Part III – Mathematics
Section-1
• This section contains FIVE questions.
• Each question has FOUR options (A), (B), (C) and (D). ONLY ONE of these four options is correct.
1. A computer producing factory has only two plants T1 and T2. Plant T1 produces 20% and plant T2 produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that
P (computer turns out to be defective given that it is produced in plant T1) = 10 P (computer turns out to be defective given that it is produced in plant T2), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T2 is
(A) 
(B) 
(C) 
(D) 
2. A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
(A) 380
(B) 320
(C) 260
(D) 95
3. The least value of α ∈ ℝ for which 
for all x > 0, is
(A) 
(B) 
(C) 
(D) 
4. Let 
Suppose α1 and β1 are the roots of the equation x2 – 2x sec θ + 1 = 0 and α2 and β2 are the roots of the equation x2 + 2x tan θ – 1 = 0. If α1 > β1 and α2 > β2, then α1 + β2 equals
(A) 2(sec θ – tan θ)
(B) 2 sec θ
(C) − 2 tan θ
(D) 0
5. Let 
The sum of all distinct solutions of the equation √3 sec x + cosec x + 2(tan x – cot x) = 0 in the set S is equal to
(A) 
(B) 
(C) 
(D) 
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.
6. Let f : (0, ∞) → ℝ be a differentiable function such that 
for all x ∈ (0, ∞) and f(1) ≠ 1. Then
(A) 
(B) 
(C) 
(D) 
7. The circle C1 : x2 + y2 = 3. with centre at O, intersects the parabola x2 = 2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 2√3 and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis, then
(A) Q2Q3 = 12
(B) R2R3 = 4√6
(C) area of the triangle OR2R3 is 6√2
(D) area of the triangle PQ2Q3 is 4√2
8. A solution curve of the differential equation 
passes through the point (1, 3).
Then the solution curve
(A) intersects y = x + 2 exactly at one point.
(B) intersects y = x + 2 exactly at two points
(C) intersects y = (x + 2)2
(D) does NOT intersect y = (x + 3)2
9. In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and 2s = x + y + z. If 
and are of incircle of the triangle XYZ is 
then
(A) area of the triangle XYZ is 6√6
(B) the radius of circumcircle of the triangle XYZ is 
(C) 
(D) 
10. Let RS be the diameter of the circle x2 + y2 = 1, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s)
(A) 
(B) 
(C) 
(D) 
11. Let 
where α ∈ ℝ, k ≠ 0 and I is the identity matrix of order 3. If 
then
(A) α = 0, k = 8
(B) 4α – k + 8 = 0
(C) det (P adj(Q)) = 29
(D) det (Q adj (P)) = 213
12. Let f : ℝ → ℝ, g : ℝ → ℝ and h : ℝ → ℝ be differentiable functions such that f(x) = x3 + 3x + 2, g(f(x)) = x and h(g(g(x))) = x for all x ∈ ℝ. Then
(A) g'(2) = 1/15
(B) h'(1) = 666
(C) h(0) = 16
(D) h(g(3)) = 36
13. Consider a pyramid OPQRS located in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the midpoint T of diagonal OQ such that TS = 3. Then
(A) the acute angle between OQ and OS is 
(B) the equation of the plane containing the triangle OQS is x − y = 0
(C) the length of the perpendicular from P to the plane containing the triangle OQS is 
(D) the perpendicular distance from O to the straight line containing RS is 
Section – 3
• This section contains FIVE questions.
• The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.
14. Let 
and r, s ∈ {1, 2, 3}. Let 
and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P2 = −I is
15. Let m be the smallest positive integer such that the coefficient of x2 in the expansion of (1 + x)2 + (1 + x)3 + … + (1 + x)49 + (1 + mx)50 is (3n + 1) 51C3 for some positive integer n. Then the value of n is
16. The total number of distinct x ∈ [0, 1] for which 
is
17. The total number of distinct x ∈ ℝ for which 
is
18. Let α, β ∈ ℝ be such that 
Then 6(α + β) equals
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