JEE (ADVANCED) 2018 PAPER 2

PART II-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 any positive integer n, define f_n : (0, ∞) → ℝ as

( Here, the inverse trigonometric function tan^–1 x assumes values in (–π/2, π/2).)

Then, which of the following statement(s) is (are) TRUE?

Answer: (D)

2. Let T be the line passing through the points P(−2, 7) and Q(2, −5). Let F₁ be the set of all pairs of circles (S₁, S₂) such that T is tangent to S₁ at P and tangent to S₂ at Q, and also such that S₁ and S₂ touch each other at a point, say, M. Let E₁ be the set representing the locus of M as the pair (S₁, S₂) varies in F₁. Let the set of all straight line segments joining a pair of distinct points of E₁ and passing through the point R(1, 1) be F₂. Let E₂ be the set of the mid-points of the line segments in the set F₂. Then, which of the following statements(s) is (are) TRUE?

(A) The point (−2, 7) lies in E₁

(B) The point (4/5, 7/5) does NOT lie in E₂

(D) The point (0, 3/2) does NOT lie in E₁

Answer: (B, D)

3. Let S be the set of all column matrices such that b₁, b₂, b₃ ∈ ℝand the system of equations (in real variables)

−x + 2y + 5z = b₁

2x – 4y + 3z = b₂

x – 2y + 2z = b₃

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each ∈ S?

(A) x + 2y + 3z = b₁, 4y + 5z = b₂ and x + 2y + 6z = b₃

(B) x + y + 3z = b₁, 5x + 2y + 6z = b₂ and −2x – y – 3z = b₃

(D) x + 2y + 5z = b₁, 2x + 3z= b₂ and x + 4y – 5z = b₃

Answer: (A, D)

4. Consider two straight lines, each of which is tangent to both the circle x² + y² = 1/2 and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0,0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is √2, then which of the following statement(s) is (are) TRUE?

(A) For the ellipse, the eccentricity is 1/√2 and the length of the latus rectum is 1

(B) For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2

(D) The area of the region bounded by the ellipse between the lines x = 1/√2 and x = 1 is

Answer: (A, C)

5. Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y ∈ ℝ, i = √–1) of the equation

where Then, which of the following statement(s) is (are) TRUE?

(A) If L has exactly one element, then |s| ≠ |t|

(B) If |s| = |t|, then L has infinitely many elements

(D) If L has more than one element, then L has infinitely many elements

Answer: (A, C, D)

6. Let f : (0, π) → ℝ be a twice differentiable function such that

If then which of the following statement(s) is (are) TRUE?

Answer: (B, C, D)

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 onscreen 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 the integral is _____ .

Answer: (2.00)

8. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the maximum possible value of the determinant of P is _____

Answer: (4.00)

9. Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of is ______.

Answer: (119.00)

10. Let f : ℝ → ℝ be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential equation then the value of

Answer: (0.40)

11. Let f : ℝ → ℝ be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f′(y) + f′(x) f(y) for all x, y ∈ ℝ. Then, the value of log_e (f(4)) is _________.

Answer: (2.00)

12. Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of PR is _____

Answer: (8.00)

13. Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O (0,0,0) is the origin. Let S(1/2, 1/2, 1/2) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If then the value of is ________.

Answer: (0.50)

14. Let X = (¹⁰C₁)² + 2(¹⁰C₂)² + 3(¹⁰C₃)² + … + 10(¹⁰C₁₀)²,

where 10Cr, r ∈ {1, 2, …, 10} denote binomial coefficients. Then, the value of is ______.

Answer: (646.00)

SECTION 3 (Maximum Marks: 12)

• This section contains FOUR (04) questions.

• Each question has TWO (02) matching lists: LIST‐I and LIST‐II.

• FOUR options are given representing matching of elements from LIST‐I and LIST‐II. ONLY ONE of these four options corresponds to a correct matching.

• For each question, choose the option corresponding to the correct matching.

• For each question, marks will be awarded according to the following marking scheme:

Full Marks : +3 If ONLY the option corresponding to the correct matching 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.

15. Let

and

(Here, the inverse trigonometric function sin^–1 x assumes values in [–π/2, π/2].)

The correct option is:

(A) P → 4; Q → 2; R → 1; S → 1

(B) P → 3; Q → 3; R → 6; S → 5

(D) P → 4; Q → 3; R → 6; S → 5

Answer: (A)

16. In a high school, a committee has to be formed from a group of 6 boys M₁, M₂, M₃, M₄, M₅, M₆ and 5 girls G₁, G₂, G3, G4, G₅.

(i) Let α₁ be the total number of ways in which the committee can be formed such that
the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let α2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

(iii) Let α3 be the total number of ways in which the committee can be formed such that
the committee has 5 members, at least 2 of them being girls.

(iv) Let α4 be the total number of ways in which the committee can be formed such that
the committee has 4 members, having at least 2 girls and such that both M₁ and G₁
are NOT in the committee together.

The correct option is:

(A) P → 4; Q → 6; R → 2; S → 1

(B) P → 1; Q → 4; R → 2; S → 3

(D) P → 4; Q → 2; R → 3; S → 1

Answer: (C)

17. Let where a > b > 0, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of 60° at one of its vertices N. Let the area of the triangle LMN be 4√3.