Part III : Mathematics
Section-1
• This section contains SIX questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.
1. Let 
and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50 – Q = I, 
equals
(A) 52
(B) 103
(C) 201
(D) 205
2. The value of 
is equal to
(A) 
(B) 
(C) 
(D) 
3. Let bi > 1 for i = 1, 2, …, 101. Suppose loge b1 loge b2, …, loge b101 are in Arithmetic Progression (A.P.) with the common difference loge 2. Suppose a1, a2, …, a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 + …+ b51 and s = a1 + a2 + … + a51, then
(A) s > t and a101 > b101
(B) s > t and a101 < b101
(C) s < t and a101 > b101
(D) s < t and a101 < b101
4. The value of 
is equal to
(A) 
(B) 
(C) 
(D) 
5. Let P be the image of the point (3, 1, 7) with respect to the plane x − y + z = 3. Then the equation of the plane passing through P and containing the straight line 
is
(A) x + y – 3z = 0
(B) 3x + z = 0
(C) x – 4y + 7z = 0
(D) 2x – y = 0
6. Area of the region 
is equal to
(A) 1/6
(B) 4/3
(C) 3/2
(D) 5/3
Section-2
• This section contains EIGHT questions.
• Each question has FOUR options (A), (B), (C) and (D). ONE OR MORE THAN ONE of these four option(s) is(are) correct.
7. Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then
(A) SP = 2√5
(B) SQ : QP = (√5 + 1) : 2
(C) the x-intercept of the normal to the parabola at P is 6
(D) the slope of the tangent to the circle at Q is 1/2
8. Let 
be a unit vector in 
Given that there exists are vector 
and 
Which of the following statement (s) is (are) correct?
(A) There is exactly one choice for such
(B) There are infinitely many choices for such
(C) If 
lies in the xy-plane then |u1| = |u2|
(D) If lies in the xz-plane then 2|u1| = |u3|
9. Let a, b ∈ ℝ and f : ℝ → ℝ be defined by f(x) = a cos (|x3 – x|) + b|x| sin(x3 + x|). Then f is |
(A) differentiable at x = 0 if a = 0 and b = 1
(B) differentiable at x = 1 if a = 1 and b = 0
(C) NOT differentiable at x = 0 if a = 1 and b = 0
(D) NOT differentiable at x = 1 if a = 1 and b = 1
10. Let 
for all x > 0. Then
(A) 
(B) 
(C) 
(D) 
11. Let f : ℝ → (0, ∞) and g : ℝ → ℝ be twice differentiable functions such that f ″ and g″ are continuous functions on ℝ. Suppose f ‘(2) = g(2) = 0, f″ (2) ≠ 0, and g″(2) ≠ 0.
If 
then
(A) f has a local minimum at x = 2
(B) f has a local maximum at x = 2
(C) f ″(2) > f(2)
(D) f(x) − f ″(x) = 0 for at least one x ∈ ℝ
12. Let a, b ∈ ℝ and a2 + b2 ≠ 0. Suppose 
where i = √−1. If z = x + iy and z ∈ S, then (x, y) lies on
(A) the circle with radius 
for a > 0, b ≠ 0
(B) the circle with radius 
for a < 0, b ≠ 0
(C) the x-axis for a ≠ 0, b = 0
(D) the y-axis for a = o, b ≠ 0
13. Let a, λ, μ ∈ ℝ. Consider the system of linear equations
ax + 2y = λ
3x – 2y = μ
Which of the following statement(s) is(are) correct?
(A) If a = −3, then the system has infinitely many solutions for all values of λ and μ
(B) If a ≠ −3, then the system has a unique solution for all values of λ and μ
(C) If λ + μ = 0, then the system has infinitely many solutions for a = −3
(D) If λ + μ ≠ 0, then the system has no solution for a = −3
14. Let 
be functions defined by f(x) = [x2 – 3] and g(x) = |x| f(x) + |4x – 7| f(x), where [y] denotes the greatest integer less than or equal to y for y ∈ ℝ. Then
(A) f is discontinuous exactly at three points in 
(B) f is discontinuous exactly at four points in
(C) g is NOT differentiable exactly at four points in 
(D) g is NOT differentiable exactly at five points in
Section-3
• 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
Football teams T1 and T2 have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T1 winning, drawing and losing a game against T2 are 
respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams T1 and T2, respectively, after two games.
15. P(X > Y) is
(A) 1/4
(B) 5/12
(C) 1/2
(D) 7/12
16. P(X = Y) is
(A) 11/36
(B) 1/3
(C) 13/36
(D) 1/2
PARAGRAPH 2
Let F1(x1, 0) and F2(x2, 0), for x1 < 0 and x2 > 0, be the foci of the ellipse 
Suppose a parabola having vertex at the origin and focus at F2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.
17. The orthocentre of the triangle F1MN is
(A) 
(B) 
(C) 
(D) 
18. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is
(A) 3 : 4
(B) 4 : 5
(C) 5 : 8
(D) 2 : 3
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