JEE (Advanced) 2025 Paper 1
SECTION 1 (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.1 L ℝ denote the set of all real numbers. Let 𝑎𝑖 , 𝑏𝑖 ∈ ℝ for 𝑖 ∈ {1, 2, 3}.
Let
Define the functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ, and ℎ: ℝ → ℝ by
𝑓(𝑥) = 𝑎1 + 10𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + 𝑥 4 ,
𝑔(𝑥) = 𝑏1 + 3𝑥 + 𝑏2 𝑥 2 + 𝑏3 𝑥 3 + 𝑥 4 ,
ℎ(𝑥) = 𝑓(𝑥 + 1) − 𝑔(𝑥 + 2).
If 𝑓(𝑥) ≠ 𝑔(𝑥) for every 𝑥 ∈ ℝ, then the coefficient of 𝑥 3 in ℎ(𝑥) is
(A) 8
(B) 2
(C) −4
(D) −6
Q.2 Three students 𝑆1 , 𝑆2 , and 𝑆3 are given a problem to solve. Consider the following events:
𝑈: At least one of 𝑆1 , 𝑆2 , and 𝑆3 can solve the problem,
𝑉: 𝑆1 can solve the problem, given that neither 𝑆2 nor 𝑆3 can solve the problem,
𝑊: 𝑆2 can solve the problem and 𝑆3 cannot solve the problem,
𝑇: 𝑆3 can solve the problem.
For any event 𝐸, let 𝑃(𝐸) denote the probability of 𝐸. If
1 1 1
𝑃(𝑈) = , 𝑃(𝑉) = , and 𝑃(𝑊) = ,
2 10 12
then 𝑃(𝑇) is equal to
(A) 13 (B) 1 (C) 19 (D) 1
36 3 60 4
1/9
,JEE (Advanced) 2025 Paper 1
Q.3 Let ℝ denote the set of all real numbers. Define the function 𝑓: ℝ → ℝ by
2 2
1
𝑓(𝑥) = { 2 − 2𝑥 − 𝑥 sin 𝑥 if 𝑥 ≠ 0,
2 if 𝑥 = 0 .
Then which one of the following statements is TRUE?
(A) The function 𝑓 is NOT differentiable at 𝑥 = 0
(B) There is a positive real number 𝛿, such that 𝑓 is a decreasing function on the interval (0, 𝛿)
(C) For any positive real number 𝛿, the function 𝑓 is NOT an increasing function on the interval
(−𝛿, 0)
(D) 𝑥 = 0 is a point of local minima of 𝑓
Q.4 0
Consider the matrix
2 0 0
𝑃 = ( 0 2 0) .
0 0 3
Let the transpose of a matrix 𝑋 be denoted by 𝑋 𝑇 . Then the number of 3 × 3 invertible matrices 𝑄
with integer entries, such that
𝑄 −1 = 𝑄 𝑇 and 𝑃𝑄 = 𝑄𝑃 ,
is
(A) 32 (B) 8 (C) 16 (D) 24
2/9
,JEE (Advanced) 2025 Paper 1
SECTION 2 (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.5 Let 𝐿1 be the line of intersection of the planes given by the equations
2𝑥 + 3𝑦 + 𝑧 = 4 and 𝑥 + 2𝑦 + 𝑧 = 5 .
Let 𝐿2 be the line passing through the point 𝑃(2, −1, 3) and parallel to 𝐿1 . Let 𝑀 denote the plane
given by the equation
2𝑥 + 𝑦 − 2𝑧 = 6 .
Suppose that the line 𝐿2 meets the plane 𝑀 at the point 𝑄. Let 𝑅 be the foot of the perpendicular
drawn from 𝑃 to the plane 𝑀 .
Then which of the following statements is (are) TRUE?
(A) The length of the line segment 𝑃𝑄 is 9√3
(B) The length of the line segment 𝑄𝑅 is 15
(C) 3
The area of Δ𝑃𝑄𝑅 is 2
√234
(D) 1
The acute angle between the line segments 𝑃𝑄 and 𝑃𝑅 is cos−1 ( )
2√3
3/9
, JEE (Advanced) 2025 Paper 1
Q.6 Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the
functions 𝑓: ℕ → ℤ and 𝑔: ℤ → ℕ defined by
(𝑛 + 1)/2 if 𝑛 is odd ,
𝑓(𝑛) = {
(4 − 𝑛)/2 if 𝑛 is even ,
and
3 + 2𝑛 if 𝑛 ≥ 0 ,
𝑔(𝑛) = {
−2𝑛 if 𝑛 < 0 .
Define (𝑔 ∘ 𝑓)(𝑛) = 𝑔(𝑓(𝑛)) for all 𝑛 ∈ ℕ, and (𝑓 ∘ 𝑔)(𝑛) = 𝑓(𝑔(𝑛)) for all 𝑛 ∈ ℤ.
Then which of the following statements is (are) TRUE?
(A) 𝑔 ∘ 𝑓 is NOT one-one and 𝑔 ∘ 𝑓 is NOT onto
(B) 𝑓 ∘ 𝑔 is NOT one-one but 𝑓 ∘ 𝑔 is onto
(C) 𝑔 is one-one and 𝑔 is onto
(D) 𝑓 is NOT one-one but 𝑓 is onto
Q.7 Let ℝ denote the set of all real numbers. Let 𝑧1 = 1 + 2𝑖 and 𝑧2 = 3𝑖 be two complex numbers,
where 𝑖 = √−1 . Let
𝑆 = {(𝑥, 𝑦) ∈ ℝ × ℝ ∶ |𝑥 + 𝑖𝑦 − 𝑧1 | = 2|𝑥 + 𝑖𝑦 − 𝑧2 | }.
Then which of the following statements is (are) TRUE?
(A) 1 10
𝑆 is a circle with centre (− 3 , 3
)
(B) 1 8
𝑆 is a circle with centre (3 , 3)
(C) √2
𝑆 is a circle with radius 3
(D) 2√2
𝑆 is a circle with radius 3
4/9
SECTION 1 (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.1 L ℝ denote the set of all real numbers. Let 𝑎𝑖 , 𝑏𝑖 ∈ ℝ for 𝑖 ∈ {1, 2, 3}.
Let
Define the functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ, and ℎ: ℝ → ℝ by
𝑓(𝑥) = 𝑎1 + 10𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + 𝑥 4 ,
𝑔(𝑥) = 𝑏1 + 3𝑥 + 𝑏2 𝑥 2 + 𝑏3 𝑥 3 + 𝑥 4 ,
ℎ(𝑥) = 𝑓(𝑥 + 1) − 𝑔(𝑥 + 2).
If 𝑓(𝑥) ≠ 𝑔(𝑥) for every 𝑥 ∈ ℝ, then the coefficient of 𝑥 3 in ℎ(𝑥) is
(A) 8
(B) 2
(C) −4
(D) −6
Q.2 Three students 𝑆1 , 𝑆2 , and 𝑆3 are given a problem to solve. Consider the following events:
𝑈: At least one of 𝑆1 , 𝑆2 , and 𝑆3 can solve the problem,
𝑉: 𝑆1 can solve the problem, given that neither 𝑆2 nor 𝑆3 can solve the problem,
𝑊: 𝑆2 can solve the problem and 𝑆3 cannot solve the problem,
𝑇: 𝑆3 can solve the problem.
For any event 𝐸, let 𝑃(𝐸) denote the probability of 𝐸. If
1 1 1
𝑃(𝑈) = , 𝑃(𝑉) = , and 𝑃(𝑊) = ,
2 10 12
then 𝑃(𝑇) is equal to
(A) 13 (B) 1 (C) 19 (D) 1
36 3 60 4
1/9
,JEE (Advanced) 2025 Paper 1
Q.3 Let ℝ denote the set of all real numbers. Define the function 𝑓: ℝ → ℝ by
2 2
1
𝑓(𝑥) = { 2 − 2𝑥 − 𝑥 sin 𝑥 if 𝑥 ≠ 0,
2 if 𝑥 = 0 .
Then which one of the following statements is TRUE?
(A) The function 𝑓 is NOT differentiable at 𝑥 = 0
(B) There is a positive real number 𝛿, such that 𝑓 is a decreasing function on the interval (0, 𝛿)
(C) For any positive real number 𝛿, the function 𝑓 is NOT an increasing function on the interval
(−𝛿, 0)
(D) 𝑥 = 0 is a point of local minima of 𝑓
Q.4 0
Consider the matrix
2 0 0
𝑃 = ( 0 2 0) .
0 0 3
Let the transpose of a matrix 𝑋 be denoted by 𝑋 𝑇 . Then the number of 3 × 3 invertible matrices 𝑄
with integer entries, such that
𝑄 −1 = 𝑄 𝑇 and 𝑃𝑄 = 𝑄𝑃 ,
is
(A) 32 (B) 8 (C) 16 (D) 24
2/9
,JEE (Advanced) 2025 Paper 1
SECTION 2 (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.5 Let 𝐿1 be the line of intersection of the planes given by the equations
2𝑥 + 3𝑦 + 𝑧 = 4 and 𝑥 + 2𝑦 + 𝑧 = 5 .
Let 𝐿2 be the line passing through the point 𝑃(2, −1, 3) and parallel to 𝐿1 . Let 𝑀 denote the plane
given by the equation
2𝑥 + 𝑦 − 2𝑧 = 6 .
Suppose that the line 𝐿2 meets the plane 𝑀 at the point 𝑄. Let 𝑅 be the foot of the perpendicular
drawn from 𝑃 to the plane 𝑀 .
Then which of the following statements is (are) TRUE?
(A) The length of the line segment 𝑃𝑄 is 9√3
(B) The length of the line segment 𝑄𝑅 is 15
(C) 3
The area of Δ𝑃𝑄𝑅 is 2
√234
(D) 1
The acute angle between the line segments 𝑃𝑄 and 𝑃𝑅 is cos−1 ( )
2√3
3/9
, JEE (Advanced) 2025 Paper 1
Q.6 Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the
functions 𝑓: ℕ → ℤ and 𝑔: ℤ → ℕ defined by
(𝑛 + 1)/2 if 𝑛 is odd ,
𝑓(𝑛) = {
(4 − 𝑛)/2 if 𝑛 is even ,
and
3 + 2𝑛 if 𝑛 ≥ 0 ,
𝑔(𝑛) = {
−2𝑛 if 𝑛 < 0 .
Define (𝑔 ∘ 𝑓)(𝑛) = 𝑔(𝑓(𝑛)) for all 𝑛 ∈ ℕ, and (𝑓 ∘ 𝑔)(𝑛) = 𝑓(𝑔(𝑛)) for all 𝑛 ∈ ℤ.
Then which of the following statements is (are) TRUE?
(A) 𝑔 ∘ 𝑓 is NOT one-one and 𝑔 ∘ 𝑓 is NOT onto
(B) 𝑓 ∘ 𝑔 is NOT one-one but 𝑓 ∘ 𝑔 is onto
(C) 𝑔 is one-one and 𝑔 is onto
(D) 𝑓 is NOT one-one but 𝑓 is onto
Q.7 Let ℝ denote the set of all real numbers. Let 𝑧1 = 1 + 2𝑖 and 𝑧2 = 3𝑖 be two complex numbers,
where 𝑖 = √−1 . Let
𝑆 = {(𝑥, 𝑦) ∈ ℝ × ℝ ∶ |𝑥 + 𝑖𝑦 − 𝑧1 | = 2|𝑥 + 𝑖𝑦 − 𝑧2 | }.
Then which of the following statements is (are) TRUE?
(A) 1 10
𝑆 is a circle with centre (− 3 , 3
)
(B) 1 8
𝑆 is a circle with centre (3 , 3)
(C) √2
𝑆 is a circle with radius 3
(D) 2√2
𝑆 is a circle with radius 3
4/9
