JEE Main 2024 — 4 Apr Shift 1 (Previous Year Paper)

In an experiment to measure focal length $(f)$ of convex lens, the least counts of the measuring scales for the position of object $(u)$ and for the position of image $(v)$ are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

1. A.$2f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$
2. B.$\frac{\Delta u}{u}+\frac{\Delta v}{v}$
3. C.$f^2\left[\frac{\Delta u}{u^2}+\frac{\Delta v}{v^2}\right]$✓
4. D.$f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$

Correct answer: C

The equation of stationary wave is: $y=2a\sin\left(\frac{2\pi nt}{\lambda}\right)\cos\left(\frac{2\pi x}{\lambda}\right)$. Which of the following is NOT correct:

1. A.The dimensions of $n/\lambda$ is [T]✓
2. B.The dimensions of $n$ is $[LT^{-1}]$
3. C.The dimensions of $x$ is [L]
4. D.The dimensions of $nt$ is [L]

Correct answer: A

A body travels 102.5 m in $n^{th}$ second and 115.0 m in $(n+2)^{th}$ second. The acceleration is:

1. A.6.25 m/s²✓
2. B.12.5 m/s²
3. C.9 m/s²
4. D.5 m/s²

Correct answer: A

The co-ordinates of a particle moving in $x-y$ plane are given by: $x=2+4t$, $y=3t+8t^2$. The motion of the particle is:

1. A.uniformly accelerated having motion along a parabolic path.✓
2. B.uniform motion along a straight line.
3. C.uniformly accelerated having motion along a straight line.
4. D.non-uniformly accelerated.

Correct answer: A

If a rubber ball falls from a height $h$ and rebounds upto the height of $h/2$. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are:

1. A.50%, $\sqrt{2gh}$✓
2. B.50%, $\sqrt{gh}$
3. C.40%, $\sqrt{2gh}$
4. D.50%, $\sqrt{\frac{gh}{2}}$

Correct answer: A

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is:

1. A.$\frac{GmM\pi^2}{L^2}$
2. B.$\frac{GMm\pi}{2L^2}$
3. C.0
4. D.$\frac{2GmM\pi}{L^2}$✓

Correct answer: D

Given below are two statements: Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $P_1-P_2=\rho g(h_2-h_1)$. Statement II : In ventury tube shown $2gh=v_1^2-v_2^2$. In the light of the above statements, choose the most appropriate answer from the options given below.

1. A.Both Statement I and Statement II are correct.
2. B.Statement I is correct but Statement II is incorrect.✓
3. C.Statement I is incorrect but Statement II is correct.
4. D.Both Statement I and Statement II are incorrect.

Correct answer: B

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are 8Ω and 10Ω respectively. After inserting in a hot bath of temperature 400°C, the resistance of platinum wire is:

1. A.10Ω
2. B.8Ω
3. C.16Ω✓
4. D.2Ω

Correct answer: C

On celcius scale the temperature of body increases by 40°C. The increase in temperature on Fahrenheit scale is:

1. A.68°F
2. B.75°F
3. C.72°F✓
4. D.70°F

Correct answer: C

P-T diagram of an ideal gas having three different densities $\rho_1,\rho_2,\rho_3$ (in three different cases) is shown in the figure. Which of the following is correct:

1. A.$\rho_1>\rho_2$✓
2. B.$\rho_2<\rho_3$
3. C.$\rho_1=\rho_2=\rho_3$
4. D.$\rho_1<\rho_2$

Correct answer: A

An infinitely long positively charged straight thread has a linear charge density λ C/m. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is:

1. A.KE increases linearly with r
2. B.KE decreases (hyperbola-like) with r
3. C.KE constant (horizontal line)✓
4. D.KE increases non-linearly with r

Correct answer: C

To measure the internal resistance of a battery, potentiometer is used. For R = 10Ω, the balance point is observed at $l$ = 500 cm and for R = 1Ω the balance point is observed at $l$ = 400 cm. The internal resistance of the battery is approximately:

1. A.0.2Ω
2. B.0.3Ω✓
3. C.0.4Ω
4. D.0.1Ω

Correct answer: B

An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then:

1. A.the electron will continue to move with uniform velocity along the axis of the solenoid.✓
2. B.the electron will be accelerated along the axis.
3. C.the electron path will be circular about the axis.
4. D.the electron will experience a force at 45° to the axis and execute a helical path.

Correct answer: A

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to: A. pure inductor. B. pure capacitor. C. pure resistor. D. combination of an inductor and capacitor. Choose the correct answer from the options given below:

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

Correct answer: D

The electric field in an electromagnetic wave is given by $\vec E=\hat i40\cos\omega(t-z/c)$ N C$^{-1}$. The magnetic field induction of this wave is (in SI unit):

1. A.$\vec B=\hat k\frac{40}{c}\cos\omega(t-z/c)$
2. B.$\vec B=\hat j40\cos\omega(t-z/c)$
3. C.$\vec B=\hat i\frac{40}{c}\cos\omega(t-z/c)$
4. D.$\vec B=\hat j\frac{40}{c}\cos\omega(t-z/c)$✓

Correct answer: D

An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25D. Focal length of each of the convex lens is:

1. A.20 cm✓
2. B.50 cm
3. C.500 cm
4. D.25 cm

Correct answer: A

Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light $(I_1<I_2)$ of same wavelengths:

1. A.Curves for $I_1$ and $I_2$ with different cut-off voltages
2. B.Both curves start at the same stopping potential $-V_0$; $I_2$ saturates higher than $I_1$✓
3. C.Curves with different stopping potentials, $I_2$ higher
4. D.Both from $-V_0$ with $I_1$ and $I_2$ crossing

Correct answer: B

Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left({}^1_0n\right)$ and uranium isotope $\left({}^{235}_{92}U\right)$ is correct:

1. A.${}^{144}_{56}\text{Ba}+{}^{89}_{36}\text{Kr}+4\,{}^1_0n$
2. B.${}^{144}_{56}\text{Ba}+{}^{89}_{36}\text{Kr}+3\,{}^1_0n$✓
3. C.${}^{140}_{56}\text{Xe}+{}^{94}_{38}\text{Sr}+3\,{}^1_0n$
4. D.${}^{153}_{51}\text{Sb}+{}^{99}_{41}\text{Nb}+3\,{}^1_0n$

Correct answer: B

The value of net resistance of the network as shown in the given figure is:

1. A.6Ω✓
2. B.(5/2)Ω
3. C.(15/4)Ω
4. D.(30/11)Ω

Correct answer: A

Two forces $\vec F_1$ and $\vec F_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec F_1$ and $\vec F_2$ is $\cos^{-1}\left(\frac1n\right)$. The value of $|n|$ is ______.

Correct answer: 6

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h_1$ and $h_2$, respectively, above the initial level. The ratio $h_1:h_2$ is $\frac{n}{10}$. The value of $n$ is ______.

Correct answer: 7

An elastic spring under tension of 3 N has a length $a$. Its length is $b$ under tension 2 N. For its length $(3a-2b)$, the value of tension will be ______ N.

Correct answer: 5

A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne/cm then the new radius is ______ cm. Take $\left(\pi=\frac{22}{7}\right)$

Correct answer: 7

An infinite plane sheet of charge having uniform surface charge density $+\sigma_s$ C/m² is placed on $x-y$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e$ C/m is placed at $z=4$ m plane and parallel to y-axis. If the magnitude values $|\sigma_s|=2|\lambda_e|$ then at point (0, 0, 2), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi\sqrt n:1$. The value of $n$ is ______.

Correct answer: 16

Twelve wires each having resistance 2Ω are joined to form a cube. A battery of 6 V emf is joined across point a and c. The voltage difference between e and f is ______ V.

Correct answer: 1

The magnetic field existing in a region is given by $\vec B=0.2(1+2x)\hat k$ T. A square loop of edge 50 cm carrying 0.5 A current is placed in $x-y$ plane with its edges parallel to the $x-y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is ______ mN.

Correct answer: 50

A alternating current at any instant is given by $i=[6+\sqrt{56}\sin(100\pi t+\pi/3)]$ A. The rms value of the current is ______ A.

Correct answer: 8

Two wavelengths $\lambda_1$ and $\lambda_2$ are used in Young's double slit experiment. $\lambda_1=450$ nm and $\lambda_2=650$ nm. The minimum order of fringe produced by $\lambda_2$ which overlaps with the fringe produced by $\lambda_1$ is $n$. The value of $n$ is ______.

Correct answer: 9

A hydrogen atom changes its state from $n=3$ to $n=2$. Due to recoil, the percentage change in the wave length of emitted light is approximately $1\times10^{-n}$. The value of $n$ is ______. [Given Rhc = 13.6 eV, hc = 1242 eV nm, h = $6.6\times10^{-34}$ J s mass of the hydrogen atom = $1.6\times10^{-27}$ kg]

Correct answer: 7

Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is ______. B, C, N, S, O, F, P, Al, Si

1. A.7
2. B.3✓
3. C.5
4. D.6

Correct answer: B

The correct order of first ionization enthalpy values of the following elements is: (A) O (B) N (C) Be (D) F (E) B. Choose the correct answer from the options given below:

1. A.E < C < A < B < D✓
2. B.C < E < A < B < D
3. C.B < D < C < E < A
4. D.A < B < D < C < E

Correct answer: A

Which one of the following molecules has maximum dipole moment?

1. A.$\text{NF}_3$
2. B.$\text{CH}_4$
3. C.$\text{PF}_5$
4. D.$\text{NH}_3$✓

Correct answer: D

Number of molecules/ions from the following in which the central atom is involved in $sp^3$ hybridization is: $\text{NO}_3^-$, $\text{BCl}_3$, $\text{ClO}_2^-$, $\text{ClO}_3$

1. A.4
2. B.3
3. C.2✓
4. D.1

Correct answer: C

Match List I with List II: List-I (Mechanism steps): (A) resonance in aniline (lone pair on N delocalised into the ring); (B) cyclohexene + H$^+$ → secondary carbocation (π electrons move towards H$^+$); (C) cyclohexene + $^-$CN → carbanion (π electrons move away from the attacking nucleophile); (D) resonance in nitrobenzene (ring π electrons delocalised onto the NO$_2$ group) List-II (Effect): (I) −E effect; (II) −R effect; (III) +E effect; (IV) +R effect Choose the correct answer from the options given below:

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

Correct answer: A

Which of the following nitrogen containing compound does not give Lassaigne's test?

1. A.Urea
2. B.Phenyl hydrazine
3. C.Glycene
4. D.Hydrazine✓

Correct answer: D

Which among the following is incorrect statement?

1. A.Electromeric effect dominates over inductive effect
2. B.The electromeric effect is, temporary effect
3. C.Hydrogen ion (H$^+$) shows negative electromeric effect✓
4. D.The organic compound shows electromeric effect in the presence of the reagent only.

Correct answer: C

(A) 1-bromo-4-(2-bromoethyl)benzene ($p$-$\text{Br-C}_6\text{H}_4\text{-CH}_2\text{CH}_2\text{Br}$) $\xrightarrow{\text{NaOH(alc)}}$ (B) $\xrightarrow{\text{HBr, Ether}}$ (C). Identify (B) and (C) and how are (A) and (C) related?

1. A.B = 4-(2-bromoethyl)phenol, C = 4-(2-hydroxyethyl)phenol; functional group isomers
2. B.B = 4-bromostyrene, C = 1-bromo-4-(2-bromoethyl)benzene; chain isomers
3. C.B = 4-bromostyrene, C = 1-bromo-4-(1-bromoethyl)benzene; position isomers✓
4. D.B = 4-(2-bromoethyl)phenol, C = 4-(1-bromoethyl)phenol; derivative

Correct answer: C

The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is: (Given : Molar Mass Na : 23 and Cl : 35.5 g mol$^{-1}$)

1. A.2
2. B.20
3. C.4
4. D.0.2✓

Correct answer: D

What pressure (bar) of $\text{H}_2$ would be required to make emf of hydrogen electrode zero in pure water at 25°C?

1. A.$10^{-7}$
2. B.0.5
3. C.1
4. D.$10^{-14}$✓

Correct answer: D

One of the commonly used electrode is calomel electrode. Under which of the following categories, calomel electrode comes?

1. A.Oxidation - Reduction electrodes
2. B.Metal ion - Metal electrodes
3. C.Gas - Ion electrodes
4. D.Metal - Insoluble Salt - Anion electrodes✓

Correct answer: D

What will be the decreasing order of basic strength of the following conjugate bases? $^-$OH, R$\overline{\text{O}}$, CH$_3$CO$\overline{\text{O}}$, C$\overline{\text{l}}$

1. A.R$\overline{\text{O}}$ > $^-$OH > CH$_3$CO$\overline{\text{O}}$ > C$\overline{\text{l}}$✓
2. B.C$\overline{\text{l}}$ > R$\overline{\text{O}}$ > $^-$OH > CH$_3$CO$\overline{\text{O}}$
3. C.$^-$OH > R$\overline{\text{O}}$ > CH$_3$CO$\overline{\text{O}}$ > C$\overline{\text{l}}$
4. D.C$\overline{\text{l}}$ > $^-$OH > R$\overline{\text{O}}$ > CH$_3$CO$\overline{\text{O}}$

Correct answer: A

The element which shows only one oxidation state other than its elemental form is:

1. A.Cobalt
2. B.Titanium
3. C.Nickel
4. D.Scandium✓

Correct answer: D

Number of complexes from the following with even number of unpaired "d" electrons is $[\text{V(H}_2\text{O)}_6]^{3+}$, $[\text{Cr(H}_2\text{O)}_6]^{2+}$, $[\text{Fe(H}_2\text{O)}_6]^{3+}$, $[\text{Ni(H}_2\text{O)}_6]^{3+}$, $[\text{Cu(H}_2\text{O)}_6]^{2+}$ [Given atomic numbers : V = 23, Cr = 24, Fe = 26, Ni = 28 Cu = 29]

1. A.2✓
2. B.1
3. C.4
4. D.5

Correct answer: A

The correct sequence of ligands in the order of decreasing field strength is:

1. A.NCS$^-$ > EDTA$^{4-}$ > CN$^-$ > CO
2. B.CO > H$_2$O > F$^-$ > S$^{2-}$✓
3. C.S$^{2-}$ > $^-$OH > EDTA$^{4-}$ > CO
4. D.$^-$OH > F$^-$ > NH$_3$ > CN$^-$

Correct answer: B

Identify the correct set of reagents or reaction conditions 'X' and 'Y' in the following set of transformation: $\text{CH}_3\text{-CH}_2\text{-CH}_2\text{-Br}\xrightarrow{'X'}$ Product $\xrightarrow{'Y'}\text{CH}_3\text{-CH(Br)-CH}_3$

1. A.X = dil.aq. NaOH, 20°C, Y = Br$_2$/CHCl$_3$
2. B.X = conc.alc. NaOH, 80°C, Y = Br$_2$/CHCl$_3$
3. C.X = dil.aq. NaOH, 20°C, Y = HBr/acetic acid
4. D.X = conc.alc. NaOH, 80°C, Y = HBr/acetic acid✓

Correct answer: D

Given below are two statements: Statements I : Acidity of α-hydrogens of aldehydes and ketones is responsible for Aldol reaction. Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product. In the light of the above statements, choose the most appropriate answer from the options given below:

1. A.Statement I is correct but Statement II is incorrect✓
2. B.Both Statement I and Statement II are correct
3. C.Both Statement I and Statement II are incorrect
4. D.Statement I is incorrect but Statement II is correct

Correct answer: A

In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to:

1. A.increase concentration of Cl$^-$ ions
2. B.increase concentration of NH$_4^+$ ions
3. C.prevent interference by phosphate ions
4. D.decrease concentration of $^-$OH ions✓

Correct answer: D

Which of the following is the correct structure of L-Glucose?

1. A.Fischer projection (C2→C5): HO left, OH right, HO left, HO left✓
2. B.OH right, HO left, OH right, OH right... as drawn (D-glucose-like pattern)
3. C.HO left on all four stereocentres
4. D.HO, HO left; OH, OH right

Correct answer: A

The de-Broglie's wavelength of an electron in the $4^{th}$ orbit is ______ $\pi a_0$. ($a_0$ = Bohr's radius)

Correct answer: 8

Number of molecules/species from the following having one unpaired electron is ______. $\text{O}_2$, $\text{O}_2^{-1}$, NO, CN$^{-1}$, $\text{O}_2^{2-}$

Correct answer: 2

The enthalpy of formation of ethane ($\text{C}_2\text{H}_6$) from ethylene by addition of hydrogen where the bond-energies of C–H, C–C, C=C, H–H are 414 kJ, 347 kJ, 615 kJ and 435 kJ respectively is ______ kJ.

Correct answer: 125

Only 2 mL of $\text{KMnO}_4$ solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid (2M) in acidic medium. The molarity of $\text{KMnO}_4$ solution should be ______ M.

Correct answer: 8

The number of different chain isomers for $\text{C}_7\text{H}_{16}$ is ______.

Correct answer: 9

2.5 g of a non-volatile, non-electrolyte is dissolved in 100 g of water at 25°C. The solution showed a boiling point elevation by 2°C. Assuming the solute concentration is negligible with respect to the solvent concentration, the vapor pressure of the resulting aqueous solution is ______ mm of Hg (nearest integer) [Given : Molal boiling point elevation constant of water $(K_b)=0.52$ K kg mol$^{-1}$, 1 atm pressure = 760 mm of Hg, molar mass of water = 18 g mol$^{-1}$]

Correct answer: 707

Consider the following transformation involving first order elementary reaction in each step at constant temperature as shown below. $A+B\underset{\text{Step 3}}{\overset{\text{Step 1}}{\rightleftharpoons}}C\xrightarrow{\text{Step 2}}P$. Some details of the above reactions are listed below. | Step | Rate constant (s$^{-1}$) | Activation energy (kJ mol$^{-1}$) | | 1 | $k_1$ | 300 | | 2 | $k_2$ | 200 | | 3 | $k_3$ | $E_{a3}$ | If the overall rate constant of the above transformation (k) is given as $k=\frac{k_1k_2}{k_3}$ and the overall activation energy $(E_a)$ is 400 kJ mol$^{-1}$, then the value of $E_{a3}$ is ______ kJ mol$^{-1}$ (nearest integer)

Correct answer: 100

Consider the following reaction $\text{MnO}_2+\text{KOH}+\text{O}_2\to A+\text{H}_2\text{O}$. Product 'A' in neutral or acidic medium disproportionate to give products 'B' and 'C' along with water. The sum of spin-only magnetic moment values of B and C is ______ BM. (nearest integer) (Given atomic number of Mn is 25)

Correct answer: 4

The number of the correct reaction(s) among the following is ______. (A) Benzene + benzoyl chloride $\xrightarrow{\text{Anhyd. AlCl}_3}$ diphenylmethane (B) Benzoyl chloride $\xrightarrow{\text{H}_2,\ \text{Pd-BaSO}_4}$ benzoic acid (C) Benzene $\xrightarrow{\text{CO, HCl, Anhyd. AlCl}_3/\text{CuCl}}$ benzaldehyde (D) Benzamide $\xrightarrow{\text{H}_3\text{O}^+,\ \Delta}$ aniline

Correct answer: 1

X g of ethylamine is subjected to reaction with NaNO$_2$/HCl followed by water; evolved dinitrogen gas which occupied 2.24 L volume at STP. X is ______ $\times10^{-1}$ g.

Correct answer: 45

If 2 and 6 are the roots of the equation $ax^2+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is:

1. A.$2x^2+11x+12=0$
2. B.$x^2+8x+12=0$✓
3. C.$4x^2+14x+12=0$
4. D.$x^2+10x+16=0$

Correct answer: B

Let α and β be the sum and the product of all the non-zero solutions of the equation $(\bar z)^2+|z|=0$, $z\in\mathbb{C}$. Then $4(\alpha^2+\beta^2)$ is equal to:

1. A.6
2. B.8
3. C.2
4. D.4✓

Correct answer: D

There are 5 points $P_1,P_2,P_3,P_4,P_5$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $P_6,P_7,\ldots,P_{11}$ on the side BC and 7 points $P_{12},P_{13},\ldots,P_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $P_1,P_2,\ldots,P_{18}$ as vertices, is:

1. A.776
2. B.796
3. C.751✓
4. D.771

Correct answer: C

Let the first three terms 2, p and q, with $q\ne2$, of a G.P. be respectively the $7^{th}$, $8^{th}$ and $13^{th}$ terms of an A.P. If the $5^{th}$ term of the G.P. is the $n^{th}$ term of the A.P., then n is equal to:

1. A.163✓
2. B.151
3. C.177
4. D.169

Correct answer: A

The sum of all rational terms in the expansion of $\left(2^{\frac15}+5^{\frac13}\right)^{15}$ is equal to:

1. A.3133✓
2. B.931
3. C.6131
4. D.633

Correct answer: A

The vertices of a triangle are A(−1, 3), B(−2, 2) and C(3, −1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

1. A.$x+y+(2-\sqrt2)=0$
2. B.$-x+y-(2-\sqrt2)=0$
3. C.$x+y-(2-\sqrt2)=0$✓
4. D.$x-y-(2+\sqrt2)=0$

Correct answer: C

A square is inscribed in the circle $x^2+y^2-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_i,y_i)$ are the vertices of the square, then $\Sigma(x_i^2+y_i^2)$ is equal to:

1. A.148
2. B.152✓
3. C.160
4. D.156

Correct answer: B

Let $\alpha,\beta\in\mathbb{R}$. Let the mean and the variance of 6 observations −3, 4, 7, −6, α, β be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

1. A.$\frac{13}{3}$✓
2. B.$\frac{16}{3}$
3. C.$\frac{11}{3}$
4. D.$\frac{14}{3}$

Correct answer: A

Let $\alpha\in(0,\infty)$ and $A=\begin{bmatrix}1&2&\alpha\\1&0&1\\0&1&2\end{bmatrix}$. If $\det\left(\text{adj}(2A-A^T)\cdot\text{adj}(A-2A^T)\right)=2^8$, then $(\det(A))^2$ is equal to:

1. A.36
2. B.16✓
3. C.1
4. D.49

Correct answer: B

If the system of equations $x+(\sqrt2\sin\alpha)y+(\sqrt2\cos\alpha)z=0$, $x+(\cos\alpha)y+(\sin\alpha)z=0$, $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha\in\left(0,\frac\pi2\right)$ is equal to:

1. A.$\frac{11\pi}{24}$
2. B.$\frac{5\pi}{24}$✓
3. C.$\frac{7\pi}{24}$
4. D.$\frac{3\pi}{4}$

Correct answer: B

If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)+\log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha,\beta]$, then $3\alpha+10\beta$ is equal to:

1. A.100
2. B.95
3. C.97✓
4. D.98

Correct answer: C

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2x^2-3x+8}{2x^2+3x+8}$ be $\frac mn$, where gcd(m, n) = 1. Then m + n is equal to:

1. A.195
2. B.201✓
3. C.217
4. D.182

Correct answer: B

Let $f:\mathbb{R}\to\mathbb{R}$ be a function given by $f(x)=\begin{cases}\frac{1-\cos2x}{x^2}, & x<0\\ \alpha, & x=0\\ \frac{\beta\sqrt{1-\cos x}}{x}, & x>0\end{cases}$, where $\alpha,\beta\in\mathbb{R}$. If f is continuous at x = 0, then $\alpha^2+\beta^2$ is equal to:

1. A.3
2. B.12✓
3. C.48
4. D.6

Correct answer: B

Let $f(x)=x^5+2e^{x/4}$ for all $x\in\mathbb{R}$. Consider a function g(x) such that $(g\circ f)(x)=x$ for all $x\in\mathbb{R}$. Then the value of $8g'(2)$ is:

1. A.2
2. B.8
3. C.4
4. D.16✓

Correct answer: D

Let $f(x)=\begin{cases}-2, & -2\le x\le0\\ x-2, & 0<x\le2\end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\int_{-2}^{2}h(x)dx$ is equal to:

1. A.1
2. B.6
3. C.4
4. D.2✓

Correct answer: D

If the solution $y=y(x)$ of the differential equation $(x^4+2x^3+3x^2+2x+2)dy-(2x^2+2x+3)dx=0$ satisfies $y(-1)=-\frac\pi4$, then $y(0)$ is equal to:

1. A.$\frac\pi2$
2. B.$-\frac\pi2$
3. C.0
4. D.$\frac\pi4$✓

Correct answer: D

Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be $(\alpha,\beta,\gamma)$. Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

1. A.165
2. B.160
3. C.155✓
4. D.150

Correct answer: C

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:

1. A.$\frac5{18}$✓
2. B.$\frac5{16}$
3. C.$\frac4{17}$
4. D.$\frac7{18}$

Correct answer: A

Let $a=1+\frac{{}^2C_2}{3!}+\frac{{}^3C_2}{4!}+\frac{{}^4C_2}{5!}+\ldots$, $b=1+\frac{{}^1C_0+{}^1C_1}{1!}+\frac{{}^2C_0+{}^2C_1+{}^2C_2}{2!}+\frac{{}^3C_0+{}^3C_1+{}^3C_2+{}^3C_3}{3!}+\ldots$. Then $\frac{2b}{a^2}$ is equal to ______.

Correct answer: 8

Let the length of the focal chord PQ of the parabola $y^2=12x$ be 15 units. If the distance of PQ from the origin is p, then $10p^2$ is equal to ______.

Correct answer: 72

If $\lim_{x\to1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt5}{n(2n)^{2/3}}$, where gcd(m, n) = 1, then 8m + 12n is equal to ______.