JEE (ADVANCED) 2018 PAPER 1
PART-III MATHEMATICS
SECTION 1 (Maximum Marks: 24)
• This section contains SIX (06) questions.
• Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s).
• For each question, choose the correct option(s) to answer the question.
• Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +𝟒 If only (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 options.
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: If first, third and fourth are the ONLY three correct options for a question with second option being an incorrect option; selecting only all the three correct options will result in +4 marks. Selecting only two of the three correct options (e.g. the first and fourth options), without selecting any incorrect option (second option in this case), will result in +2 marks. Selecting only one of the three correct options (either first or third or fourth option) ,without selecting any incorrect option (second option in this case), will result in +1 marks. Selecting any incorrect option(s) (second option in this case), with or without selection of any correct option(s) will result in -2 marks.
1. For a non-zero complex number z, let arg(z) denote the principal argument with − π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE?
(A) arg(−1 − i) =π/4 , where i = √–1
(B) The function f : ℝ → (−π, π], defined by f(t) = arg(−1 + 𝑖𝑡) for all t ∈ ℝ, is continuous at all points of ℝ, where i =√–1
(C) For any two non-zero complex numbers z1 and z2, 
is an integer multiple of 2π
(D) For any three given distinct complex numbers z1, z2 and z3, the locus of the point 𝑧 satisfying the condition 
lies on a straight line
2. In a triangle 𝑃𝑄𝑅, let ∠𝑃𝑄𝑅 = 30° and the sides 𝑃𝑄 and 𝑄𝑅 have lengths 10√3 and 10, respectively. Then, which of the following statement(s) is (are) TRUE?
(A) ∠QPR = 45°
(B) The area of the triangle PQR is 25√3 and ∠QRP = 120°
(C) The radius of the incircle of the triangle PQR is 10√3 − 15
(D) The area of the circumcircle of the triangle PQR is 100 π
3. Let P1: 2x + y − z = 3 and P2: x + 2y + z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of P1 and P2 has direction ratios 1, 2, −1
(B) The line 
is perpendicular to the line of intersection of P1 and P2
(C) The acute angle between P1 and P2 is 60°
(D) If P3 is the plane passing through the point (4, 2, −2) and perpendicular to the line of intersection of 𝑃1 and 𝑃2, then the distance of the point (2, 1, 1) from the plane P3 is 2/√3
4. F or every twice differentiable function f : ℝ → [−2, 2] with (f(0))2 + (f′(0)) 2 = 85, which of the following statement(s) is (are) TRUE?
(A) There exist r, s ∈ ℝ, where r < s, such that f is one-one on the open interval (r, s)
(B) There exists x0 ∈ (−4, 0) such that |f′(𝑥0)| ≤ 1
(C) 
(D) (D) There exists α ∈ (−4, 4) such that f(α) + f′′(α) = 0 and f′(α) ≠ 0
5. Let f :ℝ → ℝ and g : ℝ → ℝ be two non-constant differentiable functions. If f′(x) = (e(f(x)−g(x)))g′(x) for all x ∈ ℝ, and 𝑓(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?
(A) f(2) < 1 − loge 2
(B) f(2) > 1 − loge 2
(C) g(1) > 1 − loge 2
(D) g(1) < 1 − loge 2
6. Let f: [0, ∞) → ℝ be a continuous function such that 
for all x ∈ [0, ∞). Then, which of the following statement(s) is (are) TRUE?
(A) The curve y = f(x) passes through the point (1, 2)
(B) The curve y = f(x) passes through the point (2, −1)
(C) The area of the region 
(D) The area of the region
SECTION 2 (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 (in decimal notation, truncated/rounded-off to the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) 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 numerical value is entered as answer.
Zero Marks : 0 In all other cases.
7. The value of 
is ______ .
8. The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _____ .
9. Let 𝑋 be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, … , and 𝑌 be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, … . Then, the number of elements in the set X ∪ Y is _____.
10. The number of real solutions of the equation
lying in the interval 
is _____ .
(Here, the inverse trigonometric functions sin−1x and cos−1x assume values in 
and [0, π], respectively.)
11. For each positive integer n, let 
For 𝑥 ∈ ℝ, let [𝑥] be the greatest integer less than or equal to 𝑥. If 
then the value of [L] is _____
12. Let 
be two unit vectors such that 
For some x, y ∈ ℝ, let 
and the vector 
is inclined at the same angle α to both , then the value of 8 cos2 α is _____ .
13. Let a, b, c be three non-zero real numbers such that the equation
has two distinct real roots α and β with 
Then, the value of b/a is _______.
14. A farmer F1 has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F2 takes away the region which lies between the side PQ and a curve of the form y = xn (n > 1). If the area of the region taken away by the farmer F2 is exactly 30% of the area of ΔPQR, then the value of 𝑛 is _____.
SECTION 3 (Maximum Marks: 12)
• This section contains TWO (02) paragraphs. Based on each paragraph, there are TWO (02) questions.
• Each question has FOUR options. ONLY ONE of these four options corresponds to 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.
PARAGRAPH “X”
Let S be the circle in the 𝑥𝑦-plane defined by the equation x2 + y2 = 4.
(There are two questions based on PARAGRAPH “X”, the question given below is one of them)
15. Let E1E2 and F1F2 be the chords of S passing through the point P0 (1, 1) and parallel to the x-axis and the y-axis, respectively. Let G1G2 be the chord of S passing through P0 and having slope −1. Let the tangents to S at E1 and E2 meet at E3, the tangents to S at F1 and F2 meet at F3, and the tangents to S at G1 and G2 meet at G3. Then, the points E3, F3, and G3 lie on the curve
(A) x + y = 4
(B) (x − 4)2 + (y − 4)2 = 16
(C) (x − 4)(y − 4) = 4
(D) xy = 4
PARAGRAPH “X”
Let S be the circle in the 𝑥𝑦-plane defined by the equation x2 + y2 = 4.
(There are two questions based on PARAGRAPH “X”, the question given below is one of them)
16. Let P be a point on the circle 𝑆 with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve
(A) (x + y)2 = 3xy
(B) x2/3 + y2/3 = 24/3
(C) x2 + y2 = 2xy
(D) x2 + y2 = x2y2
PARAGRAPH “A”
There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initially the seat 𝑅𝑖 is allotted to the student 𝑆𝑖, 𝑖 = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.
(There are two questions based on PARAGRAPH “A”, the question given below is one of them)
17. The probability that, on the examination day, the student S1 gets the previously allotted seat R1, and NONE of the remaining students gets the seat previously allotted to him/her is
(A) 3/40
(B) 1/8
(C) 7/40
(D) 1/5
PARAGRAPH “A”
There are five students S1, S2, S3, S4 and S5 in a music class and for them there are five seats R1, R2, R3, R4 and R5 arranged in a row, where initially the seat 𝑅𝑖 is allotted to the student 𝑆𝑖, 𝑖 = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.
(There are two questions based on PARAGRAPH “A”, the question given below is one of them)
18. For i = 1, 2, 3, 4, let Ti denote the event that the students Si and Si+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T1 ∩ T2 ∩ T3 ∩ T4 is
(A) 1/15
(B) 1/10
(C) 7/60
(D) 1/5
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