JEE Advanced 2022 Paper 2

Let 𝛼 and 𝛽 be real numbers such that −𝜋 4<𝛽< 0 <𝛼<𝜋 4 . If sin (𝛼+𝛽 )=1 3 and cos(𝛼−𝛽 )=2 3 , then the greatest integer less than or equal to ( sin𝛼 cos𝛽+cos𝛽 sin𝛼+cos𝛼 sin𝛽+sin𝛽 cos𝛼 )2 is _____________.

Numerical answer type

If 𝑦(𝑥) is the solution of the differential equation 𝑥𝑑𝑦− (𝑦2−4𝑦 )𝑑𝑥= 0 for 𝑥> 0, 𝑦 (1)=2, and the slope of the curve 𝑦=𝑦(𝑥) is never zero, then the value of 10 𝑦 (√2 ) is _____________.

Numerical answer type

The greatest integer less than or equal to ∫log 2(𝑥3+1)2 1𝑑𝑥 + ∫ (2𝑥−1)1 3log29 1𝑑𝑥 is _____________.

Numerical answer type

The product of all positive real values of 𝑥 satisfying the equation 𝑥(16(log5𝑥)3−68 log5𝑥)=5−16 is __________ . SECTION 1 (Maximum marks: 24) Mathematics

Numerical answer type

If 𝛽= lim 𝑥→0𝑒𝑥3−(1−𝑥3)1 3+ ((1−𝑥2)1 2−1) sin𝑥 𝑥sin2𝑥 , then the value of 6𝛽 is ___________.

Numerical answer type

Let 𝛽 be a real number. Consider the matrix 𝐴=( 𝛽01 2 1 −2 3 1 −2) . If 𝐴7−(𝛽−1)𝐴6−𝛽𝐴5 is a singular matrix, then the value of 9𝛽 is ___________.

Numerical answer type

Consider the hyperbola 𝑥2 100−𝑦2 64=1 with foci at 𝑆 and 𝑆 1, where 𝑆 lies on the positive x-axis. Let 𝑃 be a point on the hyperbola, in the first quadrant. Let ∠𝑆𝑃𝑆 1=𝛼, with 𝛼<𝜋 2 . The straight line passing through the point 𝑆 and having the same slope as that of the tangent at 𝑃 to the hyperbola, intersects the straight line 𝑆 1𝑃 at 𝑃1. Let 𝛿 be the distance of 𝑃 from the straight line 𝑆𝑃 1, and 𝛽=𝑆 1𝑃. Then the greatest integer less than or equal to βδ 9 sin𝛼 2 is _____________.

Numerical answer type

Consider the functions 𝑓,𝑔∶ℝ→ℝ defined by 𝑓(𝑥)=𝑥2+5 12 and 𝑔 (𝑥)={ 2(1−4|𝑥| 3), |𝑥|≤3 4 , 0 , |𝑥| >3 4 . If 𝛼 is the area of the region {(𝑥,𝑦)∈ℝ×ℝ∶ |𝑥|≤3 4 , 0≤𝑦≤min {𝑓(𝑥),𝑔(𝑥)} } , then the value of 9𝛼 is _____________.

Numerical answer type

Let 𝛼=∑sin2𝑘(𝜋 6) ∞ 𝑘=1. Let 𝑔∶[0,1]→ℝ be the function defined by 𝑔(𝑥)=2𝛼𝑥+2𝛼(1−𝑥) . Then, which of the following statements is/are TRUE ?

1. The minimum value of 𝑔 (𝑥) is 27 6
2. The maximum value of 𝑔 (𝑥) is 1+21 3
3. The function 𝑔 (𝑥) attains its maximum at more than one point
4. The function 𝑔 (𝑥) attains its minimum at more than one point

Let 𝑧̅ denote the complex conjugate of a complex number 𝑧. If 𝑧 is a non-zero complex number for which both real and imaginary parts of ( 𝑧̅ )2+1 𝑧2 are integers, then which of the following is/are possible value(s) of |𝑧| ?

1. (43+3√205 2)1 4
2. (7+√33 4)1 4
3. (9+√65 4)1 4
4. (7+√13 6)1 4

Let 𝐺 be a circle of radius 𝑅> 0. Let 𝐺 1,𝐺2, … ,𝐺𝑛 be 𝑛 circles of equal radius 𝑟> 0. Suppose each of the 𝑛 circles 𝐺 1,𝐺2, … ,𝐺𝑛 touches the circle 𝐺 externally. Also, for 𝑖=1,2, … ,𝑛−1, the circle 𝐺𝑖 touches 𝐺 𝑖+1 externally, and 𝐺 𝑛 touches 𝐺 1 externally. Then, which of the following statements is/are TRUE ?

1. If 𝑛=4, then ( √2−1)𝑟<𝑅
2. If 𝑛=5, then 𝑟<𝑅
3. If 𝑛=8, then ( √2−1) 𝑟<𝑅
4. If 𝑛=12, then √2 (√3+1) 𝑟>𝑅

Let 𝑖̂, 𝑗̂ and 𝑘 ̂ be the unit vectors along the three positive coordinate axes. Let 𝑎 ⃗=3𝑖̂+𝑗̂−𝑘 ̂ , 𝑏⃗⃗=𝑖̂+𝑏2𝑗̂+𝑏3𝑘̂ , 𝑏 2,𝑏3∈ℝ , 𝑐⃗=𝑐1𝑖̂+𝑐2𝑗̂+𝑐3𝑘̂ , 𝑐1,𝑐2,𝑐3∈ℝ be three vectors such that 𝑏 2𝑏3> 0, 𝑎 ⃗⋅ 𝑏⃗⃗= 0 and ( 0−𝑐3 𝑐2 𝑐3 0−𝑐1 −𝑐2 𝑐1 0)(1 𝑏2 𝑏3)=( 3−𝑐1 1−𝑐2 −1−𝑐3) . Then, which of the following is/are TRUE ?

1. 𝑎 ⃗⋅𝑐⃗ = 0
2. 𝑏⃗⃗⋅𝑐⃗= 0
3. |𝑏⃗⃗| > √10
4. |𝑐⃗|≤√11

For 𝑥∈ℝ, let the function 𝑦(𝑥) be the solution of the differential equation 𝑑𝑦 𝑑𝑥+12𝑦= cos(𝜋 12𝑥), 𝑦 (0)= 0 . Then, which of the following statements is/are TRUE ?

1. 𝑦(𝑥) is an increasing function
2. 𝑦(𝑥) is a decreasing function
3. There exists a real number 𝛽 such that the line 𝑦=𝛽 intersects the curve 𝑦=𝑦(𝑥) at infinitely many points
4. 𝑦(𝑥) is a periodic function SECTION 3 (Maximum marks: 12)

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?

1. 21816
2. 85536
3. 12096
4. 156816

If 𝑀=( 5 2 3 2 −3 2−1 2 ) , then which of the following matrices is equal to 𝑀2022 ?

1. ( 3034 3033 −3033 −3032 )
2. ( 3034 −3033 3033 −3032 )
3. ( 3033 3032 −3032 −3031 )
4. ( 3032 3031 −3031 −3030 )

Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls. A ball is chosen randomly from Box-I; call this ball 𝑏. If 𝑏 is red then a ball is chosen randomly from Box-II, if 𝑏 is blue then a ball is chosen randomly from Box-III, and if 𝑏 is green then a ball is chosen randomly from Box-IV. The conditional probability of the event ‘one of the chosen balls is white’ given that the event ‘at least one of the chosen balls is green’ has happened, is equal to

1. 15 256
2. 3 16
3. 5 52
4. 1 8

For positive integer 𝑛, define 𝑓(𝑛)=𝑛+16+5𝑛−3𝑛2 4𝑛+3𝑛2+32+𝑛−3𝑛2 8𝑛+3𝑛2+48−3𝑛−3𝑛2 12𝑛+3𝑛2+ ⋯ +25𝑛−7𝑛2 7𝑛2 . Then, the value of lim𝑛→∞𝑓(𝑛) is equal to

1. 3+4 3log 𝑒7
2. 4−3 4log 𝑒(7 3)
3. 4−4 3log 𝑒(7 3)
4. 3+3 4log 𝑒7

Concentration of H 2SO 4 and Na 2SO 4 in a solution is 1 M and 1.8 × 10–2 M, respectively. Molar solubility of PbSO 4 in the same solution is X × 10–Y M (expressed in scientific notation). The value of Y is _________. [Given: Solubility product of PbSO 4 (K sp) = 1.6 × 10–8. For H 2SO 4, Ka1 is very large and Ka2 = 1.2 × 10–2]

Numerical answer type

An aqueous solution is prepared by dissolving 0.1 mol of an ionic salt in 1.8 kg of water at 35 ºC. The salt remains 90% dissociated in the solution. The vapour pressure of the solution is 59.724 mm of Hg. Vapor pressure of water at 35 ºC is 60.000 mm of Hg. The number of ions present per formula unit of the ionic salt is _______. SECTION 1 (Maximum marks: 24) Chemistry

Numerical answer type

Consider the strong electrolytes ZmXn, UmYp and VmXn. Limiting molar conductivity ( 0) of UmYp and VmXn are 250 and 440 S cm² mol⁻¹, respectively. The value of (m + n + p) is _______. Given: Ion Zn+ Up+ Vn+ Xm- Ym- 0 (S cm² mol⁻¹) 50.0 25.0 100.0 80.0 100.0 0 is the limiting molar conductivity of ions The plot of molar conductivity ( ) of ZmXn vs c1/2 is given below.

Numerical answer type

The reaction of Xe and O2F2 gives a Xe compound P. The number of moles of HF produced by the complete hydrolysis of 1 mol of P is _______.

Numerical answer type

Thermal decomposition of AgNO₃ produces two paramagnetic gases. The total number of electrons present in the antibonding molecular orbitals of the gas that has the higher number of unpaired electrons is _______.

Numerical answer type

The number of isomeric tetraenes (NOT containing sp-hybridized carbon atoms) that can be formed from the following reaction sequence is ________.

Numerical answer type

The number of -CH2- (methylene) groups in the product formed from the following reaction sequence is ________.

Numerical answer type

The total number of chiral molecules formed from one molecule of P on complete ozonolysis (O3, Zn/H₂O) is ________.

Numerical answer type

The correct option(s) about entropy (S) is(are) [R = gas constant, F = Faraday constant, T = Temperature]

1. For the reaction, M(s) + 2H+(aq) → H₂(g) + M2+(aq), if 𝑑𝐸𝑐𝑒𝑙𝑙 𝑑𝑇 = 𝑅 𝐹 , then the entropy change of the reaction is R (assume that entropy and internal energy changes are temperature independent ).
2. The cell reaction, Pt(s) | H₂(g, 1bar) | H+(aq, 0.01M) || H+(aq, 0.1M) | H₂(g, 1bar) | Pt(s), is an entropy driven process .
3. For racemization of an optically active compound, S > 0 .
4. S > 0, for [Ni(H₂O)6]2+ + 3 en → [Ni(en)3]2+ + 6H₂O (where en = ethylenediamine) .

The compound(s) which react(s) with NH₃ to give boron nitride (BN) is(are)

1. B
2. B2H6
3. B2O3
4. HBF4

The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range 900 – 1500 K is(are)

1. Limestone is used to remove silicate impurity.
2. Pig iron obtained from blast furnace contains about 4% carbon.
3. Coke
4. converts CO₂ to CO.
5. Exhaust gases consist of NO 2 and CO.

Considering the following reaction sequence, the correct statement(s) is(are)

1. Compounds P and Q are carboxylic acids.
2. Compound S decolorizes bromine water.
3. Compounds P and S react with hydroxylamine to give the corresponding oximes.
4. Compound R reacts with dialkylcadmium to give the corresponding tertiary alcohol.

Among the following, the correct statement(s) about polymers is(are)

1. The polymerization of chloroprene gives natural rubber.
2. Teflon is prepared from tetrafluoroethene by heating it with persulphate catalyst at high pressures.
3. PVC are thermoplastic polymers.
4. Ethene at 350-570 K temperature and 1000-2000 atm pressure in the presence of a peroxide initiator yields high density polythene. SECTION 3 (Maximum marks: 12)

Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to

1. 25
2. 35
3. 55
4. 75

The reaction of HClO 3 with HCl gives a paramagnetic gas, which upon reaction with O3 produces

1. Cl2O
2. ClO2
3. Cl2O6
4. Cl2O7

The reaction of Pb(NO3)2 and NaCl in water produces a precipitate that dissolves upon the addition of HCl of appropriate concentration. The dissolution of the precipitate is due to the formation of

1. PbCl2
2. PbCl4
3. [PbCl4]2-
4. [PbCl6]2-

Treatment of D-glucose with aqueous NaOH results in a mixture of monosaccharides, which are