JEE Main 2020 — 2 Sep Shift 1 (Previous Year Paper)

If speed V, area A and force F are chosen as fundamental units, then the dimension of Young's modulus will be:

1. A.$FA^2V^{-2}$
2. B.$FA^2V^{-3}$
3. C.$FA^{-1}V^0$✓
4. D.$FA^2V^{-1}$

Correct answer: C

Train A and train B are running on parallel tracks in the opposite directions with speeds of 36 km/hour and 72 km/hour, respectively. A person is walking in train A in the direction opposite to its motion with a speed of 1.8 km/hour. Speed (in ms$^{-1}$) of this person as observed from train B will be close to: (take the distance between the tracks as negligible)

1. A.30.5 ms$^{-1}$
2. B.28.5 ms$^{-1}$
3. C.31.5 ms$^{-1}$
4. D.29.5 ms$^{-1}$✓

Correct answer: D

A particle of mass m with an initial velocity $u\hat i$ collides perfectly elastically with a mass 3m at rest. It moves with a velocity $v\hat j$ after collision, then, v is given by:

1. A.$v=\frac{u}{\sqrt3}$
2. B.$v=\sqrt{\frac23}u$
3. C.$v=\frac{u}{\sqrt2}$✓
4. D.$v=\frac1{\sqrt6}u$

Correct answer: C

A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola $y=4Cx^2$ and rotating with angular speed $\omega$ (see figure). The value of $\omega$ is (neglect friction):

1. A.$2\sqrt{2gC}$✓
2. B.$\sqrt{\frac{2g}{C}}$
3. C.$\sqrt{\frac{2gC}{ab}}$
4. D.$2\sqrt{gC}$

Correct answer: A

A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:

1. A.$Mg\frac aR$
2. B.$Mg\sqrt{1-\left(\frac{R-a}{R}\right)^2}$✓
3. C.$Mg\sqrt{1-\frac{a^2}{R^2}}$
4. D.$Mg\sqrt{\left(\frac{R}{R-a}\right)^2-1}$

Correct answer: B

The mass density of a spherical galaxy varies as $\frac Kr$ over a large distance 'r' from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as:

1. A.$T^2\propto R^3$
2. B.$T^2\propto R$✓
3. C.$T\propto R$
4. D.$T^2\propto\frac1{R^3}$

Correct answer: B

Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass 'm' and has another weight of mass 2 m hung at a distance of 75 cm from A. The tension in the string at A is:

1. A.2 mg
2. B.0.5 mg
3. C.0.75 mg
4. D.1 mg✓

Correct answer: D

A cylindrical vessel containing a liquid is rotated about its axis so that the liquid rises at its sides as shown in the figure. The radius of vessel is 5 cm and the angular speed of rotation is $\omega$ rad s$^{-1}$. The difference in the height, h (in cm) of liquid at the centre of vessel and at the side will be:

1. A.$\frac{2\omega^2}{25g}$
2. B.$\frac{25\omega^2}{2g}$✓
3. C.$\frac{5\omega^2}{2g}$
4. D.$\frac{2\omega^2}{5g}$

Correct answer: B

A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of RT) of the mixture is:

1. A.11
2. B.13
3. C.15✓
4. D.20

Correct answer: C

Two identical strings X and Z made of same material have tension $T_X$ and $T_Z$ in them. If their fundamental frequencies are 450 Hz and 300 Hz, respectively, then the ratio $T_X/T_Z$ is:

1. A.1.5
2. B.2.25✓
3. C.1.25
4. D.0.44

Correct answer: B

A charged particle (mass m and charge q) moves along X axis with velocity $V_0$. When it passes through the origin it enters a region having uniform electric field $\vec E=-E\hat j$ which extends upto $x=d$. Equation of path of electron in the region $x>d$ is:

1. A.$y=\frac{qEd}{mV_0^2}x$
2. B.$y=\frac{qEd^2}{mV_0^2}x$
3. C.$y=\frac{qEd}{mV_0^2}(x-d)$
4. D.$y=\frac{qEd}{mV_0^2}\left(\frac d2-x\right)$✓

Correct answer: D

Consider four conducting materials copper, tungsten, mercury and aluminium with resistivity $\rho_C$, $\rho_T$, $\rho_M$ and $\rho_A$ respectively. Then:

1. A.$\rho_C>\rho_A>\rho_T$
2. B.$\rho_A>\rho_T>\rho_C$
3. C.$\rho_A>\rho_M>\rho_C$
4. D.$\rho_M>\rho_A>\rho_C$✓

Correct answer: D

A beam of protons with speed $4\times10^5$ ms$^{-1}$ enters a uniform magnetic field of 0.3 T at an angle of 60° to the magnetic field. The pitch of the resulting helical path of protons is close to: (Mass of the proton $=1.67\times10^{-27}$ kg, charge of the proton $=1.69\times10^{-19}$ C)

1. A.2 cm
2. B.5 cm
3. C.4 cm✓
4. D.12 cm

Correct answer: C

Magnetic materials used for making permanent magnets (P) and magnets in a transformer (T) have different properties of the following, which property best matches for the type of magnet required?

1. A.P: Large retentivity, large coercivity✓
2. B.P: Small retentivity, large coercivity
3. C.T: Large retentivity, large coercivity
4. D.T: Large retentivity, small coercivity

Correct answer: A

A plane electromagnetic wave, has frequency of $2.0\times10^{10}$ Hz and its energy density is $1.02\times10^{-8}$ J/m$^3$ in vacuum. The amplitude of the magnetic field of the wave is close to ($\frac1{4\pi\epsilon_0}=9\times10^9$ Nm$^2$/C$^2$ and speed of light $=3\times10^8$ ms$^{-1}$):

1. A.160 nT✓
2. B.180 nT
3. C.190 nT
4. D.150 nT

Correct answer: A

A spherical mirror is obtained as shown in the figure from a hollow glass sphere. If an object is positioned in front of the mirror, what will be the nature and magnification of the image of the object? (Figure drawn as schematic and not to scale)

1. A.Inverted, real and unmagnified
2. B.Erect, virtual and magnified
3. C.Inverted, real and magnified✓
4. D.Erect, virtual and unmagnified

Correct answer: C

Interference fringes are observed on a screen by illuminating two thin slits 1 mm apart with a light source ($\lambda=632.8$ nm). The distance between the screen and the slits is 100 cm. If a bright fringe is observed on a screen at a distance of 1.27 mm from the central bright fringe, then the path difference between the waves, which are reaching this point from the slits is close to:

1. A.1.27 $\mu$m✓
2. B.2.87 nm
3. C.2 nm
4. D.2.05 $\mu$m

Correct answer: A

In a reactor, 2 kg of $_{92}U^{235}$ fuel is fully used up in 30 days. The energy released per fission is 200 MeV. Given that the Avogadro number, $N=6.023\times10^{26}$ per kilo mole and 1 eV $=1.6\times10^{-19}$ J. The power output of the reactor is close to:

1. A.60 MW✓
2. B.125 MW
3. C.54 MW
4. D.35 MW

Correct answer: A

The least count of the main scale of a vernier callipers is 1 mm. Its vernier scale is divided into 10 divisions and coincide with 9 divisions of the main scale. When jaws are touching each other, the 7$^{th}$ division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between 3.1 cm and 3.2 cm and 4$^{th}$ VSD coincides with a main scale division. The length of the cylinder is: (VSD is vernier scale division)

1. A.3.2 cm
2. B.3.07 cm✓
3. C.3.21 cm
4. D.2.99 cm

Correct answer: B

A small block starts slipping down from a point B on an inclined plane AB, which is making an angle $\theta$ with the horizontal section BC is smooth and the remaining section CA is rough with a coefficient of friction $\mu$. It is found that the block comes to rest as it reaches the bottom (point A) of the inclined plane. If BC = 2AC, the coefficient of friction is given by $\mu=k\tan\theta$. The value of k is ________.

Correct answer: 3

An engine takes in 5 moles of air at 20°C and 1 atm, and compresses it adiabaticaly to 1/10$^{th}$ of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be X kJ. The value of X to the nearest integer is ________.

Correct answer: 46

A 5 $\mu$F capacitor is charged fully by a 220 V supply. It is then disconnected from the supply and is connected in series to another uncharged 2.5 $\mu$F capacitor. If the energy change during the charge redistribution is $\frac{X}{100}$ J then value of X to the nearest integer is ________.

Correct answer: 36

A circular coil of radius 10 cm is placed in a uniform magnetic field of $3.0\times10^{-5}$ T with its plane perpendicular to the field initially. It is rotated at constant angular speed about an axis along the diameter of coil and perpendicular to magnetic field so that it undergoes half of rotation in 0.2s. The maximum value of EMF induced (in $\mu$V) in the coil will be close to the integer ________.

Correct answer: 15

When radiation of wavelength $\lambda$ is used to illuminate a metallic surface, the stopping potential is V. When the same surface is illuminated with radiation of wavelength $3\lambda$, the stopping potential is $\frac V4$. If the threshold wavelength for the metallic surface is $n\lambda$ then value of n will be ________.

Correct answer: 9

For the following Assertion and Reason, the correct option is Assertion (A): When Cu (II) and sulphide ions are mixed, they react together extremely quickly to give a solid. Reason (R): The equilibrium constant of $\mathrm{Cu^{2+}(aq)+S^{2-}(aq)\rightleftharpoons CuS(s)}$ is high because the solubility product is low.

1. A.Both (A) and (R) are true and (R) is the explanation for (A).
2. B.Both (A) and (R) are true but (R) is not the explanation for (A).✓
3. C.(A) is false and (R) is true.
4. D.Both (A) and (R) are false.

Correct answer: B

Which of the following is used for the preparation of colloids?

1. A.Mond Process
2. B.Ostwald Process
3. C.Van Arkel Method
4. D.Bredig's Arc Method✓

Correct answer: D

If AB$_4$ molecule is a polar molecule, a possible geometry of AB$_4$ is:

1. A.Tetrahedral
2. B.Square planar
3. C.Rectangular planar
4. D.Square pyramidal✓

Correct answer: D

The figure that is not a direct manifestation of the quantum nature of atoms is:

1. A.Black-body radiation: intensity vs wavelength curves for T2 > T1
2. B.Photoelectric effect: kinetic energy of photoelectrons vs frequency for Rb, K, Na (parallel lines)
3. C.Internal energy of Ar increasing linearly with temperature (300-600 K)✓
4. D.Absorption spectrum (discrete lines with increasing wavelength)

Correct answer: C

Which one of the following graphs is not correct for ideal gas? I: d decreasing (hyperbola) with T; II: d increasing linearly with T; III: d increasing linearly with 1/T; IV: d increasing linearly with P. (d = Density, P = Pressure, T = Temperature)

1. A.I
2. B.II✓
3. C.III
4. D.IV

Correct answer: B

In general, the property (magnitudes only) that shows an opposite trend in comparison to other properties across a period is:

1. A.Electronegativity
2. B.Atomic radius✓
3. C.Ionization enthalpy
4. D.Electron gain enthalpy

Correct answer: B

While titrating dilute HCl solution with aqueous NaOH, which of the following will not be required?

1. A.Burette and porcelain tile
2. B.Pipette and distilled water
3. C.Bunsen burner and measuring cylinder✓
4. D.Clamp and phenolphthalein

Correct answer: C

The metal mainly used in devising photoelectric cells is:

1. A.Li
2. B.Na
3. C.Rb
4. D.Cs✓

Correct answer: D

On heating compound (A) gives a gas (B) which is a constituent of air. This gas when treated with H$_2$ in the presence of a catalyst gives another gas (C) which is basic in nature. (A) should not be:

1. A.NH$_4$NO$_2$
2. B.(NH$_4$)$_2$Cr$_2$O$_7$
3. C.NaN$_3$
4. D.Pb(NO$_3$)$_2$✓

Correct answer: D

For octahedral Mn(II) and tetrahedral Ni(II) complexes, consider the following statements: (I) both the complexes can be high spin. (II) Ni(II) complex can very rarely be low spin. (III) with strong field ligands, Mn(II) complexes can be low spin. (IV) aqueous solution of Mn(II) ions is yellow in color. The correct statements are:

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

Correct answer: B

Consider that a d$^6$ metal ion (M$^{2+}$) forms a complex with aqua ligands, and the spin only magnetic moment of the complex is 4.90 BM. The geometry and the crystal field stabilization energy of the complex is:

1. A.octahedral and $-2.4\Delta_0+2P$
2. B.tetrahedral and $-0.6\Delta_t$✓
3. C.octahedral and $-1.6\Delta_0$
4. D.tetrahedral and $-1.6\Delta_t+1P$

Correct answer: B

The statement that is not true about ozone is:

1. A.in the stratosphere, it forms a protective shield against UV radiation.
2. B.in the atmosphere, it is depleted by CFCs.
3. C.it is a toxic gas and its reaction with NO gives NO$_2$.
4. D.in the stratosphere, CFCs release chlorine free radicals (Cl·) which reacts with O$_3$ to give chlorine dioxide radicals.✓

Correct answer: D

The IUPAC name for the following compound is: $\mathrm{CH_3-CH(CHO)-CH=CH-CH(CH_3)-COOH}$

1. A.2,5-dimethyl-6-oxo-hex-3-enoic acid✓
2. B.6-formyl-2-methyl-hex-3-enoic acid
3. C.2,5-dimethyl-5-carboxy-hex-3-enal
4. D.2,5-dimethyl-6-carboxy-hex-3-enal

Correct answer: A

In Carius method of estimation of halogen, 0.172 g of an organic compound showed presence of 0.08 g of bromine. Which of these is the correct structure of the compound?

1. A.4-bromoaniline✓
2. B.2,4-dibromoaniline
3. C.H$_3$C–Br
4. D.H$_3$C–CH$_2$–Br

Correct answer: A

The major product in the following reaction is: 1-methyl-1-vinylcyclopentane $\xrightarrow{\text{H}_3\text{O}^+\text{, Heat}}$

1. A.1,2-dimethylcyclohexene✓
2. B.3,3-dimethylcyclohexene
3. C.1-(1-methylcyclopentyl)ethanol
4. D.1-isopropylcyclopentene

Correct answer: A

The increasing order of the following compounds towards HCN addition is: (i) 3-methoxybenzaldehyde (ii) 2-nitrobenzaldehyde (iii) 2-methoxybenzaldehyde (iv) 3-nitrobenzaldehyde

1. A.(iii) < (iv) < (i) < (ii)
2. B.(iii) < (i) < (iv) < (ii)✓
3. C.(i) < (iii) < (iv) < (ii)
4. D.(iii) < (iv) < (ii) < (i)

Correct answer: B

The major aromatic product C in the following reaction sequence will be: 2-methyl-2H-chromene (benzopyran) $\xrightarrow{\text{HBr (excess), }\Delta}$ A $\xrightarrow{\text{(i) KOH (Alc.), (ii) H}^+}$ B $\xrightarrow{\text{O}_3\text{, Zn/H}_3\text{O}^+}$ C

1. A.2-bromobenzaldehyde
2. B.2-bromophenylacetic acid
3. C.2-hydroxybenzaldehyde (salicylaldehyde)✓
4. D.2-hydroxyphenylacetic acid

Correct answer: C

Which of the following compounds will show retention in configuration on nucleophilic substitution by OH$^-$ ion?

1. A.$\mathrm{CH_3-CH(Br)-C_6H_5}$
2. B.$\mathrm{CH_3-CBr(H)-C_6H_{13}}$
3. C.$\mathrm{CH_3-CH(C_2H_5)-CH_2Br}$✓
4. D.$\mathrm{CH_3-CH(CH_3)-Br}$

Correct answer: C

Consider the following reactions: (i) Glucose + ROH $\xrightarrow{\text{dry HCl}}$ Acetal $\xrightarrow{x\text{ eq. of (CH}_3\text{CO)}_2\text{O}}$ acetyl derivative (ii) Glucose $\xrightarrow{\text{Ni/H}_2}$ A $\xrightarrow{y\text{ eq. of (CH}_3\text{CO)}_2\text{O}}$ acetyl derivative (iii) Glucose $\xrightarrow{z\text{ eq. of (CH}_3\text{CO)}_2\text{O}}$ acetyl derivative. 'x', 'y' and 'z' in these reactions are respectively.

1. A.4, 5 & 5
2. B.4, 6 & 5✓
3. C.5, 4 & 5
4. D.5, 6 & 5

Correct answer: B

The internal energy change (in J) when 90 g of water undergoes complete evaporation at 100°C is ________. (Given: $\Delta H_{vap}$ for water at 373 K = 41 kJ/mol, R = 8.314 JK$^{-1}$ mol$^{-1}$)

Correct answer: 189000

The Gibbs energy change (in J) for the given reaction at [Cu$^{2+}$] = [Sn$^{2+}$] = 1 M and 298 K is: $\mathrm{Cu(s)+Sn^{2+}(aq.)\rightarrow Cu^{2+}(aq.)+Sn(s)}$; ($E^0_{Sn^{2+}|Sn}=-0.16$ V, $E^0_{Cu^{2+}|Cu}=0.34$ V, Take F = 96500 C mol$^{-1}$)

Correct answer: 96500

The mass of gas adsorbed, x, per unit mass of adsorbate, m, was measured at various pressures, p. A graph between $\log\frac xm$ and $\log p$ gives a straight line with slope equal to 2 and the intercept equal to 0.4771. The value of $\frac xm$ at a pressure of 4 atm is: (Given log 3 = 0.4771)

Correct answer: 48

The oxidation states of iron atoms in compounds (A), (B) and (C), respectively, are x, y and z. The sum of x, y and z is ________. (A) Na$_4$[Fe(CN)$_5$(NOS)] (B) Na$_4$[FeO$_4$] (C) [Fe$_2$(CO)$_9$]

Correct answer: 6

The number of chiral carbons present in the molecule given below is ________. [A quinine-like structure: a quinoline ring bearing a –CH(OH)CH$_3$ group, linked through a CH(O–C$_6$H$_4$–CH$_3$) carbon to a quinuclidine (azabicyclo) ring that carries an ethyl group]

Correct answer: 5

If $R=\{(x,y): x,y\in\mathbb{Z}, x^2+3y^2\le8\}$ is a relation on the set of integers $\mathbb{Z}$, then the domain of $R^{-1}$ is:

1. A.$\{-2,-1,0,1,2\}$
2. B.$\{-2,-1,1,2\}$
3. C.$\{-1,0,1\}$✓
4. D.$\{0,1\}$

Correct answer: C

Let $\alpha$ and $\beta$ be the roots of the equation, $5x^2+6x-2=0$. If $S_n=\alpha^n+\beta^n$, n = 1, 2, 3, ..., then:

1. A.$6S_6+5S_5=2S_4$
2. B.$5S_6+6S_5=2S_4$✓
3. C.$5S_6+6S_5+2S_4=0$
4. D.$6S_6+5S_5+2S_4=0$

Correct answer: B

The value of $\left(\frac{1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}}{1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}}\right)^3$ is:

1. A.$\frac12(\sqrt3-i)$
2. B.$\frac12(1-i\sqrt3)$
3. C.$-\frac12(1-i\sqrt3)$
4. D.$-\frac12(\sqrt3-i)$✓

Correct answer: D

Let A be a $2\times2$ real matrix with entries from {0, 1} and $|A|\ne0$. Consider the following two statements: (P) If $A\ne I_2$, then $|A|=-1$ (Q) If $|A|=1$, then tr(A) = 2, where $I_2$ denotes $2\times2$ identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then:

1. A.(P) is true and (Q) is false
2. B.(P) is false and (Q) is true✓
3. C.Both (P) and (Q) are false
4. D.Both (P) and (Q) are true

Correct answer: B

Let S be the set of all $\lambda\in\mathbb{R}$ for which the system of linear equations $2x-y+2z=2$, $x-2y+\lambda z=-4$, $x+\lambda y+z=4$ has no solution. Then the set S

1. A.is an empty set.
2. B.is a singleton.
3. C.contains exactly two elements.✓
4. D.contains more than two elements.

Correct answer: C

Let $\alpha>0$, $\beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of x in the binomial expansion of $\left(\alpha x^{\frac19}+\beta x^{-\frac16}\right)^{10}$ is 10k, then k is equal to:

1. A.84
2. B.176
3. C.336✓
4. D.352

Correct answer: C

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in:

1. A.$[-3,\infty)$
2. B.$(-\infty,-3]\cup[9,\infty)$✓
3. C.$(-\infty,-9]\cup[3,\infty)$
4. D.$(-\infty,9]$

Correct answer: B

If $|x|<1$, $|y|<1$ and $x\ne y$, then the sum to infinity of the following series $(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+\dots$ is:

1. A.$\frac{x+y+xy}{(1+x)(1+y)}$
2. B.$\frac{x+y+xy}{(1-x)(1-y)}$
3. C.$\frac{x+y-xy}{(1-x)(1-y)}$✓
4. D.$\frac{x+y-xy}{(1+x)(1+y)}$

Correct answer: C

If a function f(x) defined by $f(x)=\begin{cases}ae^x+be^{-x}, & -1\le x<1\\ cx^2, & 1\le x\le3\\ ax^2+2cx, & 3<x\le4\end{cases}$ be continuous for some $a,b,c\in\mathbb{R}$ and $f'(0)+f'(2)=e$, then the value of a is:

1. A.$\frac{e}{e^2-3e+13}$✓
2. B.$\frac{e}{e^2+3e+13}$
3. C.$\frac{e}{e^2-3e-13}$
4. D.$\frac{1}{e^2-3e+13}$

Correct answer: A

If the tangent to the curve $y=x+\sin y$ at a point (a, b) is parallel to the line joining $\left(0,\frac32\right)$ and $\left(\frac12,2\right)$, then:

1. A.$|b-a|=1$✓
2. B.$b=a$
3. C.$|a+b|=1$
4. D.$b=\frac\pi2+a$

Correct answer: A

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to:

1. A.−12✓
2. B.−24
3. C.6
4. D.12

Correct answer: A

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

1. A.$3(4-\pi)$
2. B.$6(4-\pi)$
3. C.$6(\pi-2)$✓
4. D.$3(\pi-2)$

Correct answer: C

Let $y=y(x)$ be the solution of the differential equation, $\frac{2+\sin x}{y+1}\cdot\frac{dy}{dx}=-\cos x$, $y>0$, $y(0)=1$. If $y(\pi)=a$ and $\frac{dy}{dx}$ at $x=\pi$ is b, then the ordered pair (a, b) is equal to:

1. A.$\left(2,\frac32\right)$
2. B.(1, 1)✓
3. C.(1, −1)
4. D.(2, 1)

Correct answer: B

Let P(h, k) be a point on the curve $y=x^2+7x+2$, nearest to the line, $y=3x-3$. Then the equation of the normal to the curve at P is:

1. A.$x-3y+22=0$
2. B.$x+3y-62=0$
3. C.$x+3y+26=0$✓
4. D.$x-3y-11=0$

Correct answer: C

A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $(x_1,y_1)$. Then $x_1^2+5y_1^2$ is equal to:

1. A.10
2. B.8
3. C.6✓
4. D.5

Correct answer: C

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, $2x=3y$, $z=1$ also passes through the point:

1. A.(2, 0, −1)
2. B.(−2, 0, 1)✓
3. C.(0, 6, −2)
4. D.(0, −6, 2)

Correct answer: B

Let $X=\{x\in\mathbb{N}: 1\le x\le17\}$ and $Y=\{ax+b: x\in X$ and $a,b\in\mathbb{R}, a>0\}$. If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to:

1. A.−27
2. B.7
3. C.−7✓
4. D.9

Correct answer: C

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is:

1. A.$\frac8{17}$✓
2. B.$\frac4{17}$
3. C.$\frac25$
4. D.$\frac23$

Correct answer: A

The domain of the function $f(x)=\sin^{-1}\left(\frac{|x|+5}{x^2+1}\right)$ is $(-\infty,-a]\cup[a,\infty)$. Then a is equal to:

1. A.$\frac{\sqrt{17}}{2}$
2. B.$\frac{1+\sqrt{17}}{2}$✓
3. C.$\frac{\sqrt{17}-1}{2}$
4. D.$\frac{\sqrt{17}}{2}+1$

Correct answer: B

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is:

1. A.If I do not reach the station in time, then I will catch the train.
2. B.If I do not reach the station in time, then I will not catch the train.
3. C.If I will catch the train, then I reach the station in time.
4. D.If I will not catch the train, then I do not reach the station in time.✓

Correct answer: D

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ________.

Correct answer: 309

If $\lim_{x\to1}\frac{x+x^2+x^3+\dots+x^n-n}{x-1}=820$, $(n\in\mathbb{N})$ then the value of n is equal to ________.

Correct answer: 40

The integral $\int_0^2\left||x-1|-x\right|dx$ is equal to ________.

Correct answer: 1.5

The number of integral values of k for which the line, $3x+4y=k$ intersects the circle, $x^2+y^2-2x-4y+4=0$ at two distinct points is ________.

Correct answer: 9

Let $\vec a$, $\vec b$ and $\vec c$ be three unit vectors such that $|\vec a-\vec b|^2+|\vec a-\vec c|^2=8$. Then $|\vec a+2\vec b|^2+|\vec a+2\vec c|^2$ is equal to ________.

Correct answer: 2