PART III : MATHEMATICS
SECTION-1
• 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.
1. If 2x – y + 1 = 0 is a tangent to the hyperbola 
then which of the following CANNOT be sides of a right angled triangle?
(A) a, 4, 1
(B) a, 4, 2
(C) 2a, 8, 1
(D) 2a, 4, 1
2. If a chord, which is not a tangent, of the parabola y2 = 16x has the equation 2x + y = p, and midpoint (h, k), then which of the following is(are) possible value(s) of p, h and k?
(A) p = −2, h = 2, k = −4
(B) p = −1, h = 1, k = −3
(C) p = 2, h = 3, k = −4
(D) p = 5, h = 4, k = −3
3. Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x) = x cos (π(x + [x])) is discontinuous?
(A) x = −1
(B) x = 0
(C) x = 1
(D) x = 2
4. Let f : ℝ → (0, 1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?
(A) x9 – f(x)
(B) 
(C) 
(D) 
5. Which of the following is(are) NOT the square of a 3 × 3 matrix with real entries?
(A) 
(B) 
(C) 
(D) 
6. Let a, b, x and y be real numbers such that a – b = 1 and y ≠ 0. If the complex number z = x + iy satisfies 
then which of the following is(are) possible value(s) of x?
(A) 
(B) 
(C) 
(D) 
7. Let X and Y be two events such that P(X) = 1/3, P(X|Y) = 1/2 and P (Y|X) = 2/5. Then
(A) P(Y) = 4/15
(B) P(X’|Y) = 1/2
(C) P(X ∩ Y) = 1/5
(D) P (X ∪ Y) = 2/5
9. Let f : ℝ → ℝ be a differentiable function such that f(0) = 0, f(π/2) = 3 and f ‘(0) = 1. If 
10. For a real number α, if the system 
of linear equations, has infinitely many solutions, then 1 + α + α2 =
11. Words of length 10 are formed using the letters A, B, C, D, E, F, G,H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, y/9x =
12. The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?
SECTION-3 (Maximum Marks: 18)
• This section contains SIX questions of matching type
• This section contains TWO tables (each having 3 columns and 4 rows)
• Based on each table, there are THREE questions
• Each question has FOUR options [A], [B], [C], and [D]. ONLY ONE of these four options is correct
Answer Q.13, Q.14 and Q.15 by appropriately matching the information given in the three columns of the following table.
13. For a = √2, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1, 1), then which of the following options is the only CORRECT combination for obtaining its equation?
(A) (I) (i) (P)
(B) (I) (ii) (Q)
(C) (II) (ii) (Q)
(D) (III) (i) (P)
14. If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?
(A) (I) (ii) (Q)
(B) (II) (iv) (R)
(C) (III) (i) (P)
(D) (III) (ii) (Q)
15. The tangent to a suitable conic (Column 1) at (√3, 1/2) is found to be √3x + 2y = 4, then which of the following options is the only CORRECT combination?
(A) (IV) (iii) (S)
(B) (IV) (iv) (S)
(C) (II) (iii) (R)
(D) (II) (iv) (R)
Answer Q.16, Q.17 and Q.18 by appropriately matching the information given in the three columns of the following table.
16. Which of the following options is the only CORRECT combination?
(A) (I) (i) (P)
(B) (II) (ii) (Q)
(C) (III) (iii) (R)
(D) (IV) (iv) (S)
17. Which of the following options is the only CORRECT combination?
(A) (I) (ii) (R)
(B) (II) (iii) (S)
(C) (III) (iv) (P)
(D) (IV) (i) (S)
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