JEE Advanced Exam 2017 Paper-II Mathematics Question Paper With Answer Key

PART III : MATHEMATICS

SECTION 1 (Maximum Marks: 21)

• This section contains SEVEN questions

• Each question has FOUR options [A], [B], [C] and [D]. ONLY ONE of these four options is correct

1. The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y – 2z = 5 and 3x – 6y – 2z = 7, is

(A) 14x + 2y – 15z = 1

(B) 14x – 2y + 15z = 27

(C) 14x + 2y + 15z = 31

(D) −14x + 2y + 15z = 3

Answer: (C)

2. Let O be the origin and let PQR be an arbitrary triangle. The point S is such that

   

Then the triangle PQR and S as its

(A) centroid

(B) circumcentre

(C) incentre

(D) orthocenter

Answer: (D)

3. If y = y(x) satisfies the differential equation  x > 0 and y(0) = √7, then y(256) =

(A) 3

(B) 9

(C) 16

(D) 80

Answer: (A)

4. If f : ℝ → ℝ is a twice differentiable function such that f ” (x) > 0 for all x ∈ ℝ, and  then

(A) f ‘(1) ≤ 0

(B) 0 < f ‘(1) ≤ 1/2

(C) 1/2 < f ‘ (1) ≤ 1

(D) f ‘ (1) > 1

Answer: (D)

5. How many 3 × 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of MTM is 5?

(A) 126

(B) 198

(C) 162

(D) 135

Answer: (B)

6. Let S = {1, 2, 3, …, 9}. for k = 1, 2, …, 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =

(A) 210

(B) 252

(C) 125

(D) 126

Answer: (D)

7. Three randomly chosen non-negative integers x, y and z are found to satisfy the equation x + y + z = 10. Then the probability that z is even, is

(A) 36/55

(B) 6/11

(C) 1/2

(D) 5/11

Answer: (B)

SECTION 2 (Maximum Marks: 28)

• This section contains SEVEN questions

• Each question has FOUR options [A], [B], [C] and [D]. ONE OR MORE THAN ONE of these four options is(are) correct

8. If  then

(A) 

(B) 

(C)   

(D)   

Answer: (BONUS)

9. Let α and β be nonzero real numbers such that 2(cos β – cos α) + cos α cos β = 1. Then which of the following is/are true?

(A)   

(B) 

(C) 

(D)  

Answer: (BONUS)

10. If f : ℝ → ℝ is differentiable function such that f ‘ (x) > 2f(x) for all x ∈ ℝ, and f (0) = 1, then

(A) f (x) is increasing in (0, ∞)

(B) f(x) is decreasing in (0, ∞)

(C) f(x) > e2x in (0, ∞)

(D) f ‘ (x) < e2x in (0, ∞)

Answer: (A, C)

11. Let  for x ≠ Then

(A)  

(B)   

(C)   

(D)  

Answer: (A, D)

12. If then

(A) f ‘(x) = 0 at exactly three points in (−π, π)

(B) f ‘ (x) = 0 at more than three points in (−π, π)

(C) f(x) attains its maximum at x = 0

(D) f(x) attains its minimum at x = 0

Answer: (B, C)

13. If the line x = α divides the area of region R = {(x, y) ∈ ℝ2 : x3 ≤ y ≤ x, 0 ≤ x ≤ 1} into two equal parts, then

(A) 0 < α ≤ 1/2

(B) 1/2 < α < 1

(C) 2α4 – 4α2 + 1 = 0

(D) α4 + 4α2 – 1 = 0

Answer: (B, C)

14. If  then

(A) I > loge 99

(B) I < loge 99

(C) I < 49/50

(D) I > 49/50

Answer: (B, D)

SECTION 3 (Maximum Marks: 12)

• This section contains TWO paragraphs

• Based on each paragraph, there are TWO questions

• Each question has FOUR options [A], [B], [C], and [D]. ONLY ONE of these four options is correct

PARAGRAPH 1

Let O be the origin, and  be three unit vectors in the directions of the sides  respectively, of a triangle PQR.

15. 

(A) sin (P + Q)

(B) sin 2R

(C) sin (P + R)

(D) sin (Q + R)

Answer: (A)

16. If the triangle PQR varies, then the minimum value of cos (P + Q) + cos (Q + R) + cos (R + P)

(A) −5/3

(B) −3/2

(C) 3/2

(D) 5/3

Answer: (B)

PARAGRAPH 2

Let p, q be integers and let α, β be the roots of the equation, x2 – x – 1 = 0, where α ≠ β. For n = 0, 1, 2, …, let an = pαn + qβn

FACT : If a and b are rational numbers and a + b√5 = 0, then a = 0 = b.

17. a12 =

(A) a11 – a10

(B) a11 + a10

(C) 2a11 + a­10

(D) a11 + 2a10

Answer: (B)

18. If a4 = 28, then p + 2q = 0

(A) 21

(B) 14

(C) 7

(D) 12

Answer: (D)

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