JEE (Advanced) 2023 Paper 1
Mathematics
SECTION 1 (Maximum Marks: 12)
This section contains THREE (03) questions.
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct 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:
Full Marks : 4 ONLY if (all) the correct option(s) is(are) chosen;
Partial Marks : 3 If all the four options are correct but ONLY three options are chosen;
Partial Marks : 2 If three or more options are correct but ONLY two options are chosen, both of
which are correct;
Partial Marks : 1 If two or more options are correct but ONLY one option is chosen and it is a
correct option;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : 2 In all other cases.
For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct
answers, then
choosing ONLY (A), (B) and (D) will get 4 marks;
choosing ONLY (A) and (B) will get 2 marks;
choosing ONLY (A) and (D) will get 2 marks;
choosing ONLY (B) and (D) will get 2 marks;
choosing ONLY (A) will get +1 mark;
choosing ONLY (B) will get +1 mark;
choosing ONLY (D) will get +1 mark;
choosing no option (i.e. the question is unanswered) will get 0 marks; and
choosing any other combination of options will get 2 marks.
Q.1 Let S (0,1) (1, 2) (3, 4) and T {0,1, 2,3} . Then which of the following statements is(are)
true?
(A) There are infinitely many functions from S to T
(B) There are infinitely many strictly increasing functions from S to T
(C) The number of continuous functions from S to T is at most 120
(D) Every continuous function from S to T is differentiable
1/10
,JEE (Advanced) 2023 Paper 1
Q.2 x2 y 2
Let T1 and T2 be two distinct common tangents to the ellipse E : 1 and the parabola
6 3
P : y 2 12 x . Suppose that the tangent T1 touches P and E at the points A1 and A2 ,
respectively and the tangent T2 touches P and E at the points A4 and A3 , respectively. Then
which of the following statements is(are) true?
(A) The area of the quadrilateral A1 A2 A3 A4 is 35 square units
(B) The area of the quadrilateral A1 A2 A3 A4 is 36 square units
(C) The tangents T1 and T2 meet the x -axis at the point (3,0)
(D) The tangents T1 and T2 meet the x -axis at the point (6, 0)
Q.3 x3 5 17
Let f :[0,1] [0,1] be the function defined by f ( x) x 2 x . Consider the square
3 9 36
region S [0,1] [0,1] . Let G {( x, y) S : y f ( x)} be called the green region and
R {( x, y) S : y f ( x)} be called the red region. Let Lh {( x, h) S : x [0,1]} be the
horizontal line drawn at a height h [0,1] . Then which of the following statements is(are) true?
1 2
(A) There exists an h , such that the area of the green region above the line Lh equals the
4 3
area of the green region below the line Lh
1 2
(B) There exists an h , such that the area of the red region above the line Lh equals the
4 3
area of the red region below the line Lh
1 2
(C) There exists an h , such that the area of the green region above the line Lh equals the
4 3
area of the red region below the line Lh
1 2
(D) There exists an h , such that the area of the red region above the line Lh equals the
4 3
area of the green region below the line Lh
2/10
, JEE (Advanced) 2023 Paper 1
SECTION 2 (Maximum Marks: 12)
This section contains FOUR (04) questions.
Each question has FOUR options (A), (B), (C) and (D). 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:
Full Marks : 3 If ONLY the correct option is chosen;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : 1 In all other cases.
Q.4 1 1
Let f : (0,1) be the function defined as f ( x ) n if x , where n . Let
n 1 n
1 t
x
g : (0,1) be a function such that t
dt g ( x) 2 x for all x (0,1) . Then
x2
lim f ( x) g ( x)
x 0
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3
Q.5
Let Q be the cube with the set of vertices ( x1 , x2 , x3 ) 3 : x1 , x2 , x3 { 0, 1} . Let F be the set
of all twelve lines containing the diagonals of the six faces of the cube Q . Let S be the set of all
four lines containing the main diagonals of the cube Q ; for instance, the line passing through the
vertices (0, 0, 0) and (1,1,1) is in S . For lines 1 and 2 , let d (1 , 2 ) denote the shortest
distance between them. Then the maximum value of d (1 , 2 ) , as 1 varies over F and 2 varies
over S , is
1 1 1 1
(A) (B) (C) (D)
6 8 3 12
Q.6 x2 y2
Let X ( x, y ) : 1 and y 2 5 x . Three distinct points P, Q and R are
8 20
randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a
positive integer, is
71 73 79 83
(A) (B) (C) (D)
220 220 220 220
3/10
Mathematics
SECTION 1 (Maximum Marks: 12)
This section contains THREE (03) questions.
Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four
option(s) is(are) correct 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:
Full Marks : 4 ONLY if (all) the correct option(s) is(are) chosen;
Partial Marks : 3 If all the four options are correct but ONLY three options are chosen;
Partial Marks : 2 If three or more options are correct but ONLY two options are chosen, both of
which are correct;
Partial Marks : 1 If two or more options are correct but ONLY one option is chosen and it is a
correct option;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : 2 In all other cases.
For example, in a question, if (A), (B) and (D) are the ONLY three options corresponding to correct
answers, then
choosing ONLY (A), (B) and (D) will get 4 marks;
choosing ONLY (A) and (B) will get 2 marks;
choosing ONLY (A) and (D) will get 2 marks;
choosing ONLY (B) and (D) will get 2 marks;
choosing ONLY (A) will get +1 mark;
choosing ONLY (B) will get +1 mark;
choosing ONLY (D) will get +1 mark;
choosing no option (i.e. the question is unanswered) will get 0 marks; and
choosing any other combination of options will get 2 marks.
Q.1 Let S (0,1) (1, 2) (3, 4) and T {0,1, 2,3} . Then which of the following statements is(are)
true?
(A) There are infinitely many functions from S to T
(B) There are infinitely many strictly increasing functions from S to T
(C) The number of continuous functions from S to T is at most 120
(D) Every continuous function from S to T is differentiable
1/10
,JEE (Advanced) 2023 Paper 1
Q.2 x2 y 2
Let T1 and T2 be two distinct common tangents to the ellipse E : 1 and the parabola
6 3
P : y 2 12 x . Suppose that the tangent T1 touches P and E at the points A1 and A2 ,
respectively and the tangent T2 touches P and E at the points A4 and A3 , respectively. Then
which of the following statements is(are) true?
(A) The area of the quadrilateral A1 A2 A3 A4 is 35 square units
(B) The area of the quadrilateral A1 A2 A3 A4 is 36 square units
(C) The tangents T1 and T2 meet the x -axis at the point (3,0)
(D) The tangents T1 and T2 meet the x -axis at the point (6, 0)
Q.3 x3 5 17
Let f :[0,1] [0,1] be the function defined by f ( x) x 2 x . Consider the square
3 9 36
region S [0,1] [0,1] . Let G {( x, y) S : y f ( x)} be called the green region and
R {( x, y) S : y f ( x)} be called the red region. Let Lh {( x, h) S : x [0,1]} be the
horizontal line drawn at a height h [0,1] . Then which of the following statements is(are) true?
1 2
(A) There exists an h , such that the area of the green region above the line Lh equals the
4 3
area of the green region below the line Lh
1 2
(B) There exists an h , such that the area of the red region above the line Lh equals the
4 3
area of the red region below the line Lh
1 2
(C) There exists an h , such that the area of the green region above the line Lh equals the
4 3
area of the red region below the line Lh
1 2
(D) There exists an h , such that the area of the red region above the line Lh equals the
4 3
area of the green region below the line Lh
2/10
, JEE (Advanced) 2023 Paper 1
SECTION 2 (Maximum Marks: 12)
This section contains FOUR (04) questions.
Each question has FOUR options (A), (B), (C) and (D). 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:
Full Marks : 3 If ONLY the correct option is chosen;
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered);
Negative Marks : 1 In all other cases.
Q.4 1 1
Let f : (0,1) be the function defined as f ( x ) n if x , where n . Let
n 1 n
1 t
x
g : (0,1) be a function such that t
dt g ( x) 2 x for all x (0,1) . Then
x2
lim f ( x) g ( x)
x 0
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3
Q.5
Let Q be the cube with the set of vertices ( x1 , x2 , x3 ) 3 : x1 , x2 , x3 { 0, 1} . Let F be the set
of all twelve lines containing the diagonals of the six faces of the cube Q . Let S be the set of all
four lines containing the main diagonals of the cube Q ; for instance, the line passing through the
vertices (0, 0, 0) and (1,1,1) is in S . For lines 1 and 2 , let d (1 , 2 ) denote the shortest
distance between them. Then the maximum value of d (1 , 2 ) , as 1 varies over F and 2 varies
over S , is
1 1 1 1
(A) (B) (C) (D)
6 8 3 12
Q.6 x2 y2
Let X ( x, y ) : 1 and y 2 5 x . Three distinct points P, Q and R are
8 20
randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a
positive integer, is
71 73 79 83
(A) (B) (C) (D)
220 220 220 220
3/10
