JEE Main 2021 — 26 Aug Shift 1 (Previous Year Paper)

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation the reading on the main scale is 5 mm and the 20$^{th}$ division of the circular scale coincides with reference line. Calculate the true reading.

1. A.5.20 mm
2. B.5.25 mm
3. C.5.15 mm✓
4. D.5.00 mm

Correct answer: C

The rms speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are $V_H$, $V_O$ and $V_C$ respectively then:

1. A.$V_H=V_O=V_C$
2. B.$V_H=V_O>V_C$
3. C.$V_H>V_O>V_C$✓
4. D.$V_C>V_O>V_H$

Correct answer: C

Identify the logic operation carried out by the given circuit:

1. A.AND
2. B.OR
3. C.NAND
4. D.NOR✓

Correct answer: D

An electric appliance supplies 6000 J/min heat to the system. If the system delivers a power of 90 W. How long it would take to increase the internal energy by $2.5\times10^3$ J?

1. A.$2.5\times10^1$ s
2. B.$2.5\times10^2$ s✓
3. C.$2.4\times10^3$ s
4. D.$4.1\times10^1$ s

Correct answer: B

A particular hydrogen like ion emits radiation of frequency $2.92\times10^{15}$ Hz when it makes transition from $n=3$ to $n=1$. The frequency in Hz of radiation emitted in transition from $n=2$ to $n=1$ will be:

1. A.$4.38\times10^{15}$
2. B.$6.57\times10^{15}$
3. C.$0.44\times10^{15}$
4. D.$2.46\times10^{15}$✓

Correct answer: D

Two narrow bores of diameter 5.0 mm and 8.0 mm are joined together to form a U-shaped tube open at both ends. If this U-tube contains water, what is the difference in the level of two limbs of the tube. [Take surface tension of water $T=7.3\times10^{-2}$ Nm$^{-1}$, angle of contact = 0, $g=10$ ms$^{-2}$ and density of water $=1.0\times10^3$ kg m$^{-3}$]

1. A.5.34 mm
2. B.3.62 mm
3. C.4.97 mm
4. D.2.19 mm✓

Correct answer: D

In a photoelectric experiment ultraviolet light of wavelength 280 nm is used with lithium cathode having work function $\phi=2.5$ eV. If the wavelength of incident light is switched to 400 nm, find out the change in the stopping potential. ($h=6.63\times10^{-34}$ Js, $c=3\times10^8$ ms$^{-1}$)

1. A.1.3 V✓
2. B.1.9 V
3. C.0.6 V
4. D.1.1 V

Correct answer: A

Car B overtakes another car A at a relative speed of 40 ms$^{-1}$. How fast will the image of car B appear to move in the mirror of focal length 10 cm fitted in car A, when the car B is 1.9 m away from the car A?

1. A.0.1 ms$^{-1}$✓
2. B.0.2 ms$^{-1}$
3. C.40 ms$^{-1}$
4. D.4 ms$^{-1}$

Correct answer: A

An inductor coil stores 64 J of magnetic field energy and dissipates energy at the rate of 640 W when a current of 8 A is passed through it. If this coil is joined across an ideal battery, find the time constant of the circuit in seconds:

1. A.0.2✓
2. B.0.4
3. C.0.8
4. D.0.125

Correct answer: A

A series LCR circuit driven by 300 V at a frequency of 50 Hz contains a resistance $R=3$ k$\Omega$, an inductor of inductive reactance $X_L=250\pi\ \Omega$ and an unknown capacitor. The value of capacitance to maximize the average power should be: (take $\pi^2=10$)

1. A.400 $\mu$F
2. B.40 $\mu$F
3. C.25 $\mu$F
4. D.4 $\mu$F✓

Correct answer: D

The fractional change in the magnetic field intensity at a distance 'r' from centre on the axis of current carrying coil of radius 'a' to the magnetic field intensity at the centre of the same coil is: (Take r < a).

1. A.$\frac32\frac{a^2}{r^2}$
2. B.$\frac23\frac{a^2}{r^2}$
3. C.$\frac23\frac{r^2}{a^2}$
4. D.$\frac32\frac{r^2}{a^2}$✓

Correct answer: D

In the given figure, the emf of the cell is 2.2 V and if internal resistance is 0.6 $\Omega$. Calculate the power dissipated in the whole circuit:

1. A.2.2 W✓
2. B.4.4 W
3. C.0.65 W
4. D.1.32 W

Correct answer: A

A solid metal sphere of radius R having charge q is enclosed inside the concentric spherical shell of inner radius a and outer radius b as shown in figure. The approximate variation electric field $\vec E$ as a function of distance r from centre O is given by:

1. A.E rises from r = 0 up to R, is zero between R and a... (graph 1: rising curve up to R, gap, then decaying curve beyond b)
2. B.E is zero up to b, then decays as 1/r^2 beyond b
3. C.E is zero for r < R, decays as 1/r^2 from R to a, is zero from a to b, and decays as 1/r^2 beyond b✓
4. D.E is zero for r < R and then decays continuously as 1/r^2 from R onwards

Correct answer: C

The material filled between the plates of a parallel plate capacitor has resistivity 200 $\Omega$m. The value of capacitance of the capacitor is 2 pF. If a potential difference of 40 V is applied across the plates of the capacitor, then the value of leakage current flowing out of the capacitor is: (given the value of relative permitivity of material is 50)

1. A.9.0 mA
2. B.0.9 mA✓
3. C.0.9 $\mu$A
4. D.9.0 $\mu$A

Correct answer: B

Inside a uniform spherical shell: (a) the gravitational field is zero. (b) the gravitational potential is zero. (c) the gravitational field is same everywhere. (d) the gravitation potential is same everywhere. (e) all of the above. Choose the most appropriate answer from the options given below:

1. A.(b), (c) and (d) only
2. B.(a), (b) and (c) only
3. C.(a), (c) and (d) only✓
4. D.(e) only

Correct answer: C

The initial mass of a rocket is 1000 kg. Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of 20 ms$^{-2}$. The gases come out at a relative speed of 500 ms$^{-1}$ with respect to the rocket: [Use g = 10 m/s$^2$]

1. A.60 kg s$^{-1}$✓
2. B.$6.0\times10^2$ kg s$^{-1}$
3. C.500 kg s$^{-1}$
4. D.10 kg s$^{-1}$

Correct answer: A

What equal length of an iron wire and a copper-nickel alloy wire, each of 2 mm diameter connected parallel to give an equivalent resistance of 3 $\Omega$? (Given resistivities of iron and copper-nickel alloy wire are 12 $\mu\Omega$ cm and 51 $\mu\Omega$ cm respectively)

1. A.110 m
2. B.97 m✓
3. C.90 m
4. D.82 m

Correct answer: B

The magnitude of vectors $\vec{OA}$, $\vec{OB}$ and $\vec{OC}$ in the given figure are equal. The direction of $\vec{OA}+\vec{OB}-\vec{OC}$ with $x$-axis will be:

1. A.$\tan^{-1}\frac{(1-\sqrt3-\sqrt2)}{(1+\sqrt3+\sqrt2)}$✓
2. B.$\tan^{-1}\frac{(\sqrt3-1+\sqrt2)}{(1+\sqrt3-\sqrt2)}$
3. C.$\tan^{-1}\frac{(1+\sqrt3-\sqrt2)}{(1-\sqrt3-\sqrt2)}$
4. D.$\tan^{-1}\frac{(\sqrt3-1+\sqrt2)}{(1-\sqrt3+\sqrt2)}$

Correct answer: A

Statement I: By doping silicon semiconductor with pentavalent material, the electrons density increases. Statement II: The n-type semiconductor has net negative charge. 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 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 E, L, M and G denote the quantities as energy, angular momentum, mass and constant of gravitation respectively, then the dimensions of P in the formula $P=EL^2M^{-5}G^{-2}$ are:

1. A.$[M^1L^1T^{-2}]$
2. B.$[M^{-1}L^{-1}T^2]$
3. C.$[M^0L^1T^0]$
4. D.$[M^0L^0T^0]$✓

Correct answer: D

A uniform chain of length 3 meter and mass 3 kg overhangs a smooth table with 2 meter laying on the table. If k is the kinetic energy of the chain in joule as it completely slips off the table, then the value of k is ________. (Take g = 10 m/s$^2$)

Correct answer: 40

Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, $\frac r2$ distance from the end A of the handle will be ________ $Mr^2$.

Correct answer: 52

A soap bubble of radius 3 cm is formed inside the another soap bubble of radius 6 cm. The radius of an equivalent soap bubble which has the same excess pressure as inside the smaller bubble with respect to the atmospheric pressure is ________ cm.

Correct answer: 2

Two travelling waves produces a standing wave represented by equation. $y=1.0\text{ mm}\cos(1.57\text{ cm}^{-1})x\sin(78.5\text{ s}^{-1})t$. The node closest to the origin in the region $x>0$ will be at $x$ = ________ cm.

Correct answer: 1

Two short magnetic dipoles $m_1$ and $m_2$ each having magnetic moment of 1 Am$^2$ are placed at point O and P respectively. The distance between OP is 1 meter. The torque experienced by the magnetic dipole $m_2$ due to the presence of $m_1$ is ________ $\times10^{-7}$ Nm.

Correct answer: 1

The electric field in a plane electromagnetic wave is given by $\vec E=200\cos\left[\left(\frac{0.5\times10^3}{m}\right)x-\left(1.5\times10^{11}\frac{rad}{s}\times t\right)\right]\frac{V}{m}\hat j$. If this wave falls normally on a perfectly reflecting surface having an area of 100 cm$^2$. If the radiation pressure exerted by the E.M. wave on the surface during a 10 minute exposure is $\frac{x}{10^9}\frac{N}{m^2}$. Find the value of $x$.

Correct answer: 354

White light is passed through a double slit and interference is observed on a screen 1.5 m away. The separation between the slits is 0.3 mm. The first violet and red fringes are formed 2.0 mm and 3.5 mm away from the central white fringes. The difference in wavelengths of red and voilet light is ________ nm.

Correct answer: 300

An amplitude modulated wave is represented by $C_m(t)=10(1+0.2\cos12560t)\sin(111\times10^4t)$ volts. The modulating frequency in kHz will be ________.

Correct answer: 2

Two spherical balls having equal masses with radius of 5 cm each are thrown upwards along the same vertical direction at an interval of 3 s with the same initial velocity of 35 m/s, then these balls collide at a height of ________ m. (take g = 10 m/s$^2$)

Correct answer: 50

A source and a detector move away from each other in absence of wind with a speed of 20 m/s with respect to the ground. If the detector detects a frequency of 1800 Hz of the sound coming from the source, then the original frequency of source considering speed of sound in air 340 m/s will be ________ Hz.

Correct answer: 2025

Given below are two statements: Statement I: Frenkel defects are vacancy as well as interstitial defects. Statement II: Frenkel defect leads to colour in ionic solids due to presence of F-centres. Choose the most appropriate answer for the statements 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

Given below are two statements: Statement I: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in positive charges on the nucleus as there is no strong hold on the electron by the nucleus. Statement II: According to Bohr's model of an atom, qualitatively the magnitude of velocity of electron increases with decrease in principal quantum number. 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 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: D

Given below are two statements: Statement I: The limiting molar conductivity of KCl (strong electrolyte) is higher compared to that of CH$_3$COOH (weak electrolyte). Statement II: Molar conductivity decreases with decrease in concentration of electrolyte. 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 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: B

Which one of the following is correct for the adsorption of a gas at a given temperature on a solid surface?

1. A.$\Delta H>0, \Delta S>0$
2. B.$\Delta H<0, \Delta S>0$
3. C.$\Delta H>0, \Delta S<0$
4. D.$\Delta H<0, \Delta S<0$✓

Correct answer: D

Given below are two statements. Statement I: The choice of reducing agents for metals extraction can be made by using Ellingham diagram, a plot of $\Delta G$ vs temperature. Statement II: The value of $\Delta S$ increases from left to right in Ellingham diagram. 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 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

Which one of the following methods is most suitable for preparing deionized water?

1. A.Clark's method
2. B.Synthetic resin method✓
3. C.Calgon's method
4. D.Permutit method

Correct answer: B

What are the products formed in sequence when excess of CO$_2$ is passed in slaked lime?

1. A.CaCO$_3$, Ca(HCO$_3$)$_2$✓
2. B.Ca(HCO$_3$)$_2$, CaCO$_3$
3. C.CaO, CaCO$_3$
4. D.CaO, Ca(HCO$_3$)$_2$

Correct answer: A

The incorrect statement is:

1. A.Cl$_2$ is more reactive than ClF.✓
2. B.F$_2$ is more reactive than ClF.
3. C.On hydrolysis ClF forms HOCl and HF.
4. D.F$_2$ is a stronger oxidizing agent than Cl$_2$ in aqueous solution.

Correct answer: A

Which one of the following when dissolved in water gives coloured solution in nitrogen atmosphere?

1. A.ZnCl$_2$
2. B.CuCl$_2$✓
3. C.Cu$_2$Cl$_2$
4. D.AgCl

Correct answer: B

The conversion of hydroxyapatite occurs due to presence of F$^-$ ions in water. The correct formula of hydroxyapatite is:

1. A.[3Ca$_3$(PO$_4$)$_2$·CaF$_2$]
2. B.[Ca$_3$(PO$_4$)$_2$·CaF$_2$]
3. C.[3Ca$_3$(PO$_4$)$_2$·Ca(OH)$_2$]✓
4. D.[3Ca(OH)$_2$·CaF$_2$]

Correct answer: C

Which one of the following complexes is violet in colour?

1. A.[Fe(SCN)$_6$]$^{4-}$
2. B.[Fe(CN)$_5$NOS]$^{4-}$✓
3. C.Fe$_4$[Fe(CN)$_6$]$_3$·H$_2$O
4. D.[Fe(CN)$_6$]$^{4-}$

Correct answer: B

Excess of isobutane on reaction with Br$_2$ in presence of light at 125°C gives which one of the following, as the major product?

1. A.$\mathrm{(CH_3)_2CH-CH_2Br}$
2. B.$\mathrm{(CH_3)_3C-Br}$✓
3. C.$\mathrm{(CH_3)_2C(Br)-CH_2Br}$
4. D.$\mathrm{CH_3-CH(CH_2Br)-CH_2Br}$

Correct answer: B

The major product formed in the following reaction is: isoprene (2-methylbuta-1,3-diene) + HBr (excess) $\rightarrow$ Major Product

1. A.1,4-dibromo-2-methylbutane (Br-CH2-CH(CH3)-CH2-CH2-Br)
2. B.1-bromo-3-methylbut-2-ene ((CH3)2C=CH-CH2Br)✓
3. C.2,3-dibromo-2-methylbutane ((CH3)2CBr-CHBr-CH3)
4. D.3-bromo-3-methylbut-1-ene ((CH3)2CBr-CH=CH2)

Correct answer: B

Among the following compounds I–IV, which one forms a yellow precipitate on reacting sequentially with (i) NaOH (ii) dil. HNO$_3$ (iii) AgNO$_3$? I: 4-chlorotoluene; II: 3-chlorotoluene; III: 2-bromotoluene; IV: 2-chlorobenzyl iodide (Cl on ring, CH$_2$I side chain)

1. A.I
2. B.II
3. C.III
4. D.IV✓

Correct answer: D

The major products formed in the following reaction sequence A and B are: acetophenone $\xrightarrow{\text{Br}_2\text{, KOH}}$ A + B

1. A.A = C$_6$H$_5$COOK, B = CHBr$_3$✓
2. B.A = 3,5-dibromoacetophenone, B = 3,5-dihydroxyacetophenone
3. C.A = 2-bromo-1-(3-bromophenyl)ethanone, B = 1-(3-bromophenyl)-2-hydroxyethanone
4. D.A = C$_6$H$_5$COCBr$_3$, B = C$_6$H$_5$CHO

Correct answer: A

The correct options for the products A and B of the following reactions are: phenol $\xrightarrow{\text{Br}_2\text{ (Excess), H}_2\text{O}}$ A; phenol $\xrightarrow{\text{Br}_2\text{, CS}_2\text{, }<5^\circ\text{C}}$ B

1. A.A = 4-bromophenol, B = 4-bromophenol
2. B.A = 2,3,4-tribromophenol (as drawn), B = 2,6-dibromophenol
3. C.A = 2,4,6-tribromophenol, B = 4-bromophenol✓
4. D.A = 2,4,6-tribromophenol, B = 2,4-dibromophenol

Correct answer: C

The major product formed in the following reaction is: tryptophan $\xrightarrow{\text{SOCl}_2\text{, CH}_3\text{OH}}$ Major Product

1. A.tryptophanyl chloride (–COCl, free NH2)
2. B.tryptophan methyl ester hydrochloride (–CO2CH3, –NH2·HCl)✓
3. C.methyl ester with both the alpha-amino group and indole N-H as hydrochlorides
4. D.5-chlorotryptophan methyl ester (Cl on the benzene ring)

Correct answer: B

The correct sequential addition of reagents in the preparation of 3-nitrobenzoic acid from benzene is:

1. A.HNO$_3$/H$_2$SO$_4$, Br$_2$/AlBr$_3$, Mg/ether, CO$_2$, H$_3$O$^+$✓
2. B.Br$_2$/AlBr$_3$, NaCN, H$_3$O$^+$, HNO$_3$/H$_2$SO$_4$
3. C.Br$_2$/AlBr$_3$, HNO$_3$/H$_2$SO$_4$, NaCN, H$_3$O$^+$
4. D.Br$_2$/AlBr$_3$, HNO$_3$/H$_2$SO$_4$, Mg/ether, CO$_2$, H$_3$O$^+$

Correct answer: A

The polymer formed on heating Novolac with formaldehyde is:

1. A.Melamine
2. B.Bakelite✓
3. C.Polyester
4. D.Nylon 6,6

Correct answer: B

Given below are two statements: Statement I: In the titration between strong acid and weak base methyl orange is suitable as an indicator. Statement II: For titration of acetic acid with NaOH phenolphthalein is not a suitable indicator. 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 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

An aqueous KCl solution of density 1.20 g mL$^{-1}$ has a molality of 3.30 mol kg$^{-1}$. The molarity of the solution in mol L$^{-1}$ is ________. (Nearest integer) [Molar mass of KCl = 74.5]

Correct answer: 3

AB$_3$ is an interhalogen T-shaped molecule. The number of lone pairs of electrons on A is ________. (Integer answer)

Correct answer: 2

The Born-Haber cycle for KCl is evaluated with the following data: $\Delta_fH^\ominus$ for KCl = $-436.7$ kJ mol$^{-1}$; $\Delta_{sub}H^\ominus$ for K = 89.2 kJ mol$^{-1}$; $\Delta_{ionization}H^\ominus$ for K = 419.0 kJ mol$^{-1}$; $\Delta_{electron\ gain}H^\ominus$ for Cl$_{(g)}$ = $-348.6$ kJ mol$^{-1}$; $\Delta_{bond}H^\ominus$ for Cl$_2$ = 243.0 kJ mol$^{-1}$. The magnitude of lattice enthalpy of KCl in kJ mol$^{-1}$ is ________. (Nearest integer)

Correct answer: 718

Of the following four aqueous solutions, total number of those solutions whose freezing point is lower than that of 0.10 M C$_2$H$_5$OH is ________. (Integer answer) (i) 0.10 M Ba$_3$(PO$_4$)$_2$ (ii) 0.10 M Na$_2$SO$_4$ (iii) 0.10 M KCl (iv) 0.10 M Li$_3$PO$_4$

Correct answer: 4

The OH$^-$ concentration in a mixture of 5.0 mL of 0.0504 M NH$_4$Cl and 2 mL of 0.0210 M NH$_3$ solution is $x\times10^{-6}$ M. The value of $x$ is ________. (Nearest integer) [Given $K_w=1\times10^{-14}$ and $K_b=1.8\times10^{-5}$]

Correct answer: 3

The following data was obtained for chemical reaction given below at 975 K. $\mathrm{2NO_{(g)}+2H_{2(g)}\rightarrow N_{2(g)}+2H_2O_{(g)}}$. (A) [NO] $8\times10^{-5}$, [H$_2$] $8\times10^{-5}$, Rate $7\times10^{-9}$; (B) $24\times10^{-5}$, $8\times10^{-5}$, $2.1\times10^{-8}$; (C) $24\times10^{-5}$, $32\times10^{-5}$, $8.4\times10^{-8}$ (concentrations in mol L$^{-1}$, rate in mol L$^{-1}$s$^{-1}$). The order of the reaction with respect to NO is ________. [Integer answer]

Correct answer: 1

These are physical properties of an element (A) Sublimation enthalpy (B) Ionisation enthalpy (C) Hydration enthalpy (D) Electron gain enthalpy. The total number of above properties that affect the reduction potential is ________. (Integer answer)

Correct answer: 3

The number of 4f electrons in the ground state electronic configuration of Gd$^{2+}$ is ________. [Atomic number of Gd = 64]

Correct answer: 7

The ratio of number of water molecules in Mohr's salt and potash alum is ________ $\times10^{-1}$. (Integer answer)

Correct answer: 5

The total number of negative charge in the tetrapeptide, Gly-Glu-Asp-Tyr, at pH 12.5 will be ________. (Integer answer)

Correct answer: 4

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set:

1. A.$\{84, 86, 88, 90\}$
2. B.$\{80, 83, 86, 89\}$
3. C.$\{79, 81, 83, 85\}$✓
4. D.$\{84, 87, 90, 93\}$

Correct answer: C

The equation $\arg\left(\frac{z-1}{z+1}\right)=\frac\pi4$ represents a circle with:

1. A.centre at $(0,0)$ and radius $\sqrt2$
2. B.centre at $(0,1)$ and radius 2
3. C.centre at $(0,1)$ and radius $\sqrt2$✓
4. D.centre at $(0,-1)$ and radius $\sqrt2$

Correct answer: C

If $A=\begin{pmatrix}\frac1{\sqrt5}&\frac2{\sqrt5}\\ \frac{-2}{\sqrt5}&\frac1{\sqrt5}\end{pmatrix}$, $B=\begin{pmatrix}1&0\\i&1\end{pmatrix}$, $i=\sqrt{-1}$, and $Q=A^TBA$, then the inverse of the matrix $AQ^{2021}A^T$ is equal to:

1. A.$\begin{pmatrix}1&0\\-2021i&1\end{pmatrix}$✓
2. B.$\begin{pmatrix}\frac1{\sqrt5}&-2021\\2021&\frac1{\sqrt5}\end{pmatrix}$
3. C.$\begin{pmatrix}1&-2021i\\0&1\end{pmatrix}$
4. D.$\begin{pmatrix}1&0\\2021i&1\end{pmatrix}$

Correct answer: A

Let $\theta\in\left(0,\frac\pi2\right)$. If the system of linear equations $(1+\cos^2\theta)x+\sin^2\theta\,y+4\sin3\theta\,z=0$, $\cos^2\theta\,x+(1+\sin^2\theta)y+4\sin3\theta\,z=0$, $\cos^2\theta\,x+\sin^2\theta\,y+(1+4\sin3\theta)z=0$ has a non-trivial solution, then the value of $\theta$ is:

1. A.$\frac{\pi}{18}$
2. B.$\frac{7\pi}{18}$✓
3. C.$\frac{5\pi}{18}$
4. D.$\frac{4\pi}{9}$

Correct answer: B

The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is:

1. A.$1-\frac{2^{101}}{4^{101}-1}$✓
2. B.$1-\frac{2^{100}}{4^{100}-1}$
3. C.$1+\frac{2^{101}}{4^{101}-1}$
4. D.$1+\frac{2^{100}}{4^{101}-1}$

Correct answer: A

If ${}^{20}C_r$ is the co-efficient of $x^r$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20}r^2\,{}^{20}C_r$ is equal to:

1. A.$420\times2^{19}$
2. B.$380\times2^{19}$
3. C.$380\times2^{18}$
4. D.$420\times2^{18}$✓

Correct answer: D

If the sum of an infinite GP $a, ar, ar^2, ar^3,\dots$ is 15 and the sum of the squares of its each term is 150, then the sum of $ar^2, ar^4, ar^6,\dots$ is:

1. A.$\frac52$
2. B.$\frac12$✓
3. C.$\frac92$
4. D.$\frac{25}{2}$

Correct answer: B

Let $f(x)=\cos\left(2\tan^{-1}\sin\left(\cot^{-1}\sqrt{\frac{1-x}{x}}\right)\right)$, $0<x<1$. Then:

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

Correct answer: C

The value of $\int_{-1/\sqrt2}^{1/\sqrt2}\left(\left(\frac{x+1}{x-1}\right)^2+\left(\frac{x-1}{x+1}\right)^2-2\right)^{1/2}dx$ is:

1. A.$4\log_e(3+2\sqrt2)$
2. B.$\log_e4$
3. C.$\log_e16$✓
4. D.$2\log_e16$

Correct answer: C