4. D.Mn$^{2+}$

Correct answer: B

Which of the following is true about freons?

1. A.These are chlorofluorocarbon compounds✓
2. B.These are radicals of chlorine and chlorine monoxide
3. C.These are chemicals causing skin cancer
4. D.All radicals are called freons

Correct answer: A

The primary and secondary valencies of cobalt respectively in $[\mathrm{Co(NH_3)_5Cl}]\mathrm{Cl_2}$ are:

1. A.2 and 6
2. B.3 and 5
3. C.2 and 8
4. D.3 and 6✓

Correct answer: D

Increasing order of stability of the resonance structures is: A. $\mathrm{OHC-\overset{+}{C}H-CH=CH-\overset{-}{C}H-NH_2}$; B. $\mathrm{OHC-\overset{-}{C}H-CH=CH-\overset{+}{C}H-NH_2}$; C. $\mathrm{H-C(\overset{+}{O})=CH-CH=CH-\overset{-}{C}H-NH_2}$ (oxygen drawn with a single bond and a positive charge); D. $\mathrm{OHC-\overset{-}{C}H-CH=CH-CH=\overset{+}{N}H_2}$. Choose the correct answer from the options given below:

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

Correct answer: C

In the following given reaction, 'A' is: (cyclobutyl)C(CH$_3$)=CH$_2$ + HBr $\rightarrow$ 'A' (major product)

1. A.2-bromo-1-methylcyclohexane with CH3 and Br on adjacent ring carbons (six-membered ring)
2. B.1-bromo-1-methylcyclohexane
3. C.2-bromo-1,1-dimethylcyclopentane
4. D.1-bromo-1,2-dimethylcyclopentane✓

Correct answer: D

Assertion A: Hydrolysis of an alkyl chloride is a slow reaction but in the presence of NaI, the rate of the hydrolysis increases. Reason R: I$^-$ is a good nucleophile as well as a good leaving group. In the light of the above statements, choose the correct answer from the options given below

1. A.Both A and R are true and R is the correct explanation of A✓
2. B.Both A and R are true but R is NOT the correct explanation of A
3. C.A is true but R is false
4. D.A is false but R is true

Correct answer: A

'A' and 'B' formed in the following set of reactions are: 3-(hydroxymethyl)phenol $\xrightarrow{\text{HBr},\Delta}$ A; 3-methoxyphenol $\xrightarrow{\text{HBr},\Delta}$ B

1. A.A = 3-(bromomethyl)phenol, B = benzene-1,3-diol (resorcinol)✓
2. B.A = 3-(bromomethyl)phenol, B = 3-bromophenol
3. C.A = 3-bromobenzyl alcohol (Br replaces OH on ring), B = 1,3-dibromobenzene
4. D.A = 3-(bromomethyl)phenol, B = 2-bromo-5-methoxyphenol-type (Br on ring, OCH3 retained)

Correct answer: A

Compound (X) undergoes following sequence of reactions to give the Lactone (Y): Compound (X) $\xrightarrow{\text{(i) HCHO, KOH; (ii) KCN(alc); (iii) H}_3\text{O}^+}$ Lactone (Y) [a five-membered lactone (γ-butyrolactone ring) with OH on the carbon alpha to C=O and two CH$_3$ groups on the next (β) carbon, i.e. pantolactone]. Compound (X) is

1. A.$\mathrm{HOH_2C-C(CH_3)_2-CHO}$
2. B.$\mathrm{HO-CH(CH_3)-CH_2-CH_2-CH_2-CHO}$ (drawn as H$_2$C(OH)–CH$_2$–CH$_2$–CH$_2$–CHO)
3. C.$\mathrm{(CH_3)_2CH-CHO}$✓
4. D.$\mathrm{CH_3-CH_2-CH_2-CHO}$ (drawn as H$_2$C(CH$_3$)–CH$_2$–CHO)

Correct answer: C

'R' formed in the following sequence of reactions is: 4-chloroacetophenone $\xrightarrow{\text{NaCN, HOAc}}$ 'P' $\xrightarrow{\text{EtOH, H}^+}$ 'Q' $\xrightarrow{\text{(i) 2MeMgBr, (ii) H}_3\text{O}^+}$ 'R' (major product)

1. A.2-(4-chlorophenyl)-3-methylbutane-2,3-diol: p-Cl-C6H4-C(OH)(Me)-C(OH)Me2✓
2. B.2-(4-cyanophenyl)propan-2-ol: p-NC-C6H4-C(OH)Me2
3. C.2-(4-chlorophenyl)propan-2-ol: p-Cl-C6H4-C(OH)Me2
4. D.2-(4-cyanophenyl)-3,3-dimethylbutan-2-ol: p-NC-C6H4-C(OH)(CH3)-C(CH3)3

Correct answer: A

An ammoniacal metal salt solution gives a brilliant red precipitate on addition of dimethylglyoxime. The metal ion is:

1. A.Fe$^{2+}$
2. B.Co$^{2+}$
3. C.Ni$^{2+}$✓
4. D.Cu$^{2+}$

Correct answer: C

Given below are two statements: Statement I: Noradrenaline is a neurotransmitter. Statement II: Low level of noradrenaline is not the cause of depression in human. 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 correct
2. B.Both Statement I and Statement II are incorrect
3. C.Statement I is correct but Statement II is incorrect✓
4. D.Statement I is incorrect but Statement II is correct

Correct answer: C

5 g of NaOH was dissolved in deionized water to prepare a 450 mL stock solution. What volume (in mL) of this solution would be required to prepare 500 mL of 0.1 M solution? ________ Given: Molar Mass of Na, O and H is 23, 16 and 1 g mol$^{-1}$ respectively

Correct answer: 180

When Fe$_{0.93}$O is heated in presence of oxygen, it converts to Fe$_2$O$_3$. The number of correct statement/s from the following is ________ A. The equivalent weight of Fe$_{0.93}$O is $\frac{\text{Molecular weight}}{0.79}$ B. The number of moles of Fe$^{2+}$ and Fe$^{3+}$ in 1 mole of Fe$_{0.93}$O is 0.79 and 0.14 respectively C. Fe$_{0.93}$O is metal deficient with lattice comprising of cubic closed packed arrangement of O$^{2-}$ ions D. The % composition of Fe$^{2+}$ and Fe$^{3+}$ in Fe$_{0.93}$O is 85% and 15% respectively

Correct answer: 4

If wavelength of the first line of the Paschen series of hydrogen atom is 720 nm, then the wavelength of the second line of this series is ________ nm. (Nearest integer)

Correct answer: 492

For independent processes at 300 K: Process A: $\Delta H=-25$ kJ mol$^{-1}$, $\Delta S=-80$ J K$^{-1}$; B: $-22$, $+40$; C: $+25$, $-50$; D: $+22$, $+20$. The number of non-spontaneous processes from the following is ________

Correct answer: 2

The dissociation constant of acetic acid is $x\times10^{-5}$. When 25 mL of 0.2 M CH$_3$COONa solution is mixed with 25 mL of 0.02 M CH$_3$COOH solution, the pH of the resultant solution is found to be equal to 5. The value of $x$ is ________

Correct answer: 10

At 298 K, a 1 litre solution containing 10 mmol of Cr$_2$O$_7^{2-}$ and 100 mmol of Cr$^{3+}$ shows a pH of 3.0. Given: Cr$_2$O$_7^{2-}\rightarrow$ Cr$^{3+}$; $E^\circ$ = 1.330V and $\frac{2.303RT}{F}$ = 0.059 V. The potential for the half cell reaction is $x\times10^{-3}$ V. The value of $x$ is ________

Correct answer: 917

The number of correct statement/s from the following is ________ A. Larger the activation energy, smaller is the value of the rate constant. B. The higher is the activation energy, higher is the value of the temperature coefficient. C. At lower temperatures, increase in temperature causes more change in the value of k than at higher temperature D. A plot of ln k vs $\frac1T$ is a straight line with slope equal to $-\frac{E_a}{R}$

Correct answer: 3

The d-electronic configuration of $[\mathrm{CoCl_4}]^{2-}$ in tetrahedral crystal field is $e^mt_2^n$. Sum of "m" and "number of unpaired electrons" is ________

Correct answer: 7

Uracil is a base present in RNA with the following structure. % of N in uracil is ________ Given: Molar mass N = 14 g mol$^{-1}$, O = 16 g mol$^{-1}$, C = 12 g mol$^{-1}$, H = 1 g mol$^{-1}$

Correct answer: 25

Number of moles of AgCl formed in the following reaction is ________. [Substrate: a benzene ring bearing (i) a vinylic chloride side chain –CH=CH–Cl, (ii) an aryl Cl, and (iii) a side chain –CH(Cl)–CH$_2$–C(CH$_3$)$_2$–Cl (one benzylic secondary Cl and one tertiary Cl)] $\xrightarrow{\text{AgNO}_3}$ (A) + X AgCl$\downarrow$

Correct answer: 2

The equation $x^2-4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

1. A.exactly two solutions in $(-\infty,\infty)$
2. B.a unique solution in $(-\infty,\infty)$✓
3. C.a unique solution in $(-\infty,1)$
4. D.no solution

Correct answer: B

The relation $R=\{(a,b): \gcd(a,b)=1, 2a\ne b, a,b\in\mathbb{Z}\}$ is:

1. A.reflexive but not symmetric
2. B.symmetric but not transitive
3. C.transitive but not reflexive
4. D.neither symmetric nor transitive✓

Correct answer: D

Let $p,q\in\mathbb{R}$ and $(1-\sqrt3i)^{200}=2^{199}(p+iq)$, $i=\sqrt{-1}$. Then $p+q+q^2$ and $p-q+q^2$ are roots of the equation.

1. A.$x^2+4x+1=0$
2. B.$x^2-4x-1=0$
3. C.$x^2+4x-1=0$
4. D.$x^2-4x+1=0$✓

Correct answer: D

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$, $2x+Ny+2z=2$, $3x+3y+Nz=3$ has unique solution is $\frac k6$, then the sum of value of k and all possible values of N is

1. A.21
2. B.18
3. C.19
4. D.20✓

Correct answer: D

If A and B are two non-zero $n\times n$ matrices such that $A^2+B=A^2B$, then

1. A.$A^2B=I$
2. B.$A^2=I$ or $B=I$
3. C.$A^2B=BA^2$✓
4. D.$AB=I$

Correct answer: C

Let $\alpha$ be a root of the equation $(a-c)x^2+(b-a)x+(c-b)=0$ where a, b, c are distinct real numbers such that the matrix $\begin{bmatrix}\alpha^2&\alpha&1\\1&1&1\\a&b&c\end{bmatrix}$ is singular. Then, the value of $\frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}$ is

1. A.3✓
2. B.6
3. C.9
4. D.12

Correct answer: A

$\tan^{-1}\left(\frac{1+\sqrt3}{3+\sqrt3}\right)+\sec^{-1}\left(\sqrt{\frac{8+4\sqrt3}{6+3\sqrt3}}\right)$ is equal to:

1. A.$\frac{\pi}{2}$
2. B.$\frac{\pi}{3}$✓
3. C.$\frac{\pi}{4}$
4. D.$\frac{\pi}{6}$

Correct answer: B

Let $y=y(x)$ be the solution of the differential equation $x^3dy+(xy-1)dx=0$, $x>0$, $y\left(\frac12\right)=3-e$. Then $y(1)$ is equal to

1. A.$2-e$
2. B.e
3. C.3
4. D.1✓

Correct answer: D

The value of $\sum_{r=0}^{22}{}^{22}C_r\,{}^{23}C_r$ is

1. A.${}^{44}C_{22}$
2. B.${}^{44}C_{23}$
3. C.${}^{45}C_{23}$✓
4. D.${}^{45}C_{24}$

Correct answer: C

Let a tangent to the curve $y^2=24x$ meet the curve $xy=2$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the

1. A.directrix $4x=-3$
2. B.directrix $4x=3$✓
3. C.length of latus rectum $\frac32$
4. D.length of latus rectum 2

Correct answer: B

The distance of the point $(7,-3,-4)$ from the plane passing through the points $(2,-3,1)$, $(-1,1,-2)$ and $(3,-4,2)$ is:

1. A.4
2. B.5
3. C.$4\sqrt2$
4. D.$5\sqrt2$✓

Correct answer: D

For three positive integers p, q, r, $x^{pq^2}=y^{qr}=z^{p^2r}$ and $r=pq+1$ such that 3, $3\log_yx$, $3\log_zy$, $7\log_xz$ are in A.P. with common difference $\frac12$. Then $r-p-q$ is equal to

1. A.2✓
2. B.12
3. C.−6
4. D.6

Correct answer: A

Let $f(x)=\begin{cases}x^2\sin\left(\frac1x\right), & x\ne0\\ 0, & x=0\end{cases}$. Then at $x=0$

1. A.$f$ is continuous but not differentiable
2. B.$f'$ is continuous but not differentiable
3. C.$f$ and $f'$ both are continuous
4. D.$f$ is continuous but $f'$ is not continuous✓

Correct answer: D