SCHOLASTIC APTITUDE TEST 
1995.
MATHEMATICS
Time: Two Hours 
Max. Marks: 60

(8.30 AM  10.30 AM) 

Answers must be written either in English or the medium of
instruction of the candidate in high school.
 There will be no negative marking.
 There are FIVE parts.
 Answer all questions of a part at one place.
 Use of calculators or graph papers is not permitted.
PART A
This part contains five statements. State whether the
statements are true or false. Give reasons.
(5 x 1 = 5 Marks)
 The product of three positive integers is equal to the
product of their L.C.M and G.C.D.
 443 is prime number
 (a – b)x^{2} +(b – c)x
+(c – a) = 0 and (c – a)x^{2}
+(b – c)x +(a – b) = 0
have a common solution for all real numbers a,b,c.
 Every odd number can be expressed as a difference of two
squares.
 The graph of cuts the xaxis exactly once.
PART B
This part contains FIVE questions. Each question has only
one correct answer. Indicate your choice by marking with the
correct letter. Show your working in brief. (5 x 2 = 10 Marks)
 If a^{2}+b^{2}+c^{2} = D where a
and b are consecutive positive integers and c = ab,
then is
(A) always an even integer
(B) sometimes an odd integer and sometimes not
(C) always an odd integer
(D) sometimes rational, sometimes not
(E) always irrational
 A triangle ABC is to be constructed. Given side a (
opposite A ), B and h_{c} , the
altitude drawn from C on to AB. If N is the number of noncongruent
triangles then N is
(A) one (B) two (C)
zero (D) zero or infinite (E)
infinite
 In any triangle ABC, a and b are the sides of triangle.
Given that ( where S is the area of the
triangle), then
A) 
B) 
C) 
D) 
E) None of these 

 If a_{0,} a_{1}, .............., a_{50}
are respectively the coefficients of in
the expansion of , then is
A) odd 
B) even 
C) Zero 
D) None of these 
 The smallest value of for all real values of x is
(A) –16.25 
(B) –16 
(C) –15 
(D) –8 
(E) None of these 
PART C
This part contains FIVE incomplete statements. Complete the
statements by filling the blanks. Write only the answers in your
answer sheet strictly in the order in which the questions appear.
(5 x 2 = 10 Marks)
 The value of aÎ R for which the equation (1+a^{2})x^{2}
+2(xa)(1+ax)+1 = 0 has no real roots is
_______________________________.
 The greater of the two numbers 31^{14} and 17^{18}
is _______________.
A and B are known verticies of a triangle.
C is the unknown vertex. P is a known point in the plane
of the triangle ABC. Of the two following hypotheses,
Hypothesis
I  P is the incentre of D ABC
Hypothesis
II  P is the circumcentre of D ABC.
The hypothesis which will
enable C to be uniquely determined is ____I / II____
(Strike off the inapplicable)
 If x = 0.101001000100001 ................ is
the decimal expansion of positive x, and in the series
expansion of x the first, second and third terms are, ,
and then the 100 ^{th} term is
_______________________.
 If ABC is a triangle with AB = 7, BC = 9
and CA = n where n is a positive integer, then
possible two values of n are ____________________.
PART D
Attempt any FIVE questions. 
(5 x 5 = 25 Marks)

 Solve the equations
xy + yz + zx = 12x^{2} = 15y^{2} = 20z^{2}
 A is 2 x 2 matrix such that AB = BA
whenever B is a 2 x 2 matrix. Prove that A must
be of the form kI, where k is a real number and I the identity
matrix .
 Determine all 2 x 2 matrices A such that A^{2}=I,
where I = .
 Find an explicit formula ( in terms of nÎ N )
for f(n), if f(1)=1 and f(n)=f(n –1) + 2n – 1
 Let ABCD be a square. Let P,Q,R,S be respectively points
on AB,BC,CD,DA such that PR and QS intersect at right
angles. Show that PR=QS.
 Let ABC be any triangle. Construct parallelograms ABDE
and ACFG on the outside D ABC. Let P be any point where
the lines DE and FG intersect. Construct a parallelogram
BCHI on the outside D ABC such that PA and BI are equal
and parallel. Show that Area of ABDE + Area of ACFG =
Area of BCHI.
 A belt wraps around two pulleys which are mounted with
their centres s apart. If the radius of one pulley
is R and the radius of the other is (R+r) , show
that the length of the belt is .
PART E
Attempt any ONE question 
(1 x 10 = 10 Marks)

 Deduce pythogaras theorem from the result of problem 6 of
part D.
 Let x be any real number. Let
and . Prove that .
 Let . Show
that there exists an infinite sequence such that for all
positive integers n.