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
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
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
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
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
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
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
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) 
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) 
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, ∞)
11. Let 
for x ≠ Then
(A) 
(B) 
(C) 
(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
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
14. If 
then
(A) I > loge 99
(B) I < loge 99
(C) I < 49/50
(D) I > 49/50
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)
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
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
18. If a4 = 28, then p + 2q = 0
(A) 21
(B) 14
(C) 7
(D) 12
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