Exam (elaborations) jee

JEE (Advanced) 2025 Paper 2




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 Let 𝑥0 be the real number such that 𝑒 𝑥0 + 𝑥0 = 0 . For a given real number 𝛼, define

3𝑥𝑒 𝑥 + 3𝑥 − 𝛼𝑒 𝑥 − 𝛼𝑥
𝑔(𝑥) =
3(𝑒 𝑥 + 1)
for all real numbers 𝑥.

Then which one of the following statements is TRUE?

(A) 𝑔(𝑥)+ 𝑒 𝑥0
For 𝛼 = 2 , lim | |=0
𝑥→𝑥0 𝑥−𝑥0

(B) 𝑔(𝑥)+ 𝑒 𝑥0
For 𝛼 = 2 , lim | 𝑥−𝑥0
|=1
𝑥→𝑥0

(C) 𝑔(𝑥)+ 𝑒 𝑥0
For 𝛼 = 3 , lim | |=0
𝑥→𝑥0 𝑥−𝑥0

(D) 𝑔(𝑥)+ 𝑒 𝑥0 2
For 𝛼 = 3 , lim | 𝑥−𝑥0
|=3
𝑥→𝑥0




Q.2 Let ℝ denote the set of all real numbers. Then the area of the region

1
{(𝑥, 𝑦) ∈ ℝ × ℝ ∶ 𝑥 > 0, 𝑦 > , 5𝑥 − 4𝑦 − 1 > 0, 4𝑥 + 4𝑦 − 17 < 0 }
𝑥

is

(A) 17 (B) 33
− log 𝑒 4 − log 𝑒 4
16 8
(C) 57 (D) 17
− log 𝑒 4 − log 𝑒 4
8 2




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,JEE (Advanced) 2025 Paper 2

Q.3 The total number of real solutions of the equation

1 6 tan 𝜃
𝜃 = tan−1 (2 tan 𝜃) − sin−1 ( )
2 9 + tan2 𝜃
is
𝜋 𝜋
(Here, the inverse trigonometric functions sin−1 𝑥 and tan−1 𝑥 assume values in [− 2 , 2
] and
𝜋 𝜋
(− 2 , 2 ), respectively.)


(A) 1 (B) 2 (C) 3 (D) 5



Q.4 Let 𝑆 denote the locus of the point of intersection of the pair of lines

4𝑥 − 3𝑦 = 12𝛼 ,
4𝛼𝑥 + 3𝛼𝑦 = 12 ,

where 𝛼 varies over the set of non-zero real numbers. Let 𝑇 be the tangent to 𝑆 passing through the
3
points (𝑝, 0) and (0, 𝑞), 𝑞 > 0, and parallel to the line 4 𝑥 − 𝑦=0.
√2


Then the value of 𝑝𝑞 is

(A) −6√2 (B) −3√2 (C) −9√2 (D) −12√2




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, JEE (Advanced) 2025 Paper 2

SECTION 2 (Maximum Marks: 16)

• This section contains FOUR (04) 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 𝐼 = (
1 0
) and 𝑃 = (
2 0
). Let 𝑄 = (
𝑥 𝑦
) for some non-zero real numbers 𝑥, 𝑦, and 𝑧,
0 1 0 3 𝑧 4
for which there is a 2 × 2 matrix 𝑅 with all entries being non-zero real numbers, such that
𝑄𝑅 = 𝑅𝑃 .

Then which of the following statements is (are) TRUE?

(A) The determinant of 𝑄 − 2𝐼 is zero
(B) The determinant of 𝑄 − 6𝐼 is 12
(C) The determinant of 𝑄 − 3𝐼 is 15
(D) 𝑦𝑧 = 2




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