SCHOLASTIC APTITUDE TEST 1999

MATHEMATICS

• Answers must be written in English or the medium of instruction of the candidate in High school.
• Attempt all questions.
• Answer all the questions in the booklets provided for the purpose. No pages should be removed from the booklets.
• There is no negative marking.
• Answer all questions of section I at one place. Same applies to section II. The remaining questions can be answered in any order.
• Answers to sections I and II must be supported by mathematical reasoning.
• Use of calculators, slide rule, graph paper and logarithmic, trigonometric and statistical tables is not permitted

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

2. a is a number such that the exterior angle of a regular polygon measures 10a degrees. Then

3. f(n)=2f(n–1)+1 for all positive integers n. Then

A)	B)
C)	D)
E) None of these

4. For any triangle let S and I denote the circumcentre and the incentre respectively. Then SI is perpendicular to a side of

5. 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

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)

6. 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.
7. The coefficient of in is __________
8. 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 _____________
9. The number of positive integers less than one million (10⁶) in which the digits 5, 6, 7, 8, 9, 0 do not appear is ______________
10. 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)

11. Given any integer p, prove that integers m and n can be found such that p=3m+5n.
12. 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.
13. If a, b, c are all positive and c¹ 1, then prove that
14. Find the remainder obtained when x¹⁹⁹⁹ is divided by x²–1.
15. 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)

16. Solve the following system of 1999 equations in 1999 unknowns :
    x₁₊x₂₊x₃=0, x₂+x₃+x₄=0……., x₁₉₉₇+x₁₉₉₈+x₁₉₉₉=0,
    x₁₉₉₈+x₁₉₉₉+x₁=0, x₁₉₉₉+x₁+x₂=0
17. 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.
18. If x and y are positive numbers connected by the relation
    , prove that
    
    for any valid base of the logarithms.
19. 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).
20. Prove that 80 divides
21. 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.