Time: Two Hours

Max. Marks: 60

Section I

This section has 5 questions. Each question is provided with five alternative answers. Only one of them is correct. Indicate the correct answer by A or B or C or D or E. Order of the questions must be maintained. (5x2=10 Marks)

  1. Let R be the set of all real numbers. The number of functions satisfying the relation is
A) Infinite B) One C) Two D) Zero
E) None of the above
  1. a is a number such that the exterior angle of a regular polygon measures 10a degrees. Then
A) there is no such a B) there are infinitely many such a
B) there are precisely nine such a D) there are precisely seven such a
E) there are precisely ten such a  
  1. f(n)=2f(n–1)+1 for all positive integers n. Then
A) B)
C) D)
E) None of these  
  1. For any triangle let S and I denote the circumcentre and the incentre respectively. Then SI is perpendicular to a side of
A) any triangle B) no triangle
C) a right angled triangle D) an isosceles triangle
E) an obtuse angled triangle  
  1. If f(x)=x3+ax+b is divisible by (x-1)2 , then the remainder obtained when f(x) is divided by x+2 is
A) 1 B) 0 C) 3 D) –1
E) None of these      

Section II

This section has 5 questions. Each question is in the form of a statement with a blank. Fill the blank so that the statement is true. Maintain the order of the questions. (5x2=10 Marks)

  1. is a given line segment, H, K are points on it such that BH=HK=KC. P is a variable point such that
    (i) has the constant measure of
    a radians.
    (ii) has counter clockwise orientation
    Then the locus of the centroid of is the arc of the circle bounded by the chord with angle in the segment ____________ radians.
  2. The coefficient of in is __________
  3. n is a natural number such that
    i) the sum of its digits is divisible by 11
    ii) its units place is non-zero
    iii) its tens place is not a 9.
    Then the smallest positive integer p such that 11 divides the sum of digits of (n+p) is _____________
  4. The number of positive integers less than one million (106) in which the digits 5, 6, 7, 8, 9, 0 do not appear is ______________
  5. The roots of the polynomial are all positive and are denoted by a i, for i=1, 2, 3, ….., n. Then the roots of the polynomial
    are, in terms of
    a i, _________.

Section III

This section has 5 questions. The solutions are short and methods, easily suggested. Very long and tedious solutions may not get full marks. (5x2=10 Marks)

  1. Given any integer p, prove that integers m and n can be found such that p=3m+5n.
  2. E is the midpoint of side BC of a rectangle ABCD and F the midpoint of CD. The area of D AEF is 3 square units. Find the area of the rectangle.
  3. If a, b, c are all positive and c 1, then prove that
  4. Find the remainder obtained when x1999 is divided by x2–1.
  5. Remove the modulus :

Section IV

This section has 6 questions. The solutions involve either slightly longer computations or subtler approaches. Even incomplete solutions may get partial marks. (6x5=30 Marks)

  1. Solve the following system of 1999 equations in 1999 unknowns :
    x1+x2+x3=0, x2+x3+x4=0……., x1997+x1998+x1999=0,
    1998+x1999+x1=0, x1999+x1+x2=0
  2. Given base angles and the perimeter of a triangle, explain the method of construction of the triangle and justify the method by a proof. Use only rough sketches in your work.
  3. If x and y are positive numbers connected by the relation
    , prove that

    for any valid base of the logarithms.
  4. Let D XYZ denote the area of triangle XYZ. ABC is a triangle. E, F are points on and respectively. and intersect in O. If D EOB=4, D COF=8, D BOC=13, develop a method to estimate D ABC. (you may leave the solution at a stage where the rest is mechanical computation).
  5. Prove that 80 divides
  6. ABCD is a convex quadrilateral. Circles with AB, BC, CD, DA as a diameters are drawn. Prove that the quadrilateral is completely covered by the circles. That is, prove that there is no point inside the quadrilateral which is outside every circle.