SCHOLATIC APTITUDE TEST 1998
MATHEMATICS
Time: Two Hours (8.30
AM  10.30 AM)

Max. Marks: 60

NOTE:
 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 Aor B or C or
D or E. Order of the questions must be maintained. (5x2=10
Marks)
 p is the smallest positive such that every
positive integer greater than p can be written as
a sum of two composite numbers. Then
A) p=3 B)
p=6 C) p=10
D) p=11 E) None of these
 ABC is a triangle such that m_{a}
> a/2, where m_{a} = length
of the median through A and a = length of BC.
Then
A)
such a triangle does not exist 
B) A
is acute 
C)
A is obtuse 
D) A
is right 
E)
None of these 

 A vegetable shop keeps only four weights, one each of 1
kg, ½ kg, ¼ kg, 1/8 kg. A newly
appointed shop assistant claims that he has so far taken
once and only once each weighing possible with the
available weights. If n is the number of all
weighings possible and W the total weight of all
possible weighings, then
A) n=4,W= kg 
B) n=16,
W=30kg 
C) n=15,W=15
kg 
D) n=15,
W=7.5 kg 
E)
None of these 

 are the roots of the polynomial and Then is
independent of
A)
All a_{i} 
B)
All a_{i}, i ¹ 
C)
None of a_{i} 
D)
All a_{i} ,i even 
E)
All a_{i} ,i odd 

 Triangle ABC is isosceles, right angled at B and has area
S. A circle is constructed with B as center and BA as
radius. A semicircle is constructed externally on , that is, on
the side of AC which is opposite to that of B. Then the
area of the crescent or sickle formed between the circle
and the semicircle is
A) 2S 
B) S/2 
C) S 
D) S 
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)
 Numbers 1,2,3,…..,1998 are written in the
natural order. Numbers in odd places are stricken off to
obtain a new sequence. Numbers in odd places are stricken
off from this sequence to obtain another sequence and so
on, until only one term a is left. Then a_{=________
}
 S is the circumcircle of an equilateral triangle ABC. A
point D on S is such that C and D lie on opposite sides
of AB. Then ADB=
radians
 is a bijection from a 3element set
into a 3element set. It is given that exactly one of the
following statements A,B,C is true and the remaining two
are false: A: B: C: . Then = .
 Given that x is real, the solution set of is
.
 In the standard expansion of , the number of
terms appearing with ve sign is
.
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)
 Determine all positive integers n such that n+100
and n+168 are both perfect squares.
 Determine with proof whether integers x and y
can be found such that x+y and x^{2}+y^{2}
are consecutive integers.
 The first term of an arithmetical progression is Log a
and the second term is Log b. Express the sum to n
terms as a logarithm.
 p,q,r,s are positive real numbers. Prove that (p^{2}+p+1)(q^{2}+q+1)(r^{2}+r+1)(s^{2}+s+1)
³ 81pqrs
 With the aid of a rough sketch describe how you will draw
a direct common tangent to two circles having different
radii. (No formal proof is required).
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)
 If x+y+z=0, then prove that
(x^{2}+xy+y^{2})^{3
}+ (y^{2}+yz+z^{2})^{3
}+ (z^{2}+zx+x^{2})^{3
}= 3(x^{2}+xy+y^{2})
(y^{2}+yz+z^{2})
(z^{2}+zx+x^{2})
 Simplify till it is obtained equal to
 a,b,c are distinct and p(x) is a polynomial
in x, which leaves remainders a,b,c on
division by xa, xb, xc, respectively. Find the
remainder obtained on division of p(x) by (xa)(xb)(xc).
 A point A is taken outside a circle of radius R. Two
secants are drawn from this point: One passes through the
center, the other at a distance of R/2 from the center.
Find the area of the circular region enclosed between the
two secants.
 A right angled triangle has legs a,b, a>b. The
right angle is bisected splitting the original triangle
into two smaller triangles. Find the distance between the
orthocenters of the smaller triangles using the coordinate
geometry methods or otherwise.
 If two sides and the enclosed median of a triangle are
respectively equal to two sides and the enclosed median
of another triangle, then prove that the two triangles
are congruent.