JEE Advanced 2022 Paper 1

Considering only the principal values of the inverse trigonometric functions, the value of 3 2 cos⁻¹√2 2 +π2 + 1 4 sin−12√2 π 2 +π2 + tan−1√2 π is __________ .

Numerical answer type

Let 𝛼 be a positive real number. Let 𝑓:ℝ→ℝ and 𝑔: (𝛼,∞)→ℝ be the functions defined by 𝑓(𝑥)= sin(𝜋𝑥 12) and 𝑔 (𝑥)=2 log e ( √𝑥−√𝛼 ) log e ( 𝑒√𝑥−𝑒√𝛼 ) . Then the value of lim𝑥→𝛼+𝑓(𝑔(𝑥)) is __________.

Numerical answer type

In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____________. Mathematics

Numerical answer type

Let 𝑧 be a complex number with non-zero imaginary part. If 2 + 3𝑧+ 4𝑧2 2−3𝑧+ 4𝑧2 is a real number, then the value of |𝑧|2 is _____________.

Numerical answer type

Let 𝑧̅ denote the complex conjugate of a complex number 𝑧 and let 𝑖= √−1 . In the set of complex numbers, the number of distinct roots of the equation 𝑧̅−𝑧2=𝑖(𝑧̅+𝑧2) is _____________.

Numerical answer type

Let 𝑙1,𝑙2, … ,𝑙100 be consecutive terms of an arithmetic progression with common difference 𝑑 1, and let 𝑤1,𝑤2, … ,𝑤100 be consecutive terms of another arithmetic progression with common difference 𝑑2 , where 𝑑 1𝑑2=10. For each 𝑖=1, 2, … ,100, let 𝑅 𝑖 be a rectangle with length 𝑙 𝑖, width 𝑤 𝑖 and area 𝐴𝑖. If 𝐴51−𝐴50=1000, then the value of 𝐴 100−𝐴90 is ____________.

Numerical answer type

The number of 4-digit integers in the closed interval [2022, 4482 ] formed by using the digits 0, 2, 3, 4, 6, 7 is ____________.

Numerical answer type

Let 𝐴𝐵𝐶 be the triangle with 𝐴𝐵=1, 𝐴𝐶= 3 and ∠𝐵𝐴𝐶=π 2. If a circle of radius 𝑟> 0 touches the sides 𝐴𝐵, 𝐴𝐶 and also touches internally the circumcircle of the triangle 𝐴𝐵𝐶, then the value of 𝑟 is _____________.

Numerical answer type

Let 𝑎1,𝑎2,𝑎3, … be an arithmetic progression with 𝑎 1= 7 and common difference 8. Let 𝑇1,𝑇2,𝑇3, … be such that 𝑇 1= 3 and 𝑇 𝑛+1−𝑇𝑛=𝑎𝑛 for 𝑛≥1. Then, which of the following is/are TRUE ?

1. 𝑇20=1604
2. ∑𝑇𝑘=1051020 𝑘=1
3. 𝑇30= 3454
4. ∑𝑇𝑘=30 𝑘=1 35610

Let 𝑃1 and 𝑃2 be two planes given by 𝑃1: 10𝑥+15𝑦+12𝑧−60 = 0 , 𝑃2: −2𝑥+5𝑦+ 4𝑧−20 = 0 . Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on 𝑃 1 and 𝑃2 ?

1. 𝑥−1 0= 𝑦−1 0=𝑧−1 5
2. 𝑥−6 −5=𝑦 2=𝑧 3
3. 𝑥 −2=𝑦−4 5=𝑧 4
4. 𝑥 1=𝑦−4 −2=𝑧 3

Let 𝑆 be the reflection of a point 𝑄 with respect to the plane given by 𝑟 ⃗=− (𝑡+𝑝 )𝑖̂+𝑡𝑗̂+ (1+𝑝 )𝑘̂ where 𝑡, 𝑝 are real parameters and 𝑖̂,𝑗̂,𝑘 ̂ are the unit vectors along the three positive coordinate axes. If the position vectors of 𝑄 and 𝑆 are 10𝑖̂+15𝑗̂+ 20𝑘 ̂ and 𝛼𝑖̂+𝛽𝑗̂+𝛾𝑘 ̂ respectively, then which of the following is/are TRUE ?

1. 3 (𝛼+𝛽 )=−101
2. 3 (𝛽+𝛾 )=−71
3. 3 (𝛾+𝛼 )=−86
4. 3 (𝛼+𝛽+𝛾 )=−121

Consider the parabola 𝑦2= 4𝑥. Let 𝑆 be the focus of the parabola. A pair of tangents drawn to the parabola from the point 𝑃= (−2,1) meet the parabola at 𝑃 1 and 𝑃2. Let 𝑄1 and 𝑄2 be points on the lines𝑆𝑃1 and 𝑆𝑃2 respectively such that 𝑃𝑄 1 is perpendicular to 𝑆𝑃 1 and 𝑃𝑄2 is perpendicular to 𝑆𝑃2. Then, which of the following is/are TRUE ?

1. 𝑆𝑄 1= 2
2. 𝑄1𝑄2=3√10 5
3. 𝑃𝑄 1= 3
4. 𝑆𝑄 2=1

Let |𝑀| denote the determinant of a square matrix 𝑀. Let 𝑔: [0,𝜋 2]→ℝ be the function defined by 𝑔(𝜃)=√𝑓(𝜃)−1 +√𝑓(𝜋 2−𝜃)−1 where 𝑓(θ)=1 2|1sin𝜃 1 −sin𝜃 1sin𝜃 −1 −sin𝜃 1| + ||sin𝜋cos(𝜃+𝜋 4) tan(𝜃−𝜋 4) sin(𝜃−𝜋 4) −cos𝜋 2log𝑒(4 𝜋) cot(𝜃+𝜋 4) log 𝑒(𝜋 4) tan𝜋|| . Let 𝑝 (𝑥) be a quadratic polynomial whose roots are the maximum and minimum values of the function 𝑔 (𝜃), and 𝑝 (2)= 2−√2 . Then, which of the following is/are TRUE ?

1. 𝑝(3+√2 4)< 0
2. 𝑝(1+3√2 4)> 0
3. 𝑝(5√2−1 4)> 0
4. 𝑝(5−√2 4)< 0

Consider the following lists: List-I List-II (I) {𝑥∈[−2𝜋 3,2𝜋 3 ]: cos𝑥+ sin𝑥=1 } (P) has two elements (II) {𝑥∈[−5𝜋 18,5𝜋 18 ]: √3 tan 3𝑥=1 } (Q) has three elements (III) {𝑥∈[−6𝜋 5,6𝜋 5 ]: 2 cos (2𝑥)=√3} (R) has four elements (IV) {𝑥∈[−7𝜋 4,7𝜋 4 ]: sin𝑥−cos𝑥=1 } (S) has five elements (T) has six elements The correct option is:

1. (I)→(P); (II)→(S); (III)→(P); (IV)→(S)
2. (I)→(P); (II)→(P); (III)→(T); (IV)→(R)
3. (I)→(Q); (II)→(P); (III)→(T); (IV)→(S)
4. (I)→(Q); (II)→(S); (III)→(P); (IV)→(R)

Two players, 𝑃 1 and 𝑃2, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let 𝑥 and 𝑦 denote the readings on the die rolled by 𝑃 1 and 𝑃2 , respectively. If 𝑥>𝑦, then 𝑃 1 scores 5 points and 𝑃 2 scores 0 point. If 𝑥=𝑦, then each player scores 2 points. If 𝑥<𝑦, then 𝑃 1 scores 0 point and 𝑃 2 scores 5 points. Let 𝑋 𝑖 and 𝑌𝑖 be the total scores of 𝑃 1 and 𝑃2, respectively, after playing the 𝑖𝑡ℎ round. List-I List-II (I) Probability of (𝑋 2≥𝑌2) is (P) 3 8 (II) Probability of (𝑋 2>𝑌2) is (Q) 11 16 (III) Probability of (𝑋 3=𝑌3) is (R) 5 16 (IV) Probability of (𝑋 3>𝑌3) is (S) 355 864 (T) 77432 The correct option is:

1. (I)→(Q); (II)→(R); (III)→(T); (IV)→(S)
2. (I)→(Q); (II)→(R); (III)→(T); (IV)→(T)
3. (I)→(P); (II)→(R); (III)→(Q); (IV)→(S)
4. (I)→(P); (II)→(R); (III)→(Q); (IV)→(T)

Let 𝑝,𝑞,𝑟 be nonzero real numbers that are, respectively, the 10𝑡ℎ, 100𝑡ℎ and 1000𝑡ℎ terms of a harmonic progression. Consider the system of linear equations 𝑥+𝑦+𝑧=1 10𝑥+100𝑦+1000𝑧= 0 𝑞𝑟 𝑥+𝑝𝑟 𝑦+𝑝𝑞 𝑧= 0 . List-I List-II (I) If 𝑞 𝑟=10, then the system of linear equations has (P) 𝑥= 0, 𝑦=10 9 , 𝑧=−1 9 as a solution (II) If 𝑝 𝑟≠100, then the system of linear equations has (Q) 𝑥=10 9 , 𝑦=−1 9 , 𝑧= 0 as a solution (III) If 𝑝 𝑞≠10, then the system of linear equations has (R) infinitely many solutions (IV) If 𝑝 𝑞=10, then the system of linear equations has (S) no solution (T) at least one solution The correct option is:

1. (I)→(T); (II)→(R); (III)→(S); (IV)→(T)
2. (I)→(Q); (II)→(S); (III)→(S); (IV)→(R)
3. (I)→(Q); (II)→(R); (III)→(P); (IV)→(R)
4. (I)→(T); (II)→(S); (III)→(P); (IV)→(T)

Consider the ellipse 𝑥2 4+𝑦2 3=1 . Let 𝐻 (𝛼, 0), 0 <𝛼< 2, be a point. A straight line drawn through 𝐻 parallel to the 𝑦-axis crosses the ellipse and its auxiliary circle at points 𝐸 and 𝐹 respectively, in the first quadrant. The tangent to the ellipse at the point 𝐸 intersects the positive 𝑥-axis at a point 𝐺. Suppose the straight line joining 𝐹 and the origin makes an angle 𝜙 with the positive 𝑥-axis. List-I List-II (I) If 𝜙=𝜋 4 , then the area of the triangle 𝐹𝐺𝐻 is (P) (√3−1)4 8 (II) If 𝜙=𝜋 3 , then the area of the triangle 𝐹𝐺𝐻 is (Q) 1 (III) If 𝜙=𝜋 6 , then the area of the triangle 𝐹𝐺𝐻 is (R) 3 4 (IV) If 𝜙=𝜋 12 , then the area of the triangle 𝐹𝐺𝐻 is (S) 1 2√3 (T) 3√32 The correct option is:

1. (I)→(R); (II)→(S); (III)→(Q); (IV)→(P)
2. (I)→(R); (II)→(T); (III)→(S); (IV)→(P)
3. (I)→(Q); (II)→(T); (III)→(S); (IV)→(P)
4. (I)→(Q); (II)→(S); (III)→(Q); (IV)→(P)

2 mol of Hg(g) is combusted in a fixed volume bomb calorimeter with excess of O 2 at 298 K and 1 atm into HgO(s). During the reaction, temperature increases from 298.0 K to 312.8 K. If heat capacity of the bomb calorimeter and enthalpy of formation of Hg(g) are 20.00 kJ K⁻¹ and 61.32 kJ mol⁻¹ at 298 K, respectively, the calculated standard molar enthalpy of formation of HgO(s) at 298 K is X kJ mol⁻¹. The value of |X| is _______. [Given: Gas constant R = 8.3 J K⁻¹ mol⁻¹]

Numerical answer type

The reduction potential (E0, in V) of MnO 4−(aq)/Mn(s) is ______. [Given: 𝐸(MnO₄⁻–(aq)/MnO2(s))0= 1.68 V; 𝐸(MnO2(s)/Mn2+(aq))0= 1.21 V; 𝐸(Mn2+(aq)/Mn(s))0= –1.03 V ] Chemistry

Numerical answer type

A solution is prepared by mixing 0.01 mol each of H2CO3, NaHCO3, Na₂CO₃, and NaOH in 100 mL of water. pH of the resulting solution is _______. [Given: pKa1 and pKa2 of H2CO3 are 6.37 and 10.32, respectively ; log 2 = 0.30]

Numerical answer type

The treatment of an aqueous solution of 3.74 g of Cu(NO3)2 with excess KI results in a brown solution along with the formation of a precipitate. Passing H 2S through this brown solution gives another precipitate X. The amount of X (in g) is ________. [Given: Atomic mass of H = 1, N = 14, O = 16, S = 32, K = 39, Cu = 63, I = 127]

Numerical answer type

Dissolving 1.24 g of white phosphorous in boiling NaOH solution in an inert atmosphere gives a gas Q. The amount of CuSO4 (in g) required to completely consume the gas Q is _______. [Given: Atomic mass of H = 1, O = 16, Na = 23, P = 31, S = 32, Cu = 63]

Numerical answer type

Consider the following reaction. On estimation of bromine in 1.00 g of R using Carius method, the amount of AgBr formed (in g) is ________. [Given: Atomic mass of H = 1, C = 12, O = 16, P = 31, Br = 80, Ag = 108]

Numerical answer type

The weight percentage of hydrogen in Q, formed in the following reaction sequence, is ________. [Given: Atomic mass of H = 1, C = 12, N = 14, O = 16, S = 32, Cl = 35]

Numerical answer type

If the reaction sequence given below is carried out with 15 moles of acetylene, the amount of the product D formed (in g) is ________. The yields of A, B, C and D are given in parentheses. [Given: Atomic mass of H = 1, C = 12, O = 16, Cl = 35]

Numerical answer type

The electrochemical extraction of aluminum from bauxite ore involves

1. the reaction of Al₂O₃ with coke
2. at a temperature > 2500 ºC.
3. the neutralization of aluminate solution by passing CO 2 gas to precipitate hydrated alumina (Al 2O3·3H 2O).
4. the dissolution of Al₂O₃ in hot aqueous NaOH.
5. the electrolysis of Al 2O3 mixed with Na 3AlF 6 to give Al and CO 2.

The treatment of galena with HNO₃ produces a gas that is

1. paramagnetic
2. bent in geometry
3. an acidic oxide
4. colorless

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

1. P can be reduced to a primary alcohol using NaBH4.
2. Treating P with conc. NH4OH solution followed by acidification gives Q.
3. Treating Q with a solution of NaNO2 in aq. HCl liberates N₂.
4. P is more acidic than CH3CH2COOH.

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

Match the rate expressions in LIST-I for the decomposition of X with the corresponding profiles provided in LIST-II. X s and k are constants having appropriate units. LIST-I LIST-II (I) sk[X]rate = X +[X] under all possible initial concentrations of X (P) (II) s sk[X]rate = X +[X] where initial concentrations of X are much less than X (Q) (III) s sk[X]rate = X +[X] where initial concentrations of X are much higher than X (R) (IV) 2 s sk[X]rate = X +[X] where initial concentration of X is much higher than X (S) (T)

1. I→ P; II→ Q; III → S; IV → T
2. I→ R; II→ S; III → S; IV → T
3. I→ P; II→ Q; III → Q; IV → R
4. I→ R; II→ S; III → Q; IV → R

LIST-I contains compounds and LIST-II contains reactions LIST-I LIST-II (I) H2O₂ (P) Mg(HCO3)2 + Ca(OH)₂ → (II) Mg(OH)2 (Q) BaO2 + H₂SO₄ → (III) BaCl2 (R) Ca(OH)₂ + MgCl₂ → (IV) CaCO₃ (S) BaO2 + HCl → (T) Ca(HCO3)2 + Ca(OH)₂ → Match each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option

1. I → Q; II → P; III → S; IV → R
2. I → T; II → P; III → Q; IV → R
3. I → T; II → R; III → Q; IV → P
4. I → Q; II → R; III → S; IV → P

LIST-I contains metal species and LIST-II contains their properties. LIST-I LIST-II (I) [Cr(CN)6]4– (P) t2g orbitals contain 4 electrons (II) [RuCl6]2– (Q) µ(spin-only) = 4.9 BM (III) [Cr(H₂O)6]2+ (R) low spin complex ion (IV) [Fe(H₂O)6]2+ (S) metal ion in 4+ oxidation state (T) d4 species [Given: Atomic number of Cr = 24, Ru = 44, Fe = 26] Match each metal species in LIST-I with their properties in LIST-II, and choose the correct option

1. I → R, T; II → P, S; III → Q, T; IV → P, Q
2. I → R, S; II → P, T; III → P, Q; IV → Q, T
3. I → P, R; II → R, S; III → R, T; IV → P, T
4. I → Q, T; II → S, T; III → P, T; IV → Q, R

Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option. LIST-I LIST-II (I) Aniline (P) Sodium fusion extract of the compound on boiling with FeSO4, followed by acidification with conc. H₂SO₄, gives Prussian blue color. (II) o-Cresol (Q) Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color. (III) Cysteine (R) Addition of the compound to a saturated solution of NaHCO3 results in effervescence. (IV) Caprolactam (S) The compound reacts with bromine water to give a white precipitate. (T) Treating the compound with neutral FeCl₃ solution produces violet color.

1. I→P,Q; II→S; III→Q,R; IV→P
2. I→P; II→R,S; III→R; IV→Q,S
3. I→Q,S; II→P,T; III→P; IV→S
4. I→P,S; II→T; III→Q,R; IV→P