JEE (ADVANCED) 2019 PAPER-1

PART-1 PHYSICS

SECTION 1 (Maximum Marks: 12)

• This Section contains Four (04) Questions.

• Each question has FOUR options. ONLY ONE of these four options is the correct answer.

• For each question, Choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option is chosen;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered)

Negative Marks : −1 In all other cases.

1. Consider a spherical gaseous cloud of mass density ρ (r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r) = ρ(r)/m is

(A)

(B)

(C)

(D)

Answer: (A)

2. A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V₀. A hole with a small are α4πR² (α<<1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

(A) The potential at the center of the shell is reduced by 2αV₀

(B) The magnitude of electric field at the center of the shell is reduced by

(D) The magnitude of electric at a point, located on all in passing through the hole and shell’s center, on a distance 2R from the center of the spherical shell will be reduced by

Answer: (C)

3. A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as

T(t) = T₀(1+βt^1/4),

where β is a constant with appropriate dimension while T₀ is a constant with dimension of temperature. The heat capacity of the metal is,

(A)

(B)

(C)

(D)

Answer: (A)

4. In a radioactive sample, nuclei either decay into stable nuclei with decay constant 4.5 × 10⁻¹⁰ per year or into stable nuclei with decay constant 0.5 × 10⁻¹⁰ per year. Given that in this sample all the stable nuclei are produced by the nuclei only. In time t × 10⁹ year, if the ratio of the sum of stable nuclei to the radioactive nuclei is 99, the value of t will be, [Given : In 10 = 2.3]

(A) 1.15

(B) 9.2

(D) 4.6

Answer: (B)

SECTION 2 (Maximum Marks : 32)

• This section contains EIGHT (08) questions.

• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen.

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks : -1 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2 marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get −1 mark.

5. A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?

[Surface tension of water = 0.075 N/m, density of water = 1000 kg/m³, take g= 10 m/s²]

(A) The correction in the height of water column raised in the tube, due to weight of water contained in the meniscus, will be different for both cases.

(B) For case II, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be 3.75 cm. (Neglect the weight of the water in the meniscus)

(C) For case I, if the joint is kept at 8 cm above the water surface, the height of water column in the tube will be 7.5 cm. (Neglect the weight of the water in the meniscus)

(D) For case I, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be more than 8.75 cm. (Neglect the weight of the water in the meniscus)

Answer: (A, B, C)

6. A conducting wire of parabolic shape, initially y = x², is moving with velocity in a non-uniform magnetic field as shown in figure. If V₀, B₀, L and β are positive constant and ∆ϕ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

(A)

(B)

(D) |∆ϕ| is proportional to the length of the wire projected on the y-axis.

Answer: (B, C, D)

7. In the circuit shown, initially there is no charge on capacitors and keys S₁ and S₂ are open. The values of the capacitors are C₁ = 10 μF, C₂ = 30 μF and C₃ = C₄ = 80 μF.

Which of the statement(s) is/are correct?

(A) At time t = 0, the key S₁ is closed, the instantaneous current in the closed circuit will be 25 mA.

(B) If key S₁ is kept closed for long time such that capacitors are fully charged, the voltage across the capacitor C₁ will be 4 V.

(C) The key S₁ is kept closed for long time such that capacitors are fully charged. Now key S₂ is closed, at this time, the instantaneous current across 30 Ω resistor ( between points P and Q) will be 0.2 A (round off to 1st decimal place).

(D) If key S₁ is kept closed for long time such that capacitors are fully charged, the voltage difference between points P and Q will be 10 V.

Answer: (A, B)

8. A charged shell of radius R carries a total charge Q. Given Φ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[ϵ₀ is the permittivity of free space]

(A) If h > 2R and r > R then Φ = Q/ϵ₀

(B) If h < 8R/5 and r = 3R/5 then Φ = 0

(D) If h > 2R and r = 4R/5 then Φ = Q/5ϵ₀

Answer: (A, B, C)

9. One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-T) diagram. The correct statement(S) is/are:

[R is the gas constant]

(A) Work done in this thermodynamic cycle

(B) The above thermodynamic cycle exhibits only isochoric and adiabatic processes.

(D) The ratio of heat transfer during processes

Answer: (A, C)

10. A thin convex lens is made of two materials with refractive indices n₁ and n₂, as shown in figure. The radius of curvature of the left and right spherical surface are equal. f is the focal length of the lens when n₁ = n₂ = n. The focal lengths is f + ∆f when n₁ = n and n₂ = n + ∆ Assuming ∆n << (n – 1) and 1 < n < 2, the correct statement(s) is/are,

(A)

(B) For n = 1.5, ∆n = 10⁻³ and f = 20 cm, the value of |∆f| will be 0.02 cm (round off to 2nd decimal place).

(C)

(D) The relation between remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.

Answer: (B, C, D)

11. Let us consider a system of units in which mass and angular momentum are dimensionaless. If length has dimension of L, which of the following statement(s) is/are correct?

(A) The dimension of linear momentum is L⁻¹

(B) The dimension of energy is L⁻²

(D) The dimension of power is L⁻⁵

Answer: (A, B, C)

12. Two identical moving coil galvanometer have 10 Ω resistance full scale deflection at 2 μA current. One of them is converted into a voltmeter of 100 mV full scale reading and the other into an Ammeter of 1mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm’s law experiment with R = 1000 Ω resistor by using an ideal cell. Which of the following statement(s) is/are correct?

(A) The resistance of the Voltmeter will be 100 kΩ

(B) The resistance of the Ammeter will be 0.02 Ω (round off to 2nd decimal place)

(D) If the ideal cell is replaced by a cell having internal resistance of 5 Ω then the measured value of R will be more than 1000 Ω

Answer: (B, C)

SECTION-3 (Maximum Marks : 18)

• This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE.

• For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force where x and y are in meter and α = −1 Nm⁻¹. The work done on the particle by this force will be _____Joule.

Answer: (0.75)

14. A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm² and, length √3 m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30° and 60°, respectively. If elongation in copper wire is (∆l_C) and elongation in steel wire is (∆l_S), then the ratio

Answer: (2)

15. A train S₁, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is______.

[Speed of the sound = 330 m/s]

Answer: ((7.00 to 10.00)

16. A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness The dielectric constant of the mth layer is For a very large N (> 10³), the capacitance C is The value of α will be _______. [ϵ₀ is the permittivity of free space]

Answer: ((0.99 to 1.01))

17. A liquid at 30°C is poured very slowly into a Calorimeter that is at temperature of 110° The boiling temperature of the liquid is 80°C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be 50°C. The ratio of the Latent heat of the liquid to its specific heat will be______°C.

[Neglect the heat exchange with surrounding]

Answer: (270)

18. A planar structure of length L and width W is made of two different optical media of refractive indices n₁ = 1.5 and n₂ = 1.44 as shown in figure. If L >>W, a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle θ is varied, the maximum time taken by a ray to exit the place CD is t × 10⁻⁹ s, where t is ______.

[Speed of light c = 3 × 10⁸ m/s]

Answer: (49 to 51)

JEE (ADVANCED) 2019 PAPER 1

PART-II CHEMISTRY

SECTION 1 (Maximum Marks: 12)

• This Section contains Four (04) Questions.

•Each question has FOUR options. ONLY ONE of these four options is the correct answer.

• For each question, Choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option is chosen;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered)

Negative Marks : −1 In all other cases.

1. The green colour produced in the borax bead test of chromium (III) salt is due to

(A) Cr(BO₂)₃

(B) Cr₂(B₄O₇)₃

(D) CrB

Answer: (A)

2. Calamine, malachite, magnetite and cryolite, respectively, are

(A) ZnSO₄, CuCO₃, Fe₂O₃, AlF₃

(B) ZnSO₄, Cu(OH)₂, Fe₃O₄, Na₃AlF₆

(D) ZnCO₃, CuCO₃, Fe₂O₃, Na₃AlF₆

Answer: (C)

3. Molar conductivity (Λ_m) of aqueous solution of sodium stearate, which behaves as a strong electrolyte, is recorded at varying concentrations (c) of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution?

(A)

(B)

(C)

(D)

Answer: (C)

4. The correct of acid strength of the following carboxylic acids is

(A) III > II > I > IV

(B) I > II > III > IV

(D) II > I > IV > III

Answer: (B)

SECTION 2 (Maximum Marks : 32)

• This section contains EIGHT (08) questions.

• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen.

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks : -1 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2 marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get −1 mark.

5. A tin chloride Q undergoes the following reactions (not balanced)

Q + Cl⁻ → X

Q + Me₃N → Y

Q + CuCl₂ → Z + CuCl

X is an monoanion having pyramidal geometry. Both and Y and Z are neutral compounds. Choose the correct option(s)

(A) The central atom in X is sp³ hyrbridized

(B) There is a coordinate bond in Y

(D) The central atom in Z has one lone pair of electrons

Answer: (A, B)

6. Fusion of MnO₂ with KOH in presence of O₂ produces a salt W. Alkaline solution of W upon electrolytic oxidation yields another salt X. The manganese containing ions present in W and X. respectively, are Y and Z. Correct statement(s) is (are)

(A) In aqueous acidic solution, Y undergoes disproportionation reaction to give Z and MnO₂

(B) Both Y and Z are coloured and have tetrahedral shape

(D) In both Y and Z, π-bonding occurs between p-orbitals of oxygen and d-orbitals of manganese

Answer: (A, B, D)

7. Choose the reaction(s) from the following options, for which the standard enthalpy of reaction in equal to the standard enthalpy of formation.

(A) 2H₂(g) + O₂(g) → 2H₂O (l)

(B) 2C (g) + 3H₂ (g) → C₂H₆ (g)

(C)

(D)

Answer: (C, D)

8. Which of the following statement(s) is(are) correct regarding the root mean square speed (U_rms) and average translational kinetic energy (ε_av) of a molecule in a gas at equilibrium?

(A) U_rms is doubled when its temperature is increased four times

(B) ε_av is doubled when its temperature is increased four times

(D) U_rms inversely proportional to the square root of its molecular mass

Answer: (A, C, D)

9. Each of the following options contains a set of four molecules. Identify the option(s) where all four molecules possess permanent dipole moment at room temperature.

(A) BeCl₂, CO₂, BCl₃, CHCl₃

(B) NO₂, NH₃, POCl₃, CH₃Cl

(D) SO₂, C₆H₅Cl, H₂Se, BrF₅

Answer: (B, D)

10. In the decay sequence,

x₁, x₂, x₃ and x₄ are particles/radiation emitted by the respective isotopes. The correct option(s) is(are)

(A) x₁ will deflect towards negatively charged plate

(B) x₂ is β⁻

(D) Z is an isotope of uranium

Answer: (A, B, D)

11. Which of the following statement(s) is(are) true?

(A) Monosaccharides cannot by hydrolysed to give polyhydroxy aldehydes and ketones

(B) Oxidation of glucose with bromine water gives glutamic acid

(D) The two six-membered cyclic hemiacetal forms of D-(+)-glucose are called anomers

Answer: (A, C, D)

12. Choose the correct option(s) for the following set of reactions

(A)

(B)

(C)

(D)

Answer: (C, D)

SECTION-3 (Maximum Marks : 18)

• This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. Among B₂H₆, B₃N₃H₆, N₂O, N₂O₄, H₂S₂O₃ and H₂S₂O₈, the total number of molecules containing covalent bond between two atoms of the same kind is _____

Answer: (4)

14. At 143 K, the reaction of XeF₄ with O₂F₂ produces a xenon compound Y. The total number of lone pair(s) of electrons present on the whole molecule of Y is_____

Answer: (19)

15. For the following reaction, the equilibrium constant K_c at 298 K is 1.6 × 10¹⁷.

Fe²⁺(aq) + S²⁻ (aq) ⇌ FeS (s)

When equal volumes of 0.06 Fe²⁺ (aq) and 0.2 M S²⁻ (aq) solutions are mixed, the equilibrium concentration of Fe²⁺ (aq) is found to be Y × 10⁻¹⁷ M. The value of Y is __________.

Answer: (8.70 to 9.10)

16. On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapor pressure decreases from 650 mm Hg to 640 mm Hg. The depression of freezing point of benzene (in K) upon addition of the solute is

(Given data : Molar mass and the molal freezing point depression constant of benzene are 78 g mol⁻¹ and 5.12 K kg mol⁻¹, respectively)

Answer: (0.97 to 1.06)

17. Consider the kinetic data given in the following table for the reaction A + B + C → Product.

The rate of the reaction for [A] = 0.15 mol dm⁻³, [B] = 0.25 mol dm⁻³ and [C] = 0.15 mol dm⁻³ is found to be Y × 10⁻⁵ mol dm⁻³ s⁻¹. The value of Y is ______

Answer: (6.70 to 6.80)

18. Schemes 1 and 2 describe the conversion of P to Q and R to S, respectively. Scheme 3 describes the synthesis of T from Q and S. The total number of Br atoms in a molecule of T is ________.

Answer: (4)

JEE (ADVANCED) 2019 PAPER 1

PART-III MATHEMATICS

SECTION 1 (Maximum Marks: 12)

• This Section contains Four (04) Questions.

• Each question has FOUR options. ONLY ONE of these four options is the correct answer.

• For each question, Choose the option corresponding to the correct answer.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct option is chosen;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered)

Negative Marks : −1 In all other cases.

1. Let S be the set of all complex numbers Z satisfying |z – 2 + i| ≥ √ If the complex number z₀ is such that is the maximum of the set then the principal argument of is

(A) −π/2

(B) π/4

(D) 3π/4

Answer: (A)

2. Let

where α = α (θ) and β = β (θ) are real numbers, and I is the 2 × 2 identity matrix. If

α* is the minimum of the set {α(θ): θ ∈ [0, 2π)} and

β* is the minimum of the set {β(θ): θ ∈ [0, 2π)},

then the value of α* + β* is

(A) −37/16

(B) −31/16

(D) −17/16

Answer: (C)

3. A line y = mx + 1 intersects the circle (x – 3)² + (y + 2)² = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate −3/5, then which one of the following options is correct?

(A) −3 ≤ m < −1

(B) 2 ≤ m < 4

(D) 6 ≤ m < 8

Answer: (B)

4. The area of the region {(x, y) ; xy ≤ 8, 1 ≤ y ≤ x²} is

(A)

(B)

(D)

Answer: (A)

SECTION 2 (Maximum Marks : 32)

• This section contains EIGHT (08) questions.

• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) correct answer(s).

• For each question, choose the option(s) corresponding to (all) the correct answer(s).

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +4 If only (all) the correct option(s) is(are) chosen.

Partial Marks : +3 If all the four options are correct but ONLY three options are chosen;

Partial Marks : +2 If three or more options are correct but ONLY two options are chosen and both of which are correct;

Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);

Negative Marks : -1 In all other cases.

• For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct answers, then

choosing ONLY (A), (B) and (D) will get +4 marks;

choosing ONLY (A) and (B) will get +2 marks;

choosing ONLY (A) and (D) will get +2 marks;

choosing ONLY (B) and (D) will get +2 marks;

choosing ONLY (A) will get +1 mark;

choosing ONLY (B) will get +1 mark;

choosing ONLY (D) will get +1 mark;

choosing no option (i.e. the question is unanswered) will get 0 marks; and choosing any other combination of options will get −1 mark.

5. Let α and β be the roots of x² – x – 1 = 0, with α > β. For all positive integers n, define

Then which of the following options is/are correct?

(A) a₁ + a₂ + a₃ + … + a_n = a_n+2 – 1 for all n ≥ 1

(B)

(D)

Answer: (A, B, C)

6. Let and

where a and b are real numbers. Which of the following option is/are correct?

(A) a + b = 3

(B) (adj M)⁻¹ + adj M⁻¹ = −M

(D)

Answer: (A, B, D)

7. There are three bags B₁, B₂ and B₃. The bag B₁ contains 5 red and 5 green balls, B₂ contains 3 red and 5 green balls, and B₃ contains 5 red and 3 green balls. Bags B₁, B₂ and B₃ have probabilities 3/10, 3/10 and 4/10 respectively of being chosen. A bag is selected random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

(A) Probability that the chosen ball is green, given that the selected bag is B₃, equals 3/8

(B) Probability that the chosen ball is green equals 39/80

(D) Probability that the selected bag is B₃ and the chosen ball is green equals 3/10

Answer: (A, B)

8. In a non-right-angled triangle ∆PQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = √3, q = 1, and the radius of the circumcircle of the ∆PQR equals 1, then which of the following options is/are correct?

(A) Length of RS = √7/2

(B) Area of ∆SOE = √3/12

(D) Radius of incircle of

Answer: (A, C, D)

9. Define the collections {E₁, E₂, E₃, …} of ellipses and {R₁, R₂, R₃, …} of rectangles as follows:

R₁ : rectangle of largest area, with sides parallel to the axes, inscribed in E₁ :

E_n : ellipse of largest area inscribed in R_{n – 1}, n > 1;

R_n : rectangle of largest area, with sides parallel to the axes, inscribed in E_n, n > 1.

Then which of the following options is/are correct?

(A) The eccentricities of E₁₈ and E₁₉ are NOT equal

(B) for each positive integer N

(D) The distance of a focus from the centre in E₉ is √5/32

Answer: (B, C)

10. Let f : ℝ → ℝ be given by

Then which of the following options is/are correct?

(A) f is increasing on (−∞, 0)

(B) f ‘ has a local maximum at x = 1

(D) f ‘ is NOT differentiable at x = 1

Answer: (B, C, D)

11. Let Γ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to Γ at a point P intersect the y-axis at Y_P. If PY_P has length 1 for each point P on Γ, then which of the following options is/are correct?

(A)

(B)

(C)

(D)

Answer: (A, B)

12. Let L₁ and L₂ denote the lines

respectively. If L₃ is a line which is perpendicular to both L₁ and L₂ and cuts both of them, then which of the following options describe(s) L₃?

(A)

(B)

(C)

(D)

Answer: (A, B, C)

SECTION-3 (Maximum Marks : 18)

• This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE.

•For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual numeric keypad in the place designated to enter the answer. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

• Answer to each question will be evaluated according to the following marking scheme:

Full Marks : +3 If ONLY the correct numerical value is entered;

Zero Marks : 0 In all other cases.

13. Let ω ≠ 1 be a cube root of unity. Then the minimum of the set

{|a + bω + cω²|² : a, b, c distinct non-zero integers} equals ___

Answer: (3)

14. Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If

AP(1; 3) ∩ AP(2; 5) ∩ AP(3; 7) = AP(a; d)

then a + d equals ______

Answer: (157)

15. Let S be the sample space of all 3 × 3 matrices with entries from the set {0, 1}. Let the events E₁ and E₂ be given by

E₁ = {A ∈ S : det A = 0} and

E₂ = {A ∈ S : sum of entries of A is 7}.

If a matrix is chosen at random from S, then the conditional probability P(E₁|E₂) equals____

Answer: (0.5)

16. Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let Γ_A and Γ_B be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles Γ_A and Γ_B such that both the circles are on the same side of T. If C is the point of intersection of T and line passing through A and B, then the length of the line segment AC is _____

Answer: (10)

17. If

then 27 I² equals ______

Answer: (4)

18. Three lines are given by

Let the lines cut the plane x + y +z = 1 at the points A, B and C respectively. If the area of the triangle ABC is ∆ then the value of (6∆)² equals _____

Answer: (0.75)

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