- Attempt all questions.
- Rough work must be enclosed with answer book.
- While answering, refer to a question by its serial number as well as section heading. (eg.Q2/Sec.A)
- There is no negative marking.
- Answer each of Sections A, B, C at one place.
- Elegant solutions will be rewarded.
- Use of calculators, slide rule, graph paper and logarithmic, trigonometric and statistical tables is not permitted.
Section-A,
Section-B and Section-C must be
supported by mathematical arguments. In each of these sections order of
the questions must be maintained.
1. Real numbers x A) Such a choice is possible for
all real x
2. The consecutive sides of an equiangular hexagon measure x, y, 2, 2006, 3, 2007 units A) The hypothesis never takes
place 3. ABCD is a convex quadrilateral A) A circle can always be
circumscribed to it
4. A lattice point in a plane is one both of whose coordinates are integers. Let O be (, 1) and P any given lattice point. Then the number of lattice points Q, distinct from P, such that OP = OQ is A) 0 B) 1 C) not necessarily 0, not necessarily 1, but either 0 or 1 D) infinitely many E) none of these
5. i) f(x, y) is the polynomial f
6. (b-c)(x-a)(y-a) + (c-a)(x-b)(y-b) + (a-b)(x-c)(y-c) is
A) independent of x, but not of y B) independent of y, but not of x
SECTION-B
This section has Six Questions. In each question a blank is left. Fill in the blank.
(6x2=12 MARKS) 1. For the purpose of this
question, a square is
considered a kind of rectangle. Given the rectangle with vertices (0,
0), (0, 223), (9, 223), (9, 0), divided into 2007 unit squares by
horizontal and vertical lines. By cutting off a rectangle from the
given rectangle, we mean making cuts along horizontal and (or) vertical
lines to produce a smaller rectangle. Let m be the smallest positive
integer such that a rectangle of area ‘m’ cannot be
cut off from the
given rectangle. Then m = _________ 2. A line has an acute angled inclination and does not pass through the origin. If it makes intercepts a and b on x-, y-axes respectively, then _________ 3. If k is a positive integer,
let D 4. The digits of a positive integer m can be rearranged to form the positive integer n such that m+n is the 2007-digited number, each digit of which is 9. The number of such positive integers m is ________. 5. and are chords of a circle such
that
and
intersect in a point E outside the circle. F is a point on the minor
arc BD such that FAB = 22
6. The quadratic ax
SECTION-C (6x2=12 MARKS) 1. Explain a way of subdividing a 102 X 102 square into 2007 non-overlapping squares of integral sides. 2. ABC is a triangle. Explain how you inscribe a rhombus BDEF in the triangle such that D , E and F . 3. Equilateral triangle ABC
has centroid G. A 4. P(x) is a polynomial in x with
real coefficients. Given that the polynomial P 5. Find the homogeneous function
of 2 6. If 3yz + 2y + z + 1 = 0 and 3zx + 2z + x + 1 = 0, then prove that 3xy + 2x + y + 1 = 0.
1. x 2. Lines 3. i) ABC = 120 4. a 5. Prove that for all integers n 2, 2 6. Resolve x |