JEE Advanced 2021 Paper 2

One end of a horizontal uniform beam of weight 𝑊 and length 𝐿 is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height 𝐿 above the hinge at point O. A block of weight 𝛼𝑊 is attached at the point P of the beam, as shown in the figure (not to scale) . The rope can sustain a maximum tension of (2 √2)𝑊. Which of the following statement(s) is(are) correct?

1. The vertical component of reaction force at O does not depend on 𝛼
2. The horizontal component of reaction force at O is equal to 𝑊 for 𝛼= 0.5
3. The tension in the rope is 2 𝑊 for 𝛼= 0.5
4. The rope breaks if 𝛼> 1.5

A source, approaching with speed 𝑢 towards the open end of a stationary pipe of length 𝐿, is emitting a sound of frequency 𝑓𝑠. The farther end of the pipe is closed. The speed of sound in air is 𝑣 and 𝑓0 is the fundamental frequency of the pipe. For which of the following combination(s) of 𝑢 and 𝑓𝑠, will the sound reaching the pipe lead to a resonance?

1. 𝑢=0.8𝑣 and 𝑓𝑠=𝑓0
2. 𝑢=0.8𝑣 and 𝑓𝑠=2𝑓0
3. 𝑢=0.8𝑣 and 𝑓𝑠=0.5𝑓0
4. 𝑢=0.5𝑣 and 𝑓𝑠=1.5𝑓0

For a prism of prism angle 𝜃=60°, the refractive indices of the left half and the right half are , respectively, 𝑛1 and 𝑛2 (𝑛2≥𝑛1) as shown in the figure . The angle of incidence 𝑖 is chosen such that the incident light ray s will have minimum deviation if 𝑛1=𝑛2=𝑛=1.5. For the case of unequal refractive indices , 𝑛1=𝑛 and 𝑛2=𝑛+∆𝑛 (where ∆𝑛≪𝑛), the angle of emergence 𝑒=𝑖+∆𝑒. Which of the following statement(s) is(are) correct?

1. The value of ∆𝑒 (in radians ) is greater than that of ∆𝑛
2. ∆𝑒 is proportional to Δ𝑛
3. Δ𝑒 lies between 2.0 and 3.0 milli radians , if ∆𝑛=2.8×10−3
4. Δ𝑒 lies between 1.0 and 1.6 milli radians , if ∆𝑛=2.8×10−3

A physical quantity 𝑆⃗ is defined as 𝑆⃗=(𝐸⃗⃗×𝐵⃗⃗)/𝜇0, where 𝐸⃗⃗ is electric field, 𝐵⃗⃗ is magnetic field and 𝜇0 is the permeability of free space. The dimensions of 𝑆⃗ are the same as the dimensions of which of the following quantity(ies) ?

1. 𝐸𝑛𝑒𝑟𝑔𝑦 𝐶ℎ𝑎𝑟𝑔𝑒 × 𝐶𝑢𝑟𝑟𝑒𝑛𝑡
2. 𝐹𝑜𝑟𝑐𝑒 𝐿𝑒𝑛𝑔𝑡 ℎ × 𝑇𝑖𝑚𝑒
3. 𝐸𝑛𝑒𝑟𝑔𝑦 𝑉𝑜𝑙𝑢𝑚𝑒
4. 𝑃𝑜𝑤𝑒𝑟 𝐴𝑟𝑒𝑎

A heavy nucleus 𝑁, at rest, undergoes fission 𝑁→𝑃+𝑄, where 𝑃 and 𝑄 are two lighter nuclei. Let 𝛿=𝑀𝑁−𝑀𝑃−𝑀𝑄, where 𝑀𝑃, 𝑀𝑄 and 𝑀𝑁 are the masses of 𝑃, 𝑄 and 𝑁, respectively. 𝐸𝑃 and 𝐸𝑄 are the kinetic energies of 𝑃 and 𝑄, respectively. The speeds of 𝑃 and 𝑄 are 𝑣𝑃 and 𝑣𝑄, respective ly. If 𝑐 is the speed of light, which of the following statement(s) is(are) correct?

1. 𝐸𝑃+𝐸𝑄=𝑐2𝛿
2. 𝐸𝑃=(𝑀𝑃 𝑀𝑃+𝑀𝑄)𝑐2𝛿
3. 𝑣𝑃 𝑣𝑄=𝑀𝑄 𝑀𝑃
4. The magnitude of momentum for 𝑃 as well as 𝑄 is 𝑐√2𝜇𝛿, where 𝜇=𝑀𝑃𝑀𝑄 (𝑀𝑃+𝑀𝑄)

Two concentric circular loops, one of radius 𝑅 and the other of radius 2𝑅, lie in the xy-plane with the origin as their common center, as shown in the figure . The smaller loop carries current 𝐼1 in the anti -clockwise direction and the larger loop carries current 𝐼2 in the clockwise direction, with 𝐼2>2𝐼1. 𝐵⃗⃗(𝑥,𝑦) denotes the magnetic field at a point (𝑥,𝑦) in the xy-plane. Which of the following statement(s) is(are) correct?

1. 𝐵⃗⃗(𝑥,𝑦) is perpendicular to the xy-plane at any point in the plane
2. |𝐵⃗⃗(𝑥,𝑦)| depends on x and y only through the radial distance 𝑟=√𝑥2+𝑦2
3. |𝐵⃗⃗ (𝑥,𝑦)| is non -zero at all points for 𝑟<𝑅
4. 𝐵⃗⃗ (𝑥,𝑦) points normally outward from the xy-plane for all the points between the two loops Question Stem for Question Nos. 7 and 8 Question Stem A soft plastic bottle, filled with water of density 1 gm/cc, carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test -tube has a mass of 5 gm, and it is made of a thick glass of density 2.5 gm/cc. Initially the bottle is sealed at atmospheric pressure 𝑝0=105 Pa so that the volume of the trapped air is 𝑣0=3.3 cc. When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found tha t the test tube begins to sink at pressure 𝑝0+Δ𝑝 without changing its orientation. At this pressure, the volume of the trapped air is 𝑣0−Δ𝑣. Let Δ𝑣=𝑋 cc and Δ𝑝= 𝑌×103 Pa. SECTION 2  This section contains THREE (03) question stems.  There are TWO (02) questions corresponding to each question stem.  The answer to each question is a NUMERICAL VALUE .  For each question , enter the correct numerical value corresponding to the answer in the designated place using the mouse and the on -screen virtual numeric keypad.  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 :

The value of 𝑋 is ___ .

Numerical answer type

The value of 𝑌 is ___. Question Stem for Question Nos. 9 and 10 Question Stem A pendulum consists of a bob of mass 𝑚=0.1 kg and a massless inextensible string of length 𝐿=1.0 m. It is suspended from a fixed point at height 𝐻=0.9 m above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse 𝑃= 0.2 kg-m/s is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts o ff is 𝐽 kg-m²/s. The kinetic energy of the pendulum just after the lift - off is 𝐾 Joules.

Numerical answer type

The value of 𝐽 is ___ .

Numerical answer type

The value of 𝐾 is ___. Question Stem for Question Nos. 11 and 12 Question Stem In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C 𝜇𝐹 across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit . Take rms values of the voltages. The magnitude of the phase - angle (in degrees) between the current and the supply voltage is 𝜑. Assume, 𝜋√3≈5.

Numerical answer type

The value of C is _ __.

The value of 𝜑 is ___. SECTION 3  This section contains TWO (02) paragraphs . Based on each paragraph, there are TWO (02) questions .  Each question has FOUR options

1. ,
2. ,
3. and
4. . ONLY ONE of these four options is the correct answer.  For eac h question, choose the option correspond ing to the correct answer.  Answer to each question will be evaluated according to the following marking scheme :

In order to measure the internal resistance 𝑟1 of a cell of emf 𝐸, a meter bridge of wire resistance 𝑅0=50 Ω, a resistance 𝑅0/2, another cell of emf 𝐸/2 (internal resistance 𝑟) and a galvanometer G are used in a circuit, as shown in the figure. If the null point is found at 𝑙=72 cm, then the value of 𝑟1=___ Ω.

The distance between two stars of masses 3𝑀𝑆 and 6𝑀𝑆 is 9𝑅. Here 𝑅 is the mean distance between the centers of the Earth and the Sun, and 𝑀𝑆 is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period 𝑛𝑇, where 𝑇 is the period of Earth’s revolution around the Sun. The value of n is ___.

Numerical answer type

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals 𝑃, 𝑄 and 𝑅 are 𝐸𝑃, 𝐸𝑄 and 𝐸𝑅, respectively, and they are related by 𝐸𝑃=2𝐸𝑄=2𝐸𝑅. In this experiment, the same source of monochromatic light is used for metals 𝑃 and 𝑄 while a different source of monochromatic light is used for the metal 𝑅. The work functions for metals 𝑃, 𝑄 and 𝑅 are 4.0 eV, 4.5 eV and 5.5 eV, respectively. The energy of the incident photon used for metal 𝑅, in eV, is ___.

Numerical answer type

The reaction sequence(s) that would lead to o-xylene as the major product is(are)

1. 1. NaNO2/HCl 273 K 2. CuCN 3. DIBAL-H then H3O+ 4. N2H4, KOH heatMe NH2
2. 1. Mg, CO₂, H3O+ 2. SOCl2 3. H₂, Pd-BaSO₄ 4. Zn-Hg, HClMe Br
3. Me1. i. BH3 ii. H2O₂, NaOH 2. PBr3 3. Zn, dil. HCl
4. 1. O3, Zn/H₂O 2. N2H4, KOH, heat SECTION 1  This section contains SIX (06) questions.  Each question has FOUR options (A),
5. ,
6. and
7. . ONE OR MORE THAN ONE of these four option(s) is (are) c orrect 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 :  For example, in a question, if (A),
8. and
9. are the ONLY three options corresponding to correct answers, then choosing ONLY (A),
10. and
11. will get +4 marks; choosing ONLY (A) and
12. will get +2 marks; choosing ONLY (A) and
13. will get +2marks; choosing ONLY
14. and
15. will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY
16. will get +1 mark; choosing ONLY
17. will get +1 mark; choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks .

Correct option(s) f or the following sequence of reactions is(are) Br2P1. Q 2. H₂, Pd/CRCHCl3 KOHS foul smellingV T1. NH₃ 2. heatUWi. KMnO₄ KOH, heat ii. H3O+PhCH3 light

1. Q = KNO 2, W = LiAlH 4
2. R = benzenamine, V = KCN
3. Q = AgNO 2, R = phenylmethanamine
4. W = LiAlH 4, V = AgCN

For the following reaction 2𝐗+𝐘𝑘→𝐏 the rate of reaction is 𝑑[𝐏] 𝑑𝑡=𝑘[𝐗]. Two moles of X are mixed with one mole of Y to make 1.0 L of solution. At 50 s, 0.5 mole of Y is left in the reaction mixture . The correct statement(s) about the reaction is(are) (Use: ln 2 = 0.693)

1. The rate constant, k, of the reaction is 13.86×10−4 s–1.
2. Half -life of X is 50 s.
3. At 50 s, −𝑑[𝐗] 𝑑𝑡 = 13.86×10−3 mol L–1 s–1.
4. At 100 s, −𝑑[𝐘] 𝑑𝑡 = 3.46×10−3 mol L–1 s–1.

Some standard electrode potentials at 298 K are given below: Pb2+/Pb 0.13 V Ni2+/Ni 0.24 V Cd2+/Cd 0.40 V Fe2+/Fe 0.44 V To a solution containing 0.001 M of X2+ and 0.1 M of Y2+, the metal rods X and Y are inserted (at 298 K ) and connected by a conducting wire. This resulted in dissolution of X. The correct combination(s) of X and Y, respectively, is(are) (Given: Gas constant, R = 8.314 J K–1 mol–1, Faraday constant, F = 96500 C mol–1)

1. Cd and Ni
2. Cd and Fe
3. Ni and Pb
4. Ni and Fe

The pair(s) of complexes wherein both exhibit tetrahedral geometry is(are) (Note: py = pyridine Given: Atomic numbers of Fe, Co , Ni and Cu are 26, 27, 28 and 29, respectively )

1. [FeCl 4]– and [Fe(CO) 4]2–
2. [Co(CO) 4]– and [CoCl 4]2–
3. [Ni(CO) 4] and [Ni(CN) 4]2–
4. [Cu(py) 4]+ and [ Cu(CN)4]3–

The correct statement(s) related to oxoacids of phosphorous is(are)

1. Upon heating, H3PO 3 undergoes disproportionation reaction to produce H 3PO 4 and PH 3.
2. While H 3PO 3 can act as reducing agent, H 3PO 4 cannot .
3. H3PO 3 is a monobasic acid.
4. The H atom of P–H bond in H 3PO 3 is not ionizable in water . Question Stem for Question Nos. 7 and 8 Question Stem At 298 K, the limiting molar conductivity of a weak monobasic acid is 4×102 S cm² mol1. At 298 K, for an aqueous solution of the acid the degree of dissociation is  and the molar conductivity is 𝐲×102 S cm² mol1. At 298 K, upon 20 times dilution with water, the molar conductivity of the solution becomes 3𝐲×102 S cm² mol1.

The value of  is ___.

Numerical answer type

The value of y is ___. Question Stem for Question Nos. 9 and 10 Question Stem Reaction of x g of Sn with HCl quantitatively produced a salt. Entire amount of the salt reacted with y g of nitrobenzene in the presence of required amount of HCl to produce 1.29 g of an organic salt (quantitatively) . (Use Molar masses (in g mol–1) of H, C, N, O, Cl and Sn as 1, 12, 14, 16, 35 and 119, respectively). SECTION 2  This section contains THREE (03) question stems.  There are TWO (02) questions corresponding to each question stem.  The answer to each question is a NUMERICAL VALUE .  For each question , enter the correct numerical value corresponding to the answer in the designated place using the mouse and the on -screen virtual numeric k eypad.  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 :

Numerical answer type

The value of x is ___.

Numerical answer type

The value of y is ___. Question Stem for Question Nos. 11 and 12 Question Stem A sample (5.6 g) containing iron is completely dissolved in cold dilute HCl to prepare a 250 mL of solution. Titration of 25.0 mL of this solution requires 12.5 mL of 0.03 M KMnO 4 solution to reach the end point. Number of moles of Fe2+ present in 250 mL solution is x × 10−2 (consider complete dissolution of FeCl 2). The amount of iron present in the sample is y% by weight. (Assume: KMnO 4 reacts only with Fe2+ in the solution Use: Molar mass of iron as 56 g mol⁻¹)

Numerical answer type

The value of x is ___.

Numerical answer type

The value of y is ___. SECTION 3  This section contains TWO (02) paragraphs . Based on each paragraph, there are TWO (02) questions.  Each question has FOUR options

1. ,
2. ,
3. and
4. . 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 :

One mole of an ideal gas at 900 K, undergoes two reversible processes, I followed by II, as shown below. If the work done by the gas in the two processes are same, the value of ln𝑉3 𝑉2 is ___. (U: internal energy, S: entropy, p: pressure, V: volume, R: gas constant) (Given: molar heat capacity at constant volume, 𝐶𝑉,𝑚 of the gas is 5 2𝑅)

Numerical answer type

Consider a helium (He) atom that absorbs a photon of wavelength 330 nm. The change in the velocity (in cm s–1) of He atom after the photon absorption is ___. (Assume: Momentum is conserved when photon is absorbed. Use: Planck constant = 6.6 ×10–34 J s, Avogadro number = 6 ×1023 mol–1, Molar mass of He = 4 g mol–1)

Numerical answer type

Ozonolysis of ClO 2 produces an oxide of chlorine. The average oxidation state of chlorine in this oxide is ___.

Numerical answer type

Let 𝑆1={(𝑖,𝑗,𝑘)∶ 𝑖,𝑗,𝑘∈{1,2,…,10}}, 𝑆2={(𝑖,𝑗)∶ 1≤𝑖<𝑗+2≤10, 𝑖,𝑗∈{1,2,…,10}}, 𝑆3={(𝑖,𝑗,𝑘,𝑙)∶ 1≤𝑖<𝑗<𝑘<𝑙, 𝑖,𝑗,𝑘,𝑙∈{1,2,…,10}} and 𝑆4={(𝑖,𝑗,𝑘,𝑙)∶ 𝑖,𝑗,𝑘 and 𝑙 are distinct elements in {1,2,…,10}}. If the total number of elements in the set 𝑆𝑟 is 𝑛𝑟, 𝑟=1,2,3,4, then which of the following statement s is (are) TRUE ?

1. 𝑛1=1000
2. 𝑛2=44
3. 𝑛3=220
4. 𝑛4 12=420 SECTION 1  This section contains SIX (06) questions .  Each question has FOUR options (A),
5. ,
6. and
7. . ONE OR MORE THAN ONE of these four option(s) is (are) c orrect 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 :  For example, in a question, if (A), (B ) and
8. are the ONLY three options corresponding to correct answers , then choosing ONLY (A),
9. and
10. will get +4 marks; choosing ONLY (A) and ( B) will get +2 marks; choosing ONLY (A) and ( D) will get +2marks; choosing ONLY (B ) and ( D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY
11. will get +1 mark; choosing ONLY
12. will get +1 mark; choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option (s) will get −2 marks .

Consider a triangle 𝑃𝑄𝑅 having sides of lengths 𝑝,𝑞 and 𝑟 opposite to the angles 𝑃,𝑄 and 𝑅, respectively. Then which of the following statement s is (are) TRUE ?

1. cos𝑃≥ 1−𝑝2 2𝑞𝑟
2. cos𝑅≥(𝑞−𝑟 𝑝+𝑞)cos𝑃+(𝑝−𝑟 𝑝+𝑞)cos𝑄
3. 𝑞+𝑟 𝑝<2√sin𝑄sin𝑅 sin𝑃
4. If 𝑝<𝑞 and 𝑝<𝑟, then cos𝑄>𝑝 𝑟 and cos𝑅>𝑝 𝑞

Let 𝑓:[−𝜋 2,𝜋 2]→ℝ be a continuous function such that 𝑓(0)=1 and ∫𝑓(𝑡)𝑑𝑡=0𝜋 3 0 Then which of the following statement s is (are) TRUE ?

1. The equation 𝑓(𝑥)−3cos3𝑥=0 has at least one solution in (0,𝜋 3)
2. The equation 𝑓(𝑥)−3sin3𝑥=−6 𝜋 has at least one solution in (0,𝜋 3)
3. lim 𝑥→0𝑥∫𝑓(𝑡)𝑑𝑡𝑥 0 1−𝑒𝑥2=−1
4. lim 𝑥→0sin𝑥∫𝑓(𝑡)𝑑𝑡𝑥 0 𝑥2=−1

For any real numbers 𝛼 and 𝛽, let 𝑦𝛼,𝛽(𝑥),𝑥∈ℝ, be the solution of the differential equation 𝑑𝑦 𝑑𝑥+𝛼𝑦=𝑥𝑒𝛽𝑥, 𝑦(1)=1. Let 𝑆={𝑦𝛼,𝛽 (𝑥)∶ 𝛼,𝛽∈ℝ}. Then which of the following function s belong(s) to the set 𝑆 ?

1. 𝑓(𝑥)=𝑥2 2𝑒−𝑥+(𝑒−1 2)𝑒−𝑥
2. 𝑓(𝑥)=−𝑥2 2𝑒−𝑥+(𝑒+1 2)𝑒−𝑥
3. 𝑓(𝑥)=𝑒𝑥 2(𝑥−1 2)+(𝑒−𝑒2 4)𝑒−𝑥
4. 𝑓(𝑥)=𝑒𝑥 2(1 2−𝑥)+(𝑒+𝑒2 4)𝑒−𝑥

Let 𝑂 be the origin and 𝑂𝐴⃗⃗⃗⃗⃗ =2î+2 ĵ+k̂, 𝑂𝐵⃗⃗⃗⃗⃗ = î−2 ĵ+2k̂ and 𝑂𝐶⃗⃗⃗⃗⃗ =1 2(𝑂𝐵⃗⃗⃗⃗⃗ −𝜆𝑂𝐴⃗⃗⃗⃗⃗ ) for some 𝜆>0. If |𝑂𝐵⃗⃗⃗⃗⃗ × 𝑂𝐶⃗⃗⃗⃗⃗ |=9 2 , then which of the following statement s is (are) TRUE ?

1. Projection of 𝑂𝐶⃗⃗⃗⃗⃗ on 𝑂𝐴⃗⃗⃗⃗⃗ is −3 2
2. Area of the triangle 𝑂𝐴𝐵 is 9 2
3. Area of the triangle 𝐴𝐵𝐶 is 9 2
4. The acute a ngle between the diagonals of the parallelogram with adjacent sides 𝑂𝐴⃗⃗⃗⃗⃗ and 𝑂𝐶⃗⃗⃗⃗⃗ is 𝜋 3

Let 𝐸 denote the parabola 𝑦2=8𝑥. Let 𝑃=(−2,4), and let 𝑄 and 𝑄′ be two distinct points on 𝐸 such that the lines 𝑃𝑄 and 𝑃𝑄′ are tangents to 𝐸. Let 𝐹 be the focus of 𝐸. Then which of the following statements is (are) TRUE ?

1. The triangle 𝑃𝐹𝑄 is a right -angled triangle
2. The triangle 𝑄𝑃𝑄′ is a right -angled triangle
3. The distance between 𝑃 and 𝐹 is 5√2
4. 𝐹 lies on the line joining 𝑄 and 𝑄′ Question Stem for Question Nos. 7 and 8 Question Stem Consider the region 𝑅={(𝑥,𝑦)∈ ℝ× ℝ∶𝑥≥0 and 𝑦2≤4−𝑥}. Let ℱ be the family of all circles that are contained in 𝑅 and have centers on the 𝑥-axis. Let 𝐶 be the circle that has largest radius among th e circles in ℱ. Let (𝛼,𝛽) be a point where the circle 𝐶 meets the curve 𝑦2=4−𝑥.

The radius of the circle 𝐶 is ___ .

Numerical answer type

The value of 𝛼 is ___ . SECTION 2  This section contains THREE (03) question stems.  There are TWO (02) questions corresponding to each question stem.  The answer to each question is a NUMERICAL VALUE .  For each question , enter the correct numerical value corresponding to the answer in the designated place using the mouse and the on -screen virtual numeric keypad.  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 : Question Stem for Question Nos. 9 and 1 0 Question Stem Let 𝑓1:(0,∞)→ℝ and 𝑓2:(0,∞)→ℝ be defined by 𝑓1(𝑥)= ∫∏(𝑡−𝑗)𝑗 𝑑𝑡,𝑥>021 𝑗=1𝑥 0 and 𝑓2(𝑥)=98(𝑥−1)50−600(𝑥−1)49+2450, 𝑥>0, where, for any positive integer 𝑛 and real numbers 𝑎1,𝑎2,…,𝑎𝑛, ∏𝑎𝑖𝑛 𝑖=1 denotes the product of 𝑎1,𝑎2,…,𝑎𝑛. Let 𝑚𝑖 and 𝑛𝑖, respecti vely, denote the number of points of local minima and the number of points of local maxima of function 𝑓𝑖, 𝑖=1,2, in the interval (0,∞).

Numerical answer type

The value of 2𝑚1+3𝑛1+𝑚1𝑛1 is ___ .

Numerical answer type

The value of 6𝑚2+4𝑛2+8𝑚2𝑛2 is ___. Question Stem for Question Nos. 11 and 12 Question Stem Let 𝑔𝑖:[𝜋 8,3𝜋 8]→ℝ,𝑖=1,2, and 𝑓:[𝜋 8,3𝜋 8]→ℝ be functions such that 𝑔1(𝑥)=1, 𝑔2(𝑥)=|4𝑥−𝜋| and 𝑓(𝑥)=sin2𝑥, for all 𝑥∈[𝜋 8,3𝜋 8] Define 𝑆𝑖=∫𝑓(𝑥)⋅𝑔𝑖(𝑥) 𝑑𝑥, 𝑖=1,23𝜋 8 𝜋 8

Numerical answer type

The value of 16𝑆1 𝜋 is ___ .

Numerical answer type

The value of 48𝑆2 𝜋2 is ___ . SECTION 3  This section contains TWO (02) paragraphs . Based on each paragraph, there are TWO (02) questions .  Each question has FOUR options

1. ,
2. ,
3. and
4. . ONLY ONE of these four options is the correct answer.  For eac h question, choose the option correspond ing to the correct answer.  Answer to each question will be evaluated according to the following marking scheme :

A number is chosen at random from the set {1,2,3,…,2000}. Let 𝑝 be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500𝑝 is ___ .

Numerical answer type

Let E be the ellipse 𝑥2 16+𝑦2 9=1. For any three distinct points 𝑃,𝑄 and 𝑄′ on E, let 𝑀(𝑃,𝑄) be the mid -point of the line segment joining P and 𝑄, and 𝑀(𝑃,𝑄′) be the mid-point of the line segment joining P and 𝑄′. Then the maximum possible value of the distance between 𝑀(𝑃,𝑄) and 𝑀(𝑃,𝑄′), as 𝑃,𝑄 and 𝑄′ vary on 𝐸, is ___ .

Numerical answer type

For any real number 𝑥, let [𝑥] denote the largest integer less than or equal to 𝑥. If 𝐼=∫ [√10𝑥 𝑥+1 ] 𝑑𝑥10 0 , then the value of 9𝐼 is ___ .

Numerical answer type