MATHEMATICS2004
Time: Two Hours
(8.30 AM – 10.30 AM) 
Max. Marks: 60

NOTE:
 Attempt all questions.
 Answer to each question under Sections D, E should begin on a fresh page and
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.E)
 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.
Note: All answers to questions in SectionA,
SectionB, SectionC must be supported by mathematical arguments. In each
of these sections order of the questions must be maintained.
SECTIONA
This section has Five Questions. Each question is provided
with five alternative answers. Only one of them is the correct answer. Indicate
the correct answer by A, B, C, D, E. (5x2=10 MARKS)
 A lattice point is a point (x, y) in the plane such that x and y are
integers. The number of rectangles with corners at lattice points, sides
parallel to the axes, and center at the origin which have precisely 2004
lattice points on all of its sides put together is
A) 0 
B) 501 
C) 499 
D) 500 
E) 2004 
 The number of positive integers which divide 2004 to leave a remainder of 24
is
A) 36 
B) 20 
C) 22 
D) 21 
E) 34 
 ABCD is a parallelogram. is
parallel to AB, with
and in the interior of
respectively. Then
A) AC < is
possible if is acute
B) AC < is
possible if is right
C) AC <
whenever is obtuse
D) AC
whenever is obtuse
E) none of these
 a, b, c, d are real constants, x a real variable. Real numbers p and q exist
such that p(ax+b) + q(cx+d) = ex+f
 for every pair of reals (e, f)
 for every pair of reals (e, f), if they exist for one pair of reals (e, f)
 for every pair of reals (e, f), if they exist uniquely for one pair of
reals (e, f)
 for no pair of reals (e, f), if they do not exist for some pair of reals
(e, f)
 none of these
 By a chord of the curve y = x^{3} we mean any line joining two
distinct points on it. The number of chords which have slope –1 is
A) infinite 
B) 0 
C) 1 
D) 2 
E) None 
SECTIONB
This section has Five Questions. In each question a blank is
left. Fill in the blank. (5x2=10 MARKS)
 The descending A.P. of 4 distinct positive integers with greatest possible
last term and sum 2004 is __________
 The radii of circles C_{1}, C_{2}, …., C_{2004}
are respectively r_{1}, r_{2}, …., r_{2004}. If r_{1}
= 1 and r_{i} = r_{i1 }+ 1 for i = 2, 3,…..,2004, then r_{2004}
= _________
 f(n) = for every integer n.
p and q are integers such that p>q. The sign of f(p) – f(q) is _______
 M_{1} is the initial point of a ray in a plane. M_{i}, for
i {2, 3, ….,2004}, are
points on the ray such that M_{1}M_{2} = M_{2}M_{3}
= M_{3}M_{4} = … = M_{2003}M_{2004}. M_{1}
is (a,b) and M_{2004} is (c, d). If s_{x} and s_{y}
are respectively the sum of all xcoordinates and the sum of all
ycoordinates of M_{i} for i
{1, 2, 3,….,2004}, then (s_{x}, s_{y}) = ______
 The number of 2element sets of nonunit positive integers such that their
g.c.d. is 1 and l.c.m. is 2004 is ________
SECTIONC
 Solve in positive integers x and y the equation x^{2} + y^{2}
+ 155(x+y)+2xy = 2004.
 are distinct lines. P is a
point in the plane ABC and P B.
Explain how to draw a line through P such that if the line intersects ,
then BQ = BR.
 Prove that there is no polynomial f(x) with integral coefficients such
that f(1) = 2001 and f(3) = 2004.
 ABC is right angled at B. A
square is constructed on on
the side of opposite to that
of B. P is the center of the square. Prove that
SECTIOND
 Evaluate , where [x]
denotes the integral part of x.
 ABCD is a line segment trisected by the points B, C; P is any point on the
circle whose diameter is . If
the angles APB and CPD are respectively ,
evaluate tan.

Solve the system of equations in positive integers x, y, z:x^{2}y^{2}+z^{2} = 2004, x + y –
z = 48, xy – yz  zx = 125
 a and b are positive reals and
a line segment in a plane. For how many distinct points C in the plane will it
happen that for triangle ABC, the median and the altitude through C have lengths
a and b respectively?

Prove by induction that
SECTIONE

Find x if x+y+z+t = 1
x+3y+9z+27t = 81
x+4y+16z+64t = 256
x+167y+167^{2}z+167^{3}t = 167^{4
}
Hint: Avoid successive elimination of variables.
 The perimeter of a triangle is 2004. One side of the triangle is 21 times
the other. The shortest side is of integral length. Solve for lengths of the
sides of the triangle in every possible case.
