JEE Main 2020 — 8 Jan Shift 1 (Previous Year Paper)

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length $l$. The other end is fixed. The system is given an angular speed ω about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is:

1. A.$\frac{ml\omega^2}{k + m\omega^2}$
2. B.$\frac{ml\omega^2}{k - m\omega^2}$✓
3. C.$\frac{ml\omega^2}{k + m\omega}$
4. D.$\frac{ml\omega^2}{k - \omega m}$

Correct answer: B

The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in figure) are:

1. A.(0.75 m, 1.75 m)✓
2. B.(0.75 m, 0.75 m)
3. C.(1 m, 1.75 m)
4. D.(1.25 m, 1.50 m)

Correct answer: A

Consider a uniform rod of mass M = 4m and length $l$ pivoted about its centre. A mass m moving with velocity v making angle $\theta = \frac{\pi}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is:

1. A.$\frac37\frac{v}{l}$
2. B.$\frac{3}{7\sqrt2}\frac{v}{l}$
3. C.$\frac{3\sqrt2}{7}\frac{v}{l}$✓
4. D.$\frac47\frac{v}{l}$

Correct answer: C

Consider two solid spheres of radii $R_1$ = 1m, $R_2$ = 2m and masses $M_1$ and $M_2$, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $\frac{M_1}{M_2}$ is:

1. A.$\frac16$✓
2. B.$\frac13$
3. C.$\frac12$
4. D.$\frac23$

Correct answer: A

Consider a solid sphere of radius R and mass density $\rho(r) = \rho_0\left(1 - \frac{r^2}{R^2}\right)$, $0 < r \le R$. The minimum density of a liquid in which it will float is:

1. A.$\frac{\rho_0}{5}$
2. B.$\frac{\rho_0}{3}$
3. C.$\frac{2\rho_0}{5}$✓
4. D.$\frac{2\rho_0}{3}$

Correct answer: C

A leak proof cylinder of length 1 m, made of a metal which has very low coefficient of expansion is floating vertically in water at 0°C such that its height above the water surface is 20 cm. When the temperature of water is increased to 4°C, the height of the cylinder above the water surface becomes 21 cm. The density of water at T = 4°C, relative to the density at T = 0°C is close to:

1. A.1.04
2. B.1.01✓
3. C.1.26
4. D.1.03

Correct answer: B

A thermodynamic cycle xyzx is shown on a V-T diagram. The P-V diagram that best describes this cycle is: (Diagrams are schematic and not to scale)

1. A.P–V diagram 1 (see image)
2. B.P–V diagram 2 (see image)✓
3. C.P–V diagram 3 (see image)
4. D.P–V diagram 4 (see image)

Correct answer: B

The plot that depicts the behavior of the mean free time τ (time between two successive collisions) for the molecules of an ideal gas, as a function of temperature (T), qualitatively, is: (Graphs are schematic and not drawn to scale)

1. A.τ vs T: straight line through the origin
2. B.τ vs √T: straight line with negative slope
3. C.τ vs 1/√T: straight line through the origin✓
4. D.τ vs 1/T: straight line through the origin

Correct answer: C

Three charged particles A, B and C with charges −4q, 2q and −2q are present on the circumference of a circle of radius d. The charged particles A, C and centre O of the circle formed an equilateral triangle as shown in figure. Electric field at O along x-direction is:

1. A.$\frac{\sqrt3 q}{4\pi\epsilon_0d^2}$
2. B.$\frac{3\sqrt3 q}{4\pi\epsilon_0d^2}$
3. C.$\frac{2\sqrt3 q}{\pi\epsilon_0d^2}$
4. D.$\frac{\sqrt3 q}{\pi\epsilon_0d^2}$✓

Correct answer: D

Effective capacitance of parallel combination of two capacitors $C_1$ and $C_2$ is 10 μF. When these capacitors are individually connected to a voltage source of 1 V, the energy stored in the capacitor $C_2$ is 4 times that of $C_1$. If these capacitors are connected in series, their effective capacitance will be:

1. A.8.4 μF
2. B.4.2 μF
3. C.3.2 μF
4. D.1.6 μF✓

Correct answer: D

Proton with kinetic energy of 1 MeV moves from south to north. It gets an acceleration of $10^{12}$ m/s$^2$ by an applied magnetic field (west to east). The value of magnetic field: (Rest mass of proton is $1.6\times10^{-27}$ kg)

1. A.71 mT
2. B.7.1 mT
3. C.0.71 mT✓
4. D.0.071 mT

Correct answer: C

In finding the electric field using Gauss law the formula $|\vec E| = \frac{q_{enc}}{\epsilon_0|A|}$ is applicable. In the formula $\epsilon_0$ is permittivity of free space, A is the area of Gaussian surface and $q_{enc}$ is charge enclosed by the Gaussian surface. This equation can be used in which of the following situation?

1. A.For any choice of Gaussian surface.
2. B.Only when the Gaussian surface is an equipotential surface.
3. C.Only when the Gaussian surface is an equipotential surface and $|\vec E|$ is constant on the surface.✓
4. D.Only when $|\vec E|$ = constant on the surface.

Correct answer: C

At time t = 0 magnetic field of 1000 Gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 Gauss, in the next 5 s, then induced EMF in the loop is:

1. A.28 μV
2. B.36 μV
3. C.48 μV
4. D.56 μV✓

Correct answer: D

The critical angle of a medium for a specific wavelength, if the medium has relative permittivity 3 and relative permeability $\frac43$ for this wavelength, will be:

1. A.30°✓
2. B.45°
3. C.60°
4. D.15°

Correct answer: A

The magnifying power of a telescope with tube length 60 cm is 5. What is the focal length of its eye piece?

1. A.10 cm✓
2. B.20 cm
3. C.30 cm
4. D.40 cm

Correct answer: A

When photon of energy 4.0 eV strikes the surface of a metal A, the ejected photoelectrons have maximum kinetic energy $T_A$ eV and de-Broglie wavelength $\lambda_A$. The maximum kinetic energy of photoelectrons liberated from another metal B by photon of energy 4.50 eV is $T_B = (T_A - 1.5)$ eV. If the de-Broglie wavelength of these photoelectrons $\lambda_B = 2\lambda_A$, then the work function of metal B is:

1. A.1.5 eV
2. B.4 eV✓
3. C.3 eV
4. D.2 eV

Correct answer: B

The graph which depicts the results of Rutherford gold foil experiment with α-particles is: θ: Scattering angle; Y: Number of scattered α-particles detected (Plots are schematic and not to scale)

1. A.Y decreases linearly from θ = 0 to π
2. B.Y rises and saturates with θ
3. C.Y rises to a peak and falls (bell-shaped)
4. D.Y falls steeply from a large value at small θ and flattens towards π✓

Correct answer: D

Boolean relation at the output stage-Y for the following circuit is:

1. A.A + B
2. B.A · B
3. C.$\bar A + \bar B$
4. D.$\bar A\cdot\bar B$✓

Correct answer: D

The length of a potentiometer wire is 1200 cm and it carries a current of 60 mA. For a cell of emf 5 V and internal resistance of 20 Ω, the null point on it is found to be at 1000 cm. The resistance of whole wire is:

1. A.60 Ω
2. B.80 Ω
3. C.100 Ω✓
4. D.120 Ω

Correct answer: C

A particle is moving along the x-axis with its coordinate with time 't' given by $x(t) = 10 + 8t - 3t^2$. Another particle is moving along the y-axis with its coordinate as a function of time given by $y(t) = 5 - 8t^3$. At t = 1 s, the speed of the second particle as measured in the frame of the first particle is given as $\sqrt v$. Then v (in m/s) is ________.

Correct answer: 580

A body A, of mass m = 0.1 kg has an initial velocity of $3\hat i$ m s$^{-1}$. It collides elastically with another body, B of the same mass which has an initial velocity of $5\hat j$ m s$^{-1}$. After collision, A moves with a velocity $\vec v = 4(\hat i + \hat j)$. The energy of B after collision is written as $\frac{x}{10}$ J. The value of x is ________.

Correct answer: 1

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is 300 m/s, the frequency difference between the fundamental and second harmonic of this pipe is ________ Hz.

Correct answer: 106

Four resistances of 15 Ω, 12 Ω, 4 Ω and 10 Ω respectively in cyclic order to form Wheatstone's network. The resistance that is to be connected in parallel with the resistance of 10 Ω to balance the network is ________ Ω.

Correct answer: 10

A point object in air is in front of the curved surface of a plano-convex lens. The radius of curvature of the curved surface is 30 cm and the refractive index of the lens material is 1.5, then the focal length of the lens (in cm) is ________.

Correct answer: 60

The stoichiometry and solubility product of a salt with the solubility curve given below is, respectively:

1. A.XY, $2\times10^{-6}$ M$^3$
2. B.XY$_2$, $4\times10^{-9}$ M$^3$✓
3. C.X$_2$Y, $2\times10^{-9}$ M$^3$
4. D.XY$_2$, $1\times10^{-9}$ M$^3$

Correct answer: B

The rate of a certain biochemical reaction at physiological temperature (T) occurs $10^6$ times faster with enzyme than without. The change in the activation energy upon adding enzyme is:

1. A.−6RT
2. B.−6(2.303)RT✓
3. C.+6RT
4. D.+6(2.303)RT

Correct answer: B

As per Hardy-Schulze formulation, the flocculation values of the following for ferric hydroxide sol are in the order:

1. A.K$_3$[Fe(CN)$_6$] > AlCl$_3$ > K$_2$CrO$_4$ > KBr > KNO$_3$
2. B.AlCl$_3$ > K$_3$[Fe(CN)$_6$] > K$_2$CrO$_4$ > KBr = KNO$_3$
3. C.K$_3$[Fe(CN)$_6$] < K$_2$CrO$_4$ < AlCl$_3$ < KBr < KNO$_3$
4. D.K$_3$[Fe(CN)$_6$] < K$_2$CrO$_4$ < KBr = KNO$_3$ = AlCl$_3$✓

Correct answer: D

The predominant intermolecular forces present in ethyl acetate, a liquid, are:

1. A.hydrogen bonding and London dispersion
2. B.London dispersion, dipole-dipole and hydrogen bonding
3. C.Dipole-dipole and hydrogen bonding
4. D.London dispersion and dipole-dipole✓

Correct answer: D

For the Balmer series in the spectrum of H atom, $\bar\nu = R_H\left\{\frac{1}{n_1^2} - \frac{1}{n_2^2}\right\}$, the correct statements among (I) to (IV) are: (I) As wavelength decreases, the lines in the series converge (II) The integer $n_1$ is equal to 2 (III) The lines of longest wavelength corresponds to $n_2 = 3$ (IV) The ionization energy of hydrogen can be calculated from wave number of these lines

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

Correct answer: A

A graph of vapour pressure and temperature for three different liquids X, Y, and Z is shown below: The following inferences are made: (A) X has higher intermolecular interactions compared to Y. (B) X has lower intermolecular interactions compared to Y. (C) Z has lower intermolecular interactions compared to Y. The correct inference(s) is/are:

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

Correct answer: B

The first ionization energy (in kJ/mol) of Na, Mg, Al and Si respectively, are:

1. A.496, 577, 737, 786
2. B.786, 737, 577, 496
3. C.496, 737, 577, 786✓
4. D.496, 577, 786, 737

Correct answer: C

The strength of an aqueous NaOH solution is most accurately determined by titrating: (Note: consider that an appropriate indicator is used)

1. A.Aq. NaOH in a volumetric flask and concentrated H$_2$SO$_4$ in a conical flask
2. B.Aq. NaOH in a burette and concentrated H$_2$SO$_4$ in a conical flask
3. C.Aq. NaOH in a burette and aqueous oxalic acid in a conical flask✓
4. D.Aq. NaOH in a pipette and aqueous oxalic acid in a burette

Correct answer: C

When gypsum is heated to 393 K, it forms:

1. A.Anhydrous CaSO$_4$
2. B.Dead burnt plaster
3. C.CaSO$_4\cdot$0.5 H$_2$O✓
4. D.CaSO$_4\cdot$5 H$_2$O

Correct answer: C

The number of bonds between sulphur and oxygen atoms in S$_2$O$_8^{2-}$ and the number of bonds between sulphur and sulphur atoms in rhombic sulphur, respectively, are:

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

Correct answer: D

The third ionization enthalpy is minimum for:

1. A.Ni
2. B.Mn
3. C.Fe✓
4. D.Co

Correct answer: C

The complex that can show fac- and mer- isomers is:

1. A.[Co(NH$_3$)$_3$(NO$_2$)$_3$]✓
2. B.[CoCl$_2$(en)$_2$]
3. C.[Co(NH$_3$)$_4$Cl$_2$]$^+$
4. D.[Pt(NH$_3$)$_2$Cl$_2$]

Correct answer: A

Among the gases (a) - (e), the gases that cause greenhouse effect are: (a) CO$_2$ (b) H$_2$O (c) CFCs (d) O$_2$ (e) O$_3$

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

Correct answer: C

The most suitable reagent for the given conversion is: a decalin-type bicyclic compound bearing CONH$_2$, COCH$_3$ (acetyl), CN and CO$_2$H groups is converted into the same compound with the CO$_2$H group reduced to CH$_2$OH (other groups unchanged).

1. A.LiAlH$_4$
2. B.B$_2$H$_6$✓
3. C.H$_2$/Pd
4. D.NaBH$_4$

Correct answer: B

A flask contains a mixture of isohexane and 3-methylpentane. One of the liquids boils at 63 °C while the other boils at 60 °C. What is the best way to separate the two liquids and which one will be distilled out first?

1. A.fractional distillation, isohexane✓
2. B.fractional distillation, 3-methylpentane
3. C.simple distillation, isohexane
4. D.simple distillation, 3-methylpentane

Correct answer: A

Arrange the following compounds in increasing order of C−OH bond length: methanol, phenol, p-ethoxyphenol

1. A.methanol < phenol < p-ethoxyphenol
2. B.phenol < p-ethoxyphenol < methanol✓
3. C.phenol < methanol < p-ethoxyphenol
4. D.methanol < p-ethoxyphenol < phenol

Correct answer: B

Which of the following statement is not true for glucose?

1. A.The pentaacetate of glucose does not react with hydroxylamine to give oxime
2. B.Glucose reacts with hydroxylamine to form oxime
3. C.Glucose gives Schiff's test for aldehyde✓
4. D.Glucose exists in two crystalline forms α and β

Correct answer: C

The major products A and B in the following reactions are: (CH$_3$)$_2$CH–CN $\xrightarrow{\text{Peroxide, Heat}}$ [A]; [A] + CH$_2$=CH–CH$_2$CH$_3$ (but-1-ene) → B

1. A.A = (CH3)2C•–CN; B = the radical adduct drawn with C(CH3)2CN attached at the internal carbon of the alkene
2. B.A = (CH3)2C•–CN; B = zig-zag chain ending in C(CH3)2–CN on the other side (see image)
3. C.A = (CH3)2C•–CN; B = CH3CH2CH2CH2–C(CH3)2–CN✓
4. D.A = (CH3)2CH–CH•–CN type radical; B = corresponding chain nitrile (see image)

Correct answer: C

The decreasing order of reactivity towards dehydrohalogenation (E1) reaction of the following compounds is: (A) 1-chlorobutane (B) 4-chlorobut-1-ene (C) 2-chlorobutane (D) 3-chlorobut-1-ene

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

Correct answer: B

The major product of the following reaction is: (CH$_3$)$_2$C=CH–CH$_2$–CH$_2$–C(CH$_3$)=CH–CH$_2$OH (geraniol) $\xrightarrow{\text{dil. H}_2\text{SO}_4}$

1. A.Open-chain triol (hydration of both C=C and retention of CH2OH)
2. B.α-Terpineol: 1-methylcyclohexene ring bearing a C(CH3)2OH group✓
3. C.Cyclopentane bearing CH3, a vinyl group and C(CH3)2OH
4. D.Open-chain diol with a C(CH3)2OH terminus

Correct answer: B

Ferrous sulphate heptahydrate is used to fortify foods with iron. The amount (in grams) of the salt required to achieve 10 ppm of iron in 100 kg of wheat is ________. Atomic weight: Fe = 55.85; S = 32.00; O = 16.00

Correct answer: 4.98

What would be the electrode potential for the given half cell reaction at pH = 5? ________. 2H$_2$O → O$_2$ + 4H$^+$ + 4e$^-$; $E^0_{red}$ = 1.23 V (R = 8.314 J mol$^{-1}$ K$^{-1}$; Temp = 298 K; oxygen under std. atm. pressure of 1 bar)

Correct answer: 0.93

The volume (in mL) of 0.125 M AgNO$_3$ required to quantitatively precipitate chloride ions in 0.3 g of [Co(NH$_3$)$_6$]Cl$_3$ is ________. $M_{[Co(NH_3)_6]Cl_3}$ = 267.46 g/mol, $M_{AgNO_3}$ = 169.87 g/mol

Correct answer: 26.92

The number of chiral centres in penicillin is ________.

Correct answer: 3

The inverse function of $f(x) = \frac{8^{2x} - 8^{-2x}}{8^{2x} + 8^{-2x}}$, $x \in (-1, 1)$, is ________.

1. A.$\frac14\log_e\left(\frac{1+x}{1-x}\right)$
2. B.$\frac14(\log_8 e)\log_e\left(\frac{1-x}{1+x}\right)$
3. C.$\frac14\log_e\left(\frac{1-x}{1+x}\right)$
4. D.$\frac14(\log_8 e)\log_e\left(\frac{1+x}{1-x}\right)$✓

Correct answer: D

If the equation, $x^2 + bx + 45 = 0$ $(b \in \mathbb{R})$ has conjugate complex roots and they satisfy $|z + 1| = 2\sqrt{10}$, then:

1. A.$b^2 + b = 12$
2. B.$b^2 - b = 42$
3. C.$b^2 - b = 30$✓
4. D.$b^2 + b = 72$

Correct answer: C

For which of the following ordered pairs (μ, δ), the system of linear equations $x + 2y + 3z = 1$, $3x + 4y + 5z = \mu$, $4x + 4y + 4z = \delta$ is inconsistent?

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

Correct answer: C

Let two points be A(1, −1) and B(0, 2). If a point P(x', y') be such that the area of ΔPAB = 5 sq. units and it lies on the line, $3x + y - 4\lambda = 0$, then a value of λ is:

1. A.1
2. B.4
3. C.−3
4. D.3✓

Correct answer: D

If a, b and c are the greatest values of ${}^{19}C_p$, ${}^{20}C_q$ and ${}^{21}C_r$ respectively, then:

1. A.$\frac{a}{11} = \frac{b}{22} = \frac{c}{21}$
2. B.$\frac{a}{10} = \frac{b}{11} = \frac{c}{42}$
3. C.$\frac{a}{10} = \frac{b}{11} = \frac{c}{21}$
4. D.$\frac{a}{11} = \frac{b}{22} = \frac{c}{42}$✓

Correct answer: D

Let $f: \mathbb{R} \to \mathbb{R}$ be such that for all $x \in \mathbb{R}$ $(2^{1+x} + 2^{1-x})$, $f(x)$ and $(3^x + 3^{-x})$ are in A.P., then the minimum value of f(x) is:

1. A.0
2. B.2
3. C.3✓
4. D.4

Correct answer: C

$\lim_{x\to0}\left(\frac{3x^2+2}{7x^2+2}\right)^{1/x^2}$ is equal to:

1. A.e
2. B.e²
3. C.$\frac1e$
4. D.$\frac{1}{e^2}$✓

Correct answer: D

Let $f(x) = x\cos^{-1}(-\sin|x|)$, $x \in \left[-\frac\pi2, \frac\pi2\right]$, then which of the following is true?

1. A.f is not differentiable at x = 0
2. B.$f'(0) = -\frac\pi2$
3. C.f' is decreasing in $\left(-\frac\pi2, 0\right)$ and increasing in $\left(0, \frac\pi2\right)$✓
4. D.f' is increasing in $\left(-\frac\pi2, 0\right)$ and decreasing in $\left(0, \frac\pi2\right)$

Correct answer: C

If c is a point at which Rolle's theorem holds for the function, $f(x) = \log_e\left(\frac{x^2+\alpha}{7x}\right)$ in the interval [3, 4], where $\alpha \in \mathbb{R}$, then f''(c) is equal to:

1. A.$\frac{\sqrt3}{7}$
2. B.$\frac{1}{12}$✓
3. C.$-\frac{1}{24}$
4. D.$-\frac{1}{12}$

Correct answer: B

If $\int\frac{\cos x\,dx}{\sin^3x(1 + \sin^6x)^{2/3}} = f(x)(1 + \sin^6x)^{1/\lambda} + c$ where c is a constant of integration, then $\lambda f\left(\frac\pi3\right)$ is equal to:

1. A.$\frac98$
2. B.$-\frac98$
3. C.2
4. D.−2✓

Correct answer: D

For a > 0, let the curves $C_1: y^2 = ax$ and $C_2: x^2 = ay$ intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, $C_1$ and $C_2$, and the area of $\Delta OQR = \frac12$, then 'a' satisfies the equation:

1. A.$x^6 + 6x^3 - 4 = 0$
2. B.$x^6 - 12x^3 + 4 = 0$✓
3. C.$x^6 - 12x^3 - 4 = 0$
4. D.$x^6 - 6x^3 + 4 = 0$

Correct answer: B

Let y = y(x) be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx} + \sqrt{1-y^2} = 0$, $|x| < 1$. If $y\left(\frac12\right) = \frac{\sqrt3}{2}$, then $y\left(\frac{-1}{\sqrt2}\right)$ is equal to:

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

Correct answer: B

The locus of a point which divides the line segment joining the point (0, −1) and a point on the parabola, $x^2 = 4y$, internally in the ratio 1 : 2, is:

1. A.$4x^2 - 3y = 2$
2. B.$x^2 - 3y = 2$
3. C.$9x^2 - 12y = 8$✓
4. D.$9x^2 - 3y = 2$

Correct answer: C

Let the line y = mx and the ellipse $2x^2 + y^2 = 1$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $\left(-\frac{1}{3\sqrt2}, 0\right)$ and (0, β), then β is equal to:

1. A.$\frac{\sqrt2}{3}$✓
2. B.$\frac23$
3. C.$\frac{2}{\sqrt3}$
4. D.$\frac{2\sqrt2}{3}$

Correct answer: A

The shortest distance between the lines $\frac{x-3}{3} = \frac{y-8}{-1} = \frac{z-3}{1}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$ is:

1. A.$2\sqrt{30}$
2. B.$3\sqrt{30}$✓
3. C.3
4. D.$\frac72\sqrt{30}$

Correct answer: B

Let the volume of a parallelopiped whose coterminous edges are given by $\vec u = \hat i + \hat j + \lambda\hat k$, $\vec v = \hat i + \hat j + 3\hat k$ and $\vec w = 2\hat i + \hat j + \hat k$ be 1 cu. unit. If θ be the angle between the edges $\vec u$ and $\vec w$, then cos θ can be:

1. A.$\frac57$
2. B.$\frac{7}{6\sqrt6}$
3. C.$\frac{7}{6\sqrt3}$✓
4. D.$\frac{5}{3\sqrt3}$

Correct answer: C

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by p and then reduced by q, where p ≠ 0 and q ≠ 0. If the new mean and new s.d. become half of their original values, then q is equal to:

1. A.−5
2. B.10
3. C.−10
4. D.−20✓

Correct answer: D

Let A and B be two independent events such that $P(A) = \frac13$ and $P(B) = \frac16$. Then, which of the following is TRUE?

1. A.$P(A/(A\cup B)) = \frac14$
2. B.$P(A/B) = \frac23$
3. C.$P(A/B') = \frac13$✓
4. D.$P(A'/B') = \frac13$

Correct answer: C

Let $f(x) = (\sin(\tan^{-1}x) + \sin(\cot^{-1}x))^2 - 1$, $|x| > 1$. If $\frac{dy}{dx} = \frac12\frac{d}{dx}(\sin^{-1}(f(x)))$ and $y(\sqrt3) = \frac\pi6$, then $y(-\sqrt3)$ is equal to:

1. A.$-\frac\pi6$
2. B.$\frac{2\pi}{3}$
3. C.$\frac{5\pi}{6}$✓
4. D.$\frac\pi3$

Correct answer: C

Which one of the following is a tautology?

1. A.$P \wedge (P \vee Q)$
2. B.$P \vee (P \wedge Q)$
3. C.$(P \wedge (P \to Q)) \to Q$✓
4. D.$Q \to (P \wedge (P \to Q))$

Correct answer: C

The least positive value of 'a' for which the equation, $2x^2 + (a - 10)x + \frac{33}{2} = 2a$ has real roots is ________.

Correct answer: 8

The number of all 3 × 3 matrices A, with enteries from the set {−1, 0, 1} such that the sum of the diagonal elements of $AA^T$ is 3, is ________.

Correct answer: 672

The sum $\sum_{k=1}^{20}(1 + 2 + 3 + \ldots + k)$ is ________.

Correct answer: 1540

Let the normal at a point P on the curve $y^2 - 3x^2 + y + 10 = 0$ intersect the y-axis at $\left(0, \frac32\right)$. If m is the slope of the tangent at P to the curve, then |m| is equal to ________.

Correct answer: 4

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is ________.

Correct answer: 490