MATHEMATICS 2003
Time: Two Hours (8.30 AM – 10.30 AM) |
Max.Marks:60 |
All answers to questions in Section-A, Section-B, Section-C must be supported by mathematical arguments. In each of these sections order of the parts must be maintained.
SECTION-A
1. C is a circle with radius r. C_{1}, C_{2},
C_{3}, ....... C_{2003} are unit circles placed along the
circumference of C touching C externally. Also the pairs C_{1}, C_{2};
C_{2}, C_{3}; ........ C_{2002}, C_{2003}; C_{2003},
C_{1} touch. Then r is equal to
A)
B)
C)
D)
E) None of these
2. If n is a natural number, then is a positive integer
A) when n is even B) when n is odd C) only when n = 117 or n = 119 D) only when n = 1 or n = 3 E) none of these
3. The line segment
is completely external to a fixed circle S. A variable circle C through A, B
moves such that it intersects S in distinct points P and Q. In one position of
C, is parallel to .
Then
A) there is precisely one more position of C such that
is parallel to
B) is parallel to
for infinitely many position of C and
intersects for infinitely many
positions of C.
C) is parallel to
for all positions of C.
D) the hypothesis is wrong because there cannot be any position of C such that
is parallel to
E) None of these
4. f(x) is a polynomial of degree 2003. g(x) = f(x + 1) –
f(x). Then g(x) is a polynomial of degree m, where
A) m = 2003 B) m = 2002 C) m cannot be uniquely determined, but
D) m may be undefined in some cases E) none of these
5. D PQR is the midpoint triangle
of D ABC. Then both the triangles
A) have the same circumcentre B) have the same orthocenter
C) have the same centroid
D) are such that the circumcentre of the smaller triangle is the incentre of
the bigger triangle
E) None of these
SECTION-B
Fill in the blank. (5×2=10 MARKS)
1. (m, n) is a pair of positive integers such that i) m < n, ii) their gcd is 2003, iii) m, n are both 4-digit numbers. The number of such ordered pairs is ________
2. In a triangle ABC, AB = 12, BC = 18, CA = 25. A semicircle is inscribed in D ABC such that the diameter of the semicircle lies on . If O is the centre of teh circle, then the length AO = ___
3. If x is the recurring decimal 0.037, then x^{1/3} is the recurring decimal _____
4. The diagonals of a quadrilateral ABCD intersect in the point O. AO = OC. P is the midpoint of BD. Then D APB + D APD - D BPC - D DPC = _______
5. If x is any real number, then Int x stands for the unique integer n satisfying and Fr x stands for (x – Int x). If m is a positive integer, then = ______
SECTION-C
III. (4×2=8 MARKS)
1. Solve in positive integers m,n the equation 2^{3m+1} + 3^{2n} + 5^{m} + 5^{n} = 2003
2. Given locations P, Q, R of the points of contact of the sides of a triangle ABC with its incircle, describe the procedure to construct triangle ABC.
3. The solution set of the inequation is {x / x is real and }. Determine a : b : c.
4. ABCD is a cyclic quadrilateral. bisects . AB = 10, AD = 12, DC = 11, Determine BC.
SECTION-D
IV. (3×4=12 MARKS)
1. To each of the first two of the 4 numbers 1, 19, 203, 2003 is added x and to each of the last two y. The numbers form a G.P. Find all such ordered pairs (x, y) of real numbers.
2. Show that the 4 points A, Q, X, R are concyclic in the following figure.
3. Factorize a(b^{2} + c^{2} – a^{2}) + b(c^{2} + a^{2} – b^{2}) + c(a^{2} + b^{2} – c^{2}) – 2abc.
SECTION-E
(2×6=12 MARKS)
1. p is a plane. O is the origin. k > 0. is called a k-stretch of the plane if f_{k}(O) = 0 and for every f_{k}(A) is B where and OB = k.OA. P, Q are two points on the unit circle with origin as centre and PQ = 1. The plane is subjected alternatively to a 1/2-stretch and a 3-stretch, starting from the former. P and Q are transformed to P_{2003}, Q_{2003} respectively at the end of the 2003rd stretch. Determine the distance P_{2003} Q_{2003}.
2.
Determine the domain of f(x), in other words, determine the set of all real x for which f(x) is a real number. Sketch the graph of the function y = f(x). Determine from the graph the area of the region enclosed between the liens y = 0, x = 1 and x = 3 and the curve y = f(x).SECTION-F
(1× 8=8 MARKS)
1. a and b are both 4-digit numbers a > b and one is obtained from the other by reversing the digits. Determine b if