Mathematics
JEE (Advanced) 2022 Paper 2
SECTION 1 (Maximum marks: 24)
• This section contains EIGHT (08) questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE.
• For each question, enter the correct integer corresponding to the answer using the mouse and the on
screen virtual numeric keypad in the place designated to enter the answer.
• Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct integer is entered;
Zero Marks : 0 If the question is unanswered;
Negative Marks : −1 In all other cases.
Q.1 𝜋 𝜋 1
Let 𝛼 and 𝛽 be real numbers such that − 4 < 𝛽 < 0 < 𝛼 < 4 . If sin(𝛼 + 𝛽) = 3 and
2
cos(𝛼 − 𝛽) = 3 , then the greatest integer less than or equal to
sin 𝛼 cos 𝛽 cos 𝛼 sin 𝛽 2
( + + + )
cos 𝛽 sin 𝛼 sin 𝛽 cos 𝛼
is _____________.
Q.2 If 𝑦(𝑥) is the solution of the differential equation
𝑥𝑑𝑦 − (𝑦 2 − 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 _____________.
Q.3 The greatest integer less than or equal to
2 log2 9 1
∫ log 2 (𝑥 3 + 1) 𝑑𝑥 + ∫ (2𝑥 − 1) 3 𝑑𝑥
1 1
is _____________.
Q.4 The product of all positive real values of 𝑥 satisfying the equation
3 −68
𝑥 (16(log5 𝑥) log5 𝑥)
= 5−16
is __________ .
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,JEE (Advanced) 2022 Paper 2
Q.5 If
3 1 1
𝑒 𝑥 − (1 − 𝑥 3 )3 + ((1 − 𝑥 2 )2 − 1) sin 𝑥
𝛽 = lim ,
𝑥→0 𝑥 sin2 𝑥
then the value of 6𝛽 is ___________.
Q.6 Let 𝛽 be a real number. Consider the matrix
𝛽 0 1
𝐴=(2 1 −2) .
3 1 −2
If 𝐴7 − (𝛽 − 1)𝐴6 − 𝛽𝐴5 is a singular matrix, then the value of 9𝛽 is ___________.
Q.7 Consider the hyperbola
𝑥2 𝑦2
− =1
100 64
with foci at 𝑆 and 𝑆1 , where 𝑆 lies on the positive x-axis. Let 𝑃 be a point on the hyperbola, in the
𝜋
first quadrant. Let ∠𝑆𝑃𝑆1 = 𝛼, with 𝛼 < 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 𝑆1 𝑃 at
𝑃1 . Let 𝛿 be the distance of 𝑃 from the straight line 𝑆𝑃1 , and 𝛽 = 𝑆1 𝑃. Then the greatest integer
βδ 𝛼
less than or equal to
9
sin 2 is _____________.
Q.8 Consider the functions 𝑓, 𝑔 ∶ ℝ → ℝ defined by
4|𝑥| 3
5 2 (1 − ), |𝑥| ≤ ,
3 4
𝑓(𝑥) = 𝑥 2 + and 𝑔(𝑥) =
12 3
{ 0, |𝑥| > .
4
If 𝛼 is the area of the region
3
{(𝑥, 𝑦) ∈ ℝ × ℝ ∶ |𝑥| ≤ , 0 ≤ 𝑦 ≤ min{𝑓(𝑥), 𝑔(𝑥)} } ,
4
then the value of 9𝛼 is _____________.
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JEE (Advanced) 2022 Paper 2
SECTION 1 (Maximum marks: 24)
• This section contains EIGHT (08) questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 TO 9, BOTH INCLUSIVE.
• For each question, enter the correct integer corresponding to the answer using the mouse and the on
screen virtual numeric keypad in the place designated to enter the answer.
• Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If ONLY the correct integer is entered;
Zero Marks : 0 If the question is unanswered;
Negative Marks : −1 In all other cases.
Q.1 𝜋 𝜋 1
Let 𝛼 and 𝛽 be real numbers such that − 4 < 𝛽 < 0 < 𝛼 < 4 . If sin(𝛼 + 𝛽) = 3 and
2
cos(𝛼 − 𝛽) = 3 , then the greatest integer less than or equal to
sin 𝛼 cos 𝛽 cos 𝛼 sin 𝛽 2
( + + + )
cos 𝛽 sin 𝛼 sin 𝛽 cos 𝛼
is _____________.
Q.2 If 𝑦(𝑥) is the solution of the differential equation
𝑥𝑑𝑦 − (𝑦 2 − 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 _____________.
Q.3 The greatest integer less than or equal to
2 log2 9 1
∫ log 2 (𝑥 3 + 1) 𝑑𝑥 + ∫ (2𝑥 − 1) 3 𝑑𝑥
1 1
is _____________.
Q.4 The product of all positive real values of 𝑥 satisfying the equation
3 −68
𝑥 (16(log5 𝑥) log5 𝑥)
= 5−16
is __________ .
1/8
,JEE (Advanced) 2022 Paper 2
Q.5 If
3 1 1
𝑒 𝑥 − (1 − 𝑥 3 )3 + ((1 − 𝑥 2 )2 − 1) sin 𝑥
𝛽 = lim ,
𝑥→0 𝑥 sin2 𝑥
then the value of 6𝛽 is ___________.
Q.6 Let 𝛽 be a real number. Consider the matrix
𝛽 0 1
𝐴=(2 1 −2) .
3 1 −2
If 𝐴7 − (𝛽 − 1)𝐴6 − 𝛽𝐴5 is a singular matrix, then the value of 9𝛽 is ___________.
Q.7 Consider the hyperbola
𝑥2 𝑦2
− =1
100 64
with foci at 𝑆 and 𝑆1 , where 𝑆 lies on the positive x-axis. Let 𝑃 be a point on the hyperbola, in the
𝜋
first quadrant. Let ∠𝑆𝑃𝑆1 = 𝛼, with 𝛼 < 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 𝑆1 𝑃 at
𝑃1 . Let 𝛿 be the distance of 𝑃 from the straight line 𝑆𝑃1 , and 𝛽 = 𝑆1 𝑃. Then the greatest integer
βδ 𝛼
less than or equal to
9
sin 2 is _____________.
Q.8 Consider the functions 𝑓, 𝑔 ∶ ℝ → ℝ defined by
4|𝑥| 3
5 2 (1 − ), |𝑥| ≤ ,
3 4
𝑓(𝑥) = 𝑥 2 + and 𝑔(𝑥) =
12 3
{ 0, |𝑥| > .
4
If 𝛼 is the area of the region
3
{(𝑥, 𝑦) ∈ ℝ × ℝ ∶ |𝑥| ≤ , 0 ≤ 𝑦 ≤ min{𝑓(𝑥), 𝑔(𝑥)} } ,
4
then the value of 9𝛼 is _____________.
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