JEE (Advanced) 2022
Mathematics Paper 1
SECTION 1 (Maximum Marks: 24)
• This section contains EIGHT (08) 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 onscreen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more
than two decimal places, truncate/roundoff the value to TWO decimal places.
• Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 ONLY if the correct numerical value is entered;
Zero Marks : 0 In all other cases.
Q.1 Considering only the principal values of the inverse trigonometric functions, the value of
3 2 1 2√2 π √2
cos −1 √ 2
+ sin−1 2
+ tan−1
2 2+π 4 2+π π
is __________ .
Q.2 Let 𝛼 be a positive real number. Let 𝑓: ℝ → ℝ and 𝑔: (𝛼, ∞) → ℝ be the functions defined by
𝜋𝑥 2 log e ( √𝑥 − √𝛼 )
𝑓(𝑥) = sin ( ) and 𝑔(𝑥) = .
12 log e ( 𝑒 √𝑥 − 𝑒 √𝛼 )
Then the value of lim+ 𝑓(𝑔(𝑥)) is __________.
𝑥→𝛼
Q.3 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 _____________.
1/9
,JEE (Advanced) 2022 Paper 1
Q.4 Let 𝑧 be a complex number with non-zero imaginary part. If
2 + 3𝑧 + 4𝑧 2
2 − 3𝑧 + 4𝑧 2
is a real number, then the value of |𝑧|2 is _____________.
Q.5 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 − 𝑧 2 = 𝑖(𝑧̅ + 𝑧 2 )
is _____________.
Q.6 Let 𝑙1 , 𝑙2 , … , 𝑙100 be consecutive terms of an arithmetic progression with common difference 𝑑1 , and
let 𝑤1 , 𝑤2 , … , 𝑤100 be consecutive terms of another arithmetic progression with common difference
𝑑2 , where 𝑑1 𝑑2 = 10. For each 𝑖 = 1, 2, … , 100, let 𝑅𝑖 be a rectangle with length 𝑙𝑖 , width 𝑤𝑖 and
area 𝐴𝑖 . If 𝐴51 − 𝐴50 = 1000, then the value of 𝐴100 − 𝐴90 is ____________.
Q.7 The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits
0, 2, 3, 4, 6, 7 is ____________.
π
Q.8 Let 𝐴𝐵𝐶 be the triangle with 𝐴𝐵 = 1, 𝐴𝐶 = 3 and ∠𝐵𝐴𝐶 = . If a circle of radius 𝑟 > 0 touches
2
the sides 𝐴𝐵, 𝐴𝐶 and also touches internally the circumcircle of the triangle 𝐴𝐵𝐶, then the value
of 𝑟 is _____________.
2/9
,
, JEE (Advanced) 2022 Paper 1
Q.10 Let 𝑎1 , 𝑎2 , 𝑎3 , … be an arithmetic progression with 𝑎1 = 7 and common difference 8. Let
𝑇1 , 𝑇2 , 𝑇3 , … be such that 𝑇1 = 3 and 𝑇𝑛+1 − 𝑇𝑛 = 𝑎𝑛 for 𝑛 ≥ 1. Then, which of the following is/are
TRUE ?
(A) 𝑇20 = 1604 (B) ∑20
𝑘=1 𝑇𝑘 = 10510
(C) 𝑇30 = 3454 (D) ∑30
𝑘=1 𝑇𝑘 = 35610
Q.11 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 𝑃1
and 𝑃2 ?
𝑥−1 𝑦−1 𝑧−1
(A) = =
0 0 5
𝑥−6 𝑦 𝑧
(B) = =
−5 2 3
𝑥 𝑦−4 𝑧
(C) = =
−2 5 4
𝑥 𝑦−4 𝑧
(D) = =
1 −2 3
Q.12 Let 𝑆 be the reflection of a point 𝑄 with respect to the plane given by
𝑟⃗ = −(𝑡 + 𝑝)𝑖̂ + 𝑡𝑗̂ + (1 + 𝑝)𝑘̂
where 𝑡, 𝑝 are real parameters and 𝑖̂, 𝑗̂, 𝑘̂ are the unit vectors along the three positive coordinate
axes. If the position vectors of 𝑄 and 𝑆 are 10𝑖̂ + 15𝑗̂ + 20𝑘̂ and 𝛼𝑖̂ + 𝛽𝑗̂ + 𝛾𝑘̂ respectively, then
which of the following is/are TRUE ?
(A) 3(𝛼 + 𝛽) = −101
(B) 3(𝛽 + 𝛾) = −71
(C) 3(𝛾 + 𝛼) = −86
(D) 3(𝛼 + 𝛽 + 𝛾) = −121
4/9
Mathematics Paper 1
SECTION 1 (Maximum Marks: 24)
• This section contains EIGHT (08) 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 onscreen
virtual numeric keypad in the place designated to enter the answer. If the numerical value has more
than two decimal places, truncate/roundoff the value to TWO decimal places.
• Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 ONLY if the correct numerical value is entered;
Zero Marks : 0 In all other cases.
Q.1 Considering only the principal values of the inverse trigonometric functions, the value of
3 2 1 2√2 π √2
cos −1 √ 2
+ sin−1 2
+ tan−1
2 2+π 4 2+π π
is __________ .
Q.2 Let 𝛼 be a positive real number. Let 𝑓: ℝ → ℝ and 𝑔: (𝛼, ∞) → ℝ be the functions defined by
𝜋𝑥 2 log e ( √𝑥 − √𝛼 )
𝑓(𝑥) = sin ( ) and 𝑔(𝑥) = .
12 log e ( 𝑒 √𝑥 − 𝑒 √𝛼 )
Then the value of lim+ 𝑓(𝑔(𝑥)) is __________.
𝑥→𝛼
Q.3 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 _____________.
1/9
,JEE (Advanced) 2022 Paper 1
Q.4 Let 𝑧 be a complex number with non-zero imaginary part. If
2 + 3𝑧 + 4𝑧 2
2 − 3𝑧 + 4𝑧 2
is a real number, then the value of |𝑧|2 is _____________.
Q.5 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 − 𝑧 2 = 𝑖(𝑧̅ + 𝑧 2 )
is _____________.
Q.6 Let 𝑙1 , 𝑙2 , … , 𝑙100 be consecutive terms of an arithmetic progression with common difference 𝑑1 , and
let 𝑤1 , 𝑤2 , … , 𝑤100 be consecutive terms of another arithmetic progression with common difference
𝑑2 , where 𝑑1 𝑑2 = 10. For each 𝑖 = 1, 2, … , 100, let 𝑅𝑖 be a rectangle with length 𝑙𝑖 , width 𝑤𝑖 and
area 𝐴𝑖 . If 𝐴51 − 𝐴50 = 1000, then the value of 𝐴100 − 𝐴90 is ____________.
Q.7 The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits
0, 2, 3, 4, 6, 7 is ____________.
π
Q.8 Let 𝐴𝐵𝐶 be the triangle with 𝐴𝐵 = 1, 𝐴𝐶 = 3 and ∠𝐵𝐴𝐶 = . If a circle of radius 𝑟 > 0 touches
2
the sides 𝐴𝐵, 𝐴𝐶 and also touches internally the circumcircle of the triangle 𝐴𝐵𝐶, then the value
of 𝑟 is _____________.
2/9
,
, JEE (Advanced) 2022 Paper 1
Q.10 Let 𝑎1 , 𝑎2 , 𝑎3 , … be an arithmetic progression with 𝑎1 = 7 and common difference 8. Let
𝑇1 , 𝑇2 , 𝑇3 , … be such that 𝑇1 = 3 and 𝑇𝑛+1 − 𝑇𝑛 = 𝑎𝑛 for 𝑛 ≥ 1. Then, which of the following is/are
TRUE ?
(A) 𝑇20 = 1604 (B) ∑20
𝑘=1 𝑇𝑘 = 10510
(C) 𝑇30 = 3454 (D) ∑30
𝑘=1 𝑇𝑘 = 35610
Q.11 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 𝑃1
and 𝑃2 ?
𝑥−1 𝑦−1 𝑧−1
(A) = =
0 0 5
𝑥−6 𝑦 𝑧
(B) = =
−5 2 3
𝑥 𝑦−4 𝑧
(C) = =
−2 5 4
𝑥 𝑦−4 𝑧
(D) = =
1 −2 3
Q.12 Let 𝑆 be the reflection of a point 𝑄 with respect to the plane given by
𝑟⃗ = −(𝑡 + 𝑝)𝑖̂ + 𝑡𝑗̂ + (1 + 𝑝)𝑘̂
where 𝑡, 𝑝 are real parameters and 𝑖̂, 𝑗̂, 𝑘̂ are the unit vectors along the three positive coordinate
axes. If the position vectors of 𝑄 and 𝑆 are 10𝑖̂ + 15𝑗̂ + 20𝑘̂ and 𝛼𝑖̂ + 𝛽𝑗̂ + 𝛾𝑘̂ respectively, then
which of the following is/are TRUE ?
(A) 3(𝛼 + 𝛽) = −101
(B) 3(𝛽 + 𝛾) = −71
(C) 3(𝛾 + 𝛼) = −86
(D) 3(𝛼 + 𝛽 + 𝛾) = −121
4/9
