JEE Main 2020 — 7 Jan Shift 2 (Previous Year Paper)

The dimension of $\frac{B^2}{2\mu_0}$, where B is magnetic field and $\mu_0$ is the magnetic permeability of vacuum, is:

1. A.$ML^2T^{-2}$
2. B.$MLT^{-2}$
3. C.$ML^2T^{-1}$
4. D.$ML^{-1}T^{-2}$✓

Correct answer: D

A mass of 10 kg is suspended by a rope of length 4 m, from the ceiling. A force F is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of 45° with the vertical. Then F equals: (Take $g = 10\ \mathrm{m\,s^{-2}}$ and the rope to be massless)

1. A.90 N
2. B.70 N
3. C.100 N✓
4. D.75 N

Correct answer: C

An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator ($g = 10\ \mathrm{m/s^2}$) must be at least:

1. A.56300 W
2. B.66000 W✓
3. C.48000 W
4. D.62360 W

Correct answer: B

Mass per unit area of a circular disc of radius a depends on the distance r from its centre as $\sigma(r) = A + Br$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:

1. A.$2\pi a^4\left(\frac{A}{4} + \frac{aB}{5}\right)$✓
2. B.$\pi a^4\left(\frac{A}{4} + \frac{aB}{5}\right)$
3. C.$2\pi a^4\left(\frac{A}{4} + \frac{B}{5}\right)$
4. D.$2\pi a^4\left(\frac{aA}{4} + \frac{B}{5}\right)$

Correct answer: A

A box weighs 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $g = 10\ \mathrm{m\,s^{-2}}$ at the north pole and the radius of the earth = 6400 km):

1. A.194.32 N
2. B.195.66 N
3. C.195.32 N✓
4. D.194.66 N

Correct answer: C

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is:

1. A.$\frac{\sqrt3}{2}$
2. B.$\frac{3}{4}$
3. C.$\frac{81}{256}$
4. D.$\frac{9}{16}$✓

Correct answer: D

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $T_1$ and $T_2$. The temperature of the hot reservoir of the first engine is $T_1$ and the temperature of the cold reservoir of the second engine is $T_2$. T is temperature of the sink of first engine which is also the source for the second engine. How is T related to $T_1$ and $T_2$, if both the engines perform equal amount of work?

1. A.$T = \frac{T_1+T_2}{2}$✓
2. B.$T = \sqrt{T_1T_2}$
3. C.$T = \frac{2T_1T_2}{T_1+T_2}$
4. D.$T = 0$

Correct answer: A

A stationary observer receives sound from two identical tuning forks, one of which approaches and the other one recedes with the same speed (much less than the speed of sound). The observer hears 2 beats/sec. The oscillation frequency of each tuning fork is $\nu_0 = 1400$ Hz and the velocity of sound in air is 350 m/s. The speed of each tuning fork is close to:

1. A.1 m/s
2. B.$\frac{1}{2}$ m/s
3. C.$\frac{1}{4}$ m/s✓
4. D.$\frac{1}{8}$ m/s

Correct answer: C

A particle of mass m and charge q has an initial velocity $\vec v = v_0\hat j$. If an electric field $\vec E = E_0\hat i$ and magnetic field $\vec B = B_0\hat i$ act on the particle, its speed will double after a time:

1. A.$\frac{2mv_0}{qE_0}$
2. B.$\frac{\sqrt2 mv_0}{qE_0}$
3. C.$\frac{3mv_0}{qE_0}$
4. D.$\frac{\sqrt3 mv_0}{qE_0}$✓

Correct answer: D

In a building there are 15 bulbs of 45 W, 15 bulbs of 100 W, 15 small fans of 10 W and 2 heaters of 1 kW. The voltage of electric main is 220 V. The minimum fuse capacity (rated value) of the building will be:

1. A.10 A
2. B.20 A✓
3. C.15 A
4. D.25 A

Correct answer: B

A planar loop of wire rotates in a uniform magnetic field. Initially, at t = 0, the plane of the loop is perpendicular to the magnetic field. If it rotates with a period of 10 s about an axis in its plane then the magnitude of induced emf will be maximum and minimum, respectively at:

1. A.2.5 s and 5.0 s✓
2. B.5.0 s and 7.5 s
3. C.5.0 s and 10.0 s
4. D.2.5 s and 7.5 s

Correct answer: A

The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity, co-ercivity and saturation, respectively, of the material are:

1. A.1.0 T, 50 A/m and 1.5 T✓
2. B.1.5 T, 50 A/m and 1.0 T
3. C.150 A/m, 1.0 T and 1.5 T
4. D.1.5 T, 50 A/m and 1.0 T

Correct answer: A

The electric field of a plane electromagnetic wave is given by $\vec E = E_0\frac{\hat i + \hat j}{\sqrt2}\cos(kz + \omega t)$. At t = 0, a positively charged particle is at the point $(x, y, z) = \left(0, 0, \frac{\pi}{k}\right)$. If its instantaneous velocity at (t = 0) is $v_0\hat k$, the force acting on it due to the wave is:

1. A.parallel to $\frac{\hat i + \hat j}{\sqrt2}$
2. B.antiparallel to $\frac{\hat i + \hat j}{\sqrt2}$✓
3. C.zero
4. D.parallel to $\hat k$

Correct answer: B

A thin lens made of glass (refractive index = 1.5) of focal length f = 16 cm is immersed in a liquid of refractive index 1.42. If its focal length in liquid is $f_l$, then the ratio $f_l/f$ is closest to the integer:

1. A.1
2. B.5
3. C.9✓
4. D.17

Correct answer: C

In a Young's double slit experiment, the separation between the slits is 0.15 mm. In the experiment, a source of light of wavelength 589 nm is used and the interference pattern is observed on a screen kept 1.5 m away. The separation between the successive bright fringes on the screen is:

1. A.6.9 mm
2. B.5.9 mm✓
3. C.4.9 mm
4. D.3.9 mm

Correct answer: B

An electron (of mass m) and a photon have the same energy E in the range of a few eV. The ratio of the de-Broglie wavelength associated with the electron and the wavelength of the photon is (c = speed of light in vacuum)

1. A.$c(2mE)^{1/2}$
2. B.$\frac{1}{c}\left(\frac{2E}{m}\right)^{1/2}$
3. C.$\left(\frac{E}{2m}\right)^{1/2}$
4. D.$\frac{1}{c}\left(\frac{E}{2m}\right)^{1/2}$✓

Correct answer: D

The activity of a radioactive sample falls from 700 s$^{-1}$ to 500 s$^{-1}$ in 30 minutes. Its half life is close to:

1. A.52 min
2. B.62 min✓
3. C.66 min
4. D.72 min

Correct answer: B

In the figure, potential difference between A and B is:

1. A.zero
2. B.5 V
3. C.10 V✓
4. D.15 V

Correct answer: C

The sum of two forces $\vec P$ and $\vec Q$ is $\vec R$ such that $|\vec R| = |\vec P|$. The angle θ (in degrees) that the resultant of $2\vec P$ and $\vec Q$ will make with $\vec Q$ is, ________.

Correct answer: 90

M grams of steam at 100°C is mixed with 200 g of ice at its melting point in a thermally insulated container. If it produces liquid water at 40°C [heat of vaporization of water is 540 cal/g and heat of fusion of ice is 80 cal/g], the value of M is ________.

Correct answer: 40

A 60 pF capacitor is fully charged by a 20 V supply. It is then disconnected from the supply and is connected to another uncharged 60 pF capacitor in parallel. The electrostatic energy that is lost in this process by the time the charge is redistributed between them is (in nJ) ________.

Correct answer: 6

Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is $\mu = 0.4$, the maximum possible value of $100\times\frac{b}{a}$ for box not to topple before moving is ________.

Correct answer: 50

The balancing length for a cell is 560 cm in a potentiometer experiment. When an external resistance of 10 Ω is connected in parallel to the cell, the balancing length changes by 60 cm. If the internal resistance of the cell is $\frac{N}{10}$ Ω, where N is an integer then value of N is ________.

Correct answer: 12

For the reaction 2H$_2$(g) + 2NO(g) → N$_2$(g) + 2H$_2$O(g) the observed rate expression is, rate = $k_f$[NO]$^2$[H$_2$]. The rate expression for the reverse reaction is:

1. A.$k_b$[N$_2$][H$_2$O]
2. B.$k_b$[N$_2$][H$_2$O]$^2$
3. C.$k_b$[N$_2$][H$_2$O]$^2$/[NO]
4. D.$k_b$[N$_2$][H$_2$O]$^2$/[H$_2$]✓

Correct answer: D

Two open beakers one containing a solvent and the other containing a mixture of that solvent with a non volatile solute are together sealed in a container. Over time:

1. A.the volume of the solution increases and the volume of the solvent decreases✓
2. B.the volume of the solution decreases and the volume of the solvent increases
3. C.the volume of the solution does not change and the volume of the solvent decreases
4. D.the volume of the solution and the solvent does not change

Correct answer: A

The equation that is incorrect is:

1. A.$(\Lambda^0_m)_{NaBr} - (\Lambda^0_m)_{NaCl} = (\Lambda^0_m)_{KBr} - (\Lambda^0_m)_{KCl}$
2. B.$(\Lambda^0_m)_{KCl} - (\Lambda^0_m)_{NaCl} = (\Lambda^0_m)_{KBr} - (\Lambda^0_m)_{NaBr}$
3. C.$(\Lambda^0_m)_{NaBr} - (\Lambda^0_m)_{NaI} = (\Lambda^0_m)_{KBr} - (\Lambda^0_m)_{NaBr}$✓
4. D.$(\Lambda^0_m)_{H_2O} = (\Lambda^0_m)_{HCl} + (\Lambda^0_m)_{NaOH} - (\Lambda^0_m)_{NaCl}$

Correct answer: C

The ammonia (NH$_3$) released on quantitative reaction of 0.6 g urea (NH$_2$CONH$_2$) with sodium hydroxide (NaOH) can be neutralized by:

1. A.100 ml of 0.1 N HCl
2. B.200 ml of 0.2 N HCl
3. C.100 ml of 0.2 N HCl✓
4. D.200 ml of 0.4 N HCl

Correct answer: C

Identify the correct labels of A, B and C in the following graph from the options given below: Root mean square speed ($V_{rms}$); most probable speed ($V_{mp}$); Average speed ($V_{av}$)

1. A.A - $V_{rms}$; B - $V_{mp}$; C - $V_{av}$
2. B.A - $V_{mp}$; B - $V_{av}$; C - $V_{rms}$✓
3. C.A - $V_{av}$; B - $V_{rms}$; C - $V_{mp}$
4. D.A - $V_{mp}$; B - $V_{rms}$; C - $V_{av}$

Correct answer: B

The bond order and the magnetic characteristics of CN$^-$ are:

1. A.$2\frac12$, diamagnetic
2. B.3, paramagnetic
3. C.$2\frac12$, paramagnetic
4. D.3, diamagnetic✓

Correct answer: D

Within each pair of elements F & Cl, S & Se, and Li & Na, respectively, the elements that release more energy upon an electron gain are:

1. A.Cl, Se and Na
2. B.F, S and Li
3. C.F, Se and Na
4. D.Cl, S and Li✓

Correct answer: D

The refining method used when the metal and the impurities have low and high melting temperatures, respectively, is:

1. A.distillation
2. B.liquation✓
3. C.vapour phase refining
4. D.zone refining

Correct answer: B

Among statements (a)-(d), the correct ones are: (a) Decomposition of hydrogen peroxide gives dioxygen. (b) Like hydrogen peroxide, compounds, such as KClO$_3$, Pb(NO$_3$)$_2$ and NaNO$_3$ when heated liberate dioxygen. (c) 2-Ethylanthraquinone is useful for the industrial preparation of hydrogen peroxide. (d) Hydrogen peroxide is used for the manufacture of sodium perborate.

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

Correct answer: D

In the following reactions, products (A) and (B), respectively, are: NaOH (hot and conc.) + Cl$_2$ → (A) + side products; Ca(OH)$_2$ (dry) + Cl$_2$ → (B) + side products

1. A.NaOCl and Ca(OCl)$_2$
2. B.NaClO$_3$ and Ca(OCl)$_2$✓
3. C.NaClO$_3$ and Ca(ClO$_3$)$_2$
4. D.NaOCl and Ca(ClO$_3$)$_2$

Correct answer: B

The redox reaction among the following is:

1. A.formation of ozone from atmospheric oxygen in the presence of sunlight
2. B.reaction of H$_2$SO$_4$ with NaOH
3. C.combination of dinitrogen with dioxygen at 2000 K✓
4. D.reaction of [Co(H$_2$O)$_6$]Cl$_3$ with AgNO$_3$

Correct answer: C

The number of possible optical isomers for the complexes MA$_2$B$_2$ with sp$^3$ and dsp$^2$ hybridized metal atom, respectively, is: Note: A and B are unidentate neutral and unidentate monoanionic ligands, respectively.

1. A.2 and 2
2. B.0 and 2
3. C.0 and 1
4. D.0 and 0✓

Correct answer: D

Among the statements (a)-(d), the incorrect ones are: (a) Octahedral Co(III) complexes with strong field ligands have very high magnetic moments (b) When $\Delta_0 < P$, the d-electron configuration of Co(III) in an octahedral complex is $t_{2g}^4e_g^2$ (c) Wavelength of light absorbed by [Co(en)$_3$]$^{3+}$ is lower than that of [CoF$_6$]$^{3-}$ (d) If the $\Delta_0$ for an octahedral complex of Co(III) is 18,000 cm$^{-1}$, the $\Delta_t$ for its tetrahedral complex with the same ligand will be 16,000 cm$^{-1}$

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

Correct answer: C

A chromatography column, packed with silica gel as stationary phase, was used to separate a mixture of compounds consisting of (A) benzanilide (B) aniline and (C) acetophenone. When the column is eluted with a mixture of solvents, hexane : ethyl acetate (20:80), the sequence of obtained compounds is:

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

Correct answer: C

In the following reaction sequence, p-toluidine (4-methylaniline) $\xrightarrow{\text{Ac}_2\text{O}}$ A $\xrightarrow{\text{Br}_2/\text{AcOH}}$ B, the major product B is:

1. A.2-bromo-4-methylacetanilide (Br ortho to NHCOCH3, CH3 para)✓
2. B.2-acetyl-4-methylacetanilide (COCH3 ortho to NHCOCH3, CH3 para)
3. C.3-bromo-4-methylacetanilide (Br ortho to CH3)
4. D.2-bromo-4-(bromomethyl)acetanilide (Br ortho to NHCOCH3, CH2Br para)

Correct answer: A

The correct order of stability for the following alkoxides is: (A) CH$_3$–CH(NO$_2$)–O$^-$ (B) CH$_2$=C(NO$_2$)–O$^-$ (C) O$_2$N–CH=CH–O$^-$

1. A.(C) > (A) > (B)
2. B.(C) > (B) > (A)✓
3. C.(B) > (A) > (C)
4. D.(B) > (C) > (A)

Correct answer: B

In the following reaction sequence, structures of A and B, respectively will be: chromane bearing a CH$_2$Br group on the benzene ring (see figure) $\xrightarrow{\text{HBr},\ \Delta}$ A $\xrightarrow{\text{Na, Ether}}$ (Intramolecular Product) B

1. A.A: 2-(3-bromopropyl)-phenol bearing the ring CH2Br; B: a phenol fused to a six-membered carbocycle (5,6,7,8-tetrahydronaphthol type)✓
2. B.A: same bromopropyl phenol with ring CH2Br; B: a phenol fused to a five-membered ring carrying a methyl (indane type)
3. C.A: aryl bromide with a 3-hydroxypropyl chain and ring CH2Br; B: bromo-substituted fused bicyclic compound (ring drawn with a substituent)
4. D.A: aryl bromide with a 3-hydroxypropyl chain and ring CH2Br; B: a CH2Br-substituted benzopyran (chromane)

Correct answer: A

Which of the following statements is correct?

1. A.Gluconic acid can form cyclic (acetal/hemiacetal) structure
2. B.Gluconic acid is a partial oxidation product of glucose✓
3. C.Gluconic acid is a dicarboxylic acid
4. D.Gluconic acid is obtained by oxidation of glucose with HNO$_3$

Correct answer: B

Consider the following reactions: (a) benzene + chlorobenzene $\xrightarrow{\text{anhyd. AlCl}_3}$ biphenyl (b) benzene + Cl$_2$ (excess) $\xrightarrow{\text{anhyd. AlCl}_3,\ \text{dark}}$ hexachlorobenzene (c) benzene + CH$_2$=CH–Cl $\xrightarrow{\text{anhyd. AlCl}_3}$ styrene (C$_6$H$_5$–CH=CH$_2$) (d) benzene + CH$_2$=CH–CH$_2$Cl $\xrightarrow{\text{anhyd. AlCl}_3}$ allylbenzene (C$_6$H$_5$–CH$_2$–CH=CH$_2$). Which of these reactions are possible?

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

Correct answer: B

For the following reactions: CH$_3$CH$_2$CH$_2$Br + Z$^-$ $\xrightarrow{k_s,\ \text{Substitution}}$ CH$_3$CH$_2$CH$_2$Z + Br$^-$ and CH$_3$CH$_2$CH$_2$Br + Z$^-$ $\xrightarrow{k_e,\ \text{elimination}}$ CH$_3$CH=CH$_2$ + HZ + Br$^-$, where, Z$^-$ = CH$_3$CH$_2$O$^-$ (A) or (CH$_3$)$_3$C–O$^-$ (B), $k_s$ and $k_e$, are, respectively, the rate constants for substitution and elimination, and $\mu = \frac{k_s}{k_e}$, the correct option is ________.

1. A.$\mu_A > \mu_B$ and $k_e(A) > k_e(B)$
2. B.$\mu_A > \mu_B$ and $k_e(B) > k_e(A)$✓
3. C.$\mu_B > \mu_A$ and $k_e(A) > k_e(B)$
4. D.$\mu_B > \mu_A$ and $k_e(B) > k_e(A)$

Correct answer: B

3 g of acetic acid is added to 250 mL of 0.1 M HCl and the solution made up to 500 mL. To 20 mL of this solution $\frac{1}{2}$ mL of 5 M NaOH is added. The pH of the solution is ________. [Given: pKa of acetic acid = 4.75, molar mass of acetic acid = 60 g/mol, log 3 = 0.4771] Neglect any changes in volume.

Correct answer: 5.23

The flocculation value of HCl for arsenic sulphide sol. is 30 m mol L$^{-1}$. If H$_2$SO$_4$ is used for the flocculation of arsenic sulphide, the amount, in grams, of H$_2$SO$_4$ in 250 ml required for the above purpose is ________. (molecular mass of H$_2$SO$_4$ = 98 g/mol)

Correct answer: 0.37

Consider the following reactions: NaCl + K$_2$Cr$_2$O$_7$ + H$_2$SO$_4$ (Conc.) → (A) + side products; (A) + NaOH → (B) + Side products; (B) + H$_2$SO$_4$ (dilute) + H$_2$O$_2$ → (C) + Side products. The sum of the total number of atoms in one molecule each of (A), (B) and (C) is ________.

Correct answer: 18

The number of sp$^2$ hybridised carbons present in "Aspartame" is ________.

Correct answer: 9

Let α and β be the roots of the equation $x^2 - x - 1 = 0$. If $p_k = (\alpha)^k + (\beta)^k$, $k \ge 1$, then which one of the following statements is not true?

1. A.$p_5 = 11$
2. B.$p_3 = p_5 - p_4$
3. C.$(p_1 + p_2 + p_3 + p_4 + p_5) = 26$
4. D.$p_5 = p_2\cdot p_3$✓

Correct answer: D

If $\frac{3 + i\sin\theta}{4 - i\cos\theta}$, $\theta \in [0, 2\pi]$, is a real number, then an argument of $\sin\theta + i\cos\theta$ is:

1. A.$-\tan^{-1}\left(\frac34\right)$
2. B.$\pi - \tan^{-1}\left(\frac34\right)$
3. C.$\pi - \tan^{-1}\left(\frac43\right)$✓
4. D.$\tan^{-1}\left(\frac43\right)$

Correct answer: C

Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two 3 × 3 real matrices such that $b_{ij} = (3)^{(i+j-2)}a_{ji}$, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

1. A.1/3
2. B.1/9✓
3. C.1/81
4. D.3

Correct answer: B

The number of ordered pairs (r, k) for which $6\cdot{}^{35}C_r = (k^2 - 3)\cdot{}^{36}C_{r+1}$, where k is an integer, is:

1. A.6
2. B.4✓
3. C.3
4. D.2

Correct answer: B

The coefficient of $x^7$ in the expression $(1+x)^{10} + x(1+x)^9 + x^2(1+x)^8 + \ldots + x^{10}$ is:

1. A.120
2. B.210
3. C.330✓
4. D.420

Correct answer: C

Let $a_1, a_2, a_3, \ldots$ be a G. P. such that $a_1 < 0$, $a_1 + a_2 = 4$ and $a_3 + a_4 = 16$. If $\sum_{i=1}^{9} a_i = 4\lambda$, then λ is equal to:

1. A.−513
2. B.$\frac{511}{3}$
3. C.−171✓
4. D.171

Correct answer: C

If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \ldots$ is (102)m, then m is equal to:

1. A.25
2. B.20✓
3. C.10
4. D.5

Correct answer: B

Let $y = y(x)$ be a function of x satisfying $y\sqrt{1-x^2} = k - x\sqrt{1-y^2}$ where k is a constant and $y\left(\frac12\right) = -\frac14$. Then $\frac{dy}{dx}$ at $x = \frac12$, is equal to:

1. A.$\frac{2}{\sqrt5}$
2. B.$-\frac{\sqrt5}{4}$
3. C.$-\frac{\sqrt5}{2}$✓
4. D.$\frac{\sqrt5}{2}$

Correct answer: C

The value of c in the Lagrange's mean value theorem for the function $f(x) = x^3 - 4x^2 + 8x + 11$, when $x \in [0, 1]$ is:

1. A.$\frac{4-\sqrt7}{3}$✓
2. B.$\frac{\sqrt7-2}{3}$
3. C.$\frac{4-\sqrt5}{3}$
4. D.$\frac{2}{3}$

Correct answer: A

Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If $\lim_{x\to0}\left(2 + \frac{f(x)}{x^3}\right) = 4$, then which one of the following is not true?

1. A.f is an odd function.
2. B.x = 1 is a point of maxima and x = −1 is a point of minimum of f.
3. C.f(1) − 4f(−1) = 4.
4. D.x = 1 is a point of minima and x = −1 is a point of maxima of f.✓

Correct answer: D

The value of α for which $4\alpha\int_{-1}^{2} e^{-\alpha|x|}\,dx = 5$, is:

1. A.$\log_e\left(\frac43\right)$
2. B.$\log_e 2$✓
3. C.$\log_e\sqrt2$
4. D.$\log_e\left(\frac32\right)$

Correct answer: B

The area (in sq. units) of the region $\{(x, y) \in \mathbb{R}^2 \mid 4x^2 \le y \le 8x + 12\}$ is:

1. A.$\frac{124}{3}$
2. B.$\frac{125}{3}$
3. C.$\frac{127}{3}$
4. D.$\frac{128}{3}$✓

Correct answer: D

Let $y = y(x)$ be the solution curve of the differential equation, $(y^2 - x)\frac{dy}{dx} = 1$, satisfying $y(0) = 1$. This curve intersects the x-axis at a point whose abscissa is:

1. A.2
2. B.2 + e
3. C.2 − e✓
4. D.−e

Correct answer: C

Let the tangents drawn from the origin to the circle, $x^2 + y^2 - 8x - 4y + 16 = 0$ touch it at the points A and B. The $(AB)^2$ is equal to:

1. A.$\frac{32}{5}$
2. B.$\frac{52}{5}$
3. C.$\frac{56}{5}$
4. D.$\frac{64}{5}$✓

Correct answer: D

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is:

1. A.3x − 2y = 0
2. B.2x − 3y = 0
3. C.7x − 5y = 0
4. D.5x − 7y = 0✓

Correct answer: D

If $3x + 4y = 12\sqrt2$ is a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{9} = 1$ for some $a \in \mathbb{R}$, then the distance between the foci of the ellipse is:

1. A.$2\sqrt2$
2. B.$2\sqrt7$✓
3. C.$2\sqrt5$
4. D.4

Correct answer: B

Let $\vec a$, $\vec b$ and $\vec c$ be three unit vectors such that $\vec a + \vec b + \vec c = \vec 0$. If $\lambda = \vec a\cdot\vec b + \vec b\cdot\vec c + \vec c\cdot\vec a$ and $\vec d = \vec a\times\vec b + \vec b\times\vec c + \vec c\times\vec a$, then the ordered pair, $(\lambda, \vec d)$ is equal to:

1. A.$\left(-\frac32, 3\vec a\times\vec b\right)$✓
2. B.$\left(\frac32, 3\vec a\times\vec c\right)$
3. C.$\left(-\frac32, 3\vec c\times\vec b\right)$
4. D.$\left(\frac32, 3\vec b\times\vec c\right)$

Correct answer: A

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac14$. If the probability that at most two machines will be out of service on the same day is $\left(\frac34\right)^3k$, then k is equal to: