Part III : Mathematics
Section I : Single Correct Answer Type
This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
1. The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x – y + z = 3 and at a distance 2/√3 from the point (3, 1, −1) is
(A) 5x – 11y + z = 17
(B) √2x + y = 3√2 – 1
(C) x + y + z = √3
(D) x – √2y = 1 – √2
2. Let PQR be a triangle of area ∆ with a = 2, b = 7/2 and c = 5/2, where a, b and c are the lengths of the sides of the triangle opposite to the angles at P, Q and R respectively. Then 
equals
(A) 
(B) 
(C) 
(D) 
3. If 
are vectors such that
and 
then a possible value of 
is
(A) 0
(B) 3
(C) 4
(D) 8
4. If P is a 3 × 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix 
such that
(A) 
(B) PX = X
(C) PX = 2X
(D) PX = −X
5. Let α(a) and β(a) be the roots of the equation 
where a > −1.
Then 
are
(A) 
(B) 
(C) 
(D) 
6. Four fair dice D1, D2 D3 and D4, each having six faces numbered 1, 2, 3, 4, 5 and 6, are rolled simultaneously. The probability that D4 shows a number appearing on One of D1, D2 and D3 is
(A) 
(B) 
(C) 
(D) 
7. The value of the integral
is
(A) 
(B) 
(C) 
(D) 
8. Let a1, a2, a3, … be in harmonic progression with a1 = 5 and a20 = 25. The least positive integer n for which an < 0 is
(A) 22
(B) 23
(C) 24
(D) 25
Section II : Paragraph Type
This section contains 6 multiple choices question relating to three paragraphs with two questions on each paragraph. Each question has four choices (A) , (B), (C) and (D) out of the ONLY ONE is correct.
Paragraph for Questions 9 and 10
Let an denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let bn = the number of such n-digit integers ending with digit 1 and cn = the number of such n-digit integers ending with digit 0.
9. The value of b6 is
(A) 7
(B) 8
(C) 9
(D) 11
10. Which of the following is correct ?
(A) a17 = a16 + a15
(B) c17 ≠ c16 + c15
(C) b17 ≠ b16 + c16
(D) a17 = c17 + b16
Paragraph for Question 11 and 12
Let f(x) = (1 – x)2 sin2x + x2 for all x ∈ R, and let 
for all x ∈ (1, ∞)
11. Which of the following is true?
(A) g is increasing on (1, ∞)
(B) g is decreasing on (1, ∞)
(C) g is increasing on (1, 2) and decreasing on (2, ∞)
(D) g is decreasing on (1, 2) and increasing on (2, ∞)
12. Consider the statements :
P : There exists some x ∈ R such that f(x) + 2x = 2(1 + x2)
Q : There exists some x ∈ R such that 2f(x) + 1 = 2x(1 + x)
Then
(A) both P and Q are true
(B) P is true and Q is false
(C) P is false and Q is true
(D) both P and Q are false
Paragraph for Questions 13 and 14
A tangent PT is drawn to the circle x2 + y2 = 4 at the point P(√3, 1). A straight line L, perpendicular to PT is a tangent to the circle (x – 3)2 + y2 = 1
13. A possible equation of L is
(A) x – √3y = 1
(B) x + √3y = 1
(C) x – √3y = −1
(D) x + √3y = 5
14. A common tangent of the two circles is
(A) x = 4
(B) y = 2
(C) x + √3y = 4
(D) x + 2√2y = 6
Section III : Multiple Correct Answer(s) Type
This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.
15. For every integer n, let an and bn real numbers. Let function f : R → R be given by
, for all integers n.
If f is continuous, then which of the following hold(s) for all n?
(A) an – 1 – bn – 1 = 0
(B) an – bn = 0
(C) an – bn + 1 = 1
(D) an – 1 – bn = −1
16. If 
for all x ∈ (0, ∞), then
(A) f has a local maximum at x = 2
(B) f is decreasing on (2, 3)
(C) there exists some c ∈ (0, ∞) such that f″ (c) = 0
(D) f has a local minimum at x = 3
17. If the straight lines 
and are coplanar, then the plane (s) containing these two lines is(are)
(A) y + 2z = −1
(B) y + z = −1
(C) y – z = −1
(D) y – 2z = 1
18. Let X and Y be two events such that 
and 
Which of the following is (are) correct?
(A) P (X ∪ Y) = 2/3
(B) X and Y are independent
(C) X and Y are not independent
(D) P(XC ∩ C) = 1/3
19. If the adjoint of a 3 × 3 matrix P is 
then the possible value(s) of the determinant of P is (are)
(A) −2
(B) −1
(C) 1
(D) 2
20. Let f : (−1, 1) → R be such that 
for 
Then the value(s) of 
is (are)
(A) 
(B) 
(C) 
(D) 
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