Part II : Mathematics
SECTION−I
Single Correct Choice Type
This section contains 8 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.
1. Let P(3, 2, 6) be a point in space and Q be a point on the line Then the value of μ for which the vector
is parallel to the plane x – 4y + 3z = 1 is
(A) 1/4
(B) −1/4
(C) 1/8
(D) −1/8
2. Tangents drawn from the point P (1, 8) to the circle x2 + y2 − 6x − 4y − 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is
(A) x2 + y2 + 4x − 6y + 19 = 0
(B) x2 + y2 − 4x − 10y + 19 = 0
(C) x2 + y2 − 2x + 6y − 29 = 0
(D) x2 + y2 − 6x − 4y + 19 = 0
3. Let f be a non-negative function defined on the interval [0, 1]. If 0 ≤ x ≤ 1, and f(0) = 0, then
(A)
(B)
(C)
(D)
4. Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation is
(A) 48
(B) 32
(C) 40
(D) 80
5. The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2 = 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is
(A)
(B)
(C)
(D)
6. Ifare unit vectors such that
then
(A)
(B)
(C)
(D)
7. Let z = cosθ + i sinθ. Then the value of at θ = 2° is
(A)
(B)
(C)
(D)
8. The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
(A) 55
(B) 66
(C) 77
(D) 88
SECTION−II
Multiple Correct Choice Type
This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct.
9. Area of the region bounded by the curve y = ex and lines x = 0 and y = e is
(A)
(B)
(C)
(D)
10. Let a > 0. If L is finite, then
(A) a = 2
(B) a = 1
(C) L = 1/64
(D) L = 1/32
11. In a triangle ABC with fixed base BC, the vertex A moves such that If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then
(A) b + c = 4a
(B) b + c = 2a
(C) locus of point A is an ellipse
(D) locus of point A is a pair of straight lines
12. If then
(A)
(B)
(C)
(D)
SECTION−III
Comprehension Type
This section contains 2 groups of questions. Each group has 3 multiple choice questions based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.
Paragraph for question Nos. 33 to 35
Let 𝒜 be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
13. The number of matrices in 𝒜 is
(A) 12
(B) 6
(C) 9
(D) 3
14. The number of matrices A in 𝒜 for which the system of linear equations has a unique solution, is
(A) less than 4
(B) at least 4 but less than 4
(C) atleast 7 but less than 10
(D) at least 10
15. The number of matrices A in 𝒜 for which the system of linear equations is inconsistent, is
(A) 0
(B) more than 2
(C) 2
(D) 1
Paragraph for question Nos. 16 to 18
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
16. The probability that X = 3 equals
(A)
(B)
(C)
(D)
17. The probability that X ≥ 3 equals
(A)
(B)
(C)
(D)
18. The conditional probability that X ≥ 6 given X > 3 equals
(A)
(B)
(C)
(D)
SECTION − IV
Matrix − Match Type
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statement in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement (s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example:
If the correct matches are A − p, s and t; B − q and r; C − p and q; and D −s and t; then choose the correct answer.
19. Match the statements/expressions in Column I with the open intervals in Column II.
(A) (A) → (r, s, t) ; (B) → (q, r) ; (C) → (p, s, q, t) ; (D) → (r)
(B) (A) → (s, p, r) ; (B) → (r, p) ; (C) → (s, t, r, q) ; (D) → (t)
(C) (A) → (p, q, s) ; (B) → (p, t) ; (C) → (p, q, r, t) ; (D) → (s)
(D) (A) → (t, r) ; (B) → (s, r, q) ; (C) → (s, p, t, q) ; (D) → (p)
20. Match conics in Column I with the statement/expressions in Column II.
(A) (A) → (p) ; (B) → (s, t) ; (C) → (r) ; (D) → (q, s)
(B) (A) → (r) ; (B) → (q, r) ; (C) → (s) ; (D) → (t, p)
(C) (A) → (s) ; (B) → (q, r) ; (C) → (t) ; (D) → (s, p)
(D) (A) → (t) ; (B) → (q, p) ; (C) → (p) ; (D) → (t, r)
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