SCHOLASTIC APTITUDE TEST 1996
MATHEMATICS
Time: Two Hours  Max. Marks: 60 
(8.30 AM  10.30 AM) 
NOTE:
PARTA
I. This section contains (7) questions. Each question has one or more than one correct answer(s). Indicate the answer(s) by A,B,C,D with brief reasoning. Order of the questions must be maintained. (7 x 2 =14 MARKS)
A) n+1 is always a perfect square  B) n is never a perfect square 
C) n is always divisible by 24  D) n+12 is always a perfect square 
A) All real values of x  B) No real value of x 
C) All real values of x except x=0  D) No real value of x except x=0 
E) All real values of x except x=1 
A)  B)  C)  D) None of these 










A) n=4  B) n=10  C) n=12  D) n=50 
A) not a function from X X  B) an one to one function from X X 
C) an onto function of X X  D) R is a transitive relation on X 
II. This section contains (6) incomplete statements. complete the statements by filling the blanks. Write your answers in your answer book in the order in which the statements are given below. (6 x 2=12 MARKS)
i. If for some angle q and for some real number x it is given that sinq = x^{2}2x+2 then x should be equal to _______________ and q should be equal to ________________ degrees
ii. If a, b, c, d are four distinct integers between 1 to 10 such that for a traingle whose sides are in the ratios a:b:c, the altitudes are d:c:b. Then greatest of these four integers is _____________ and the smallest is __________________
iii. In a cylinder, AB is a diameter of the base circle and C is the centre of the top circle. Given that the volume of the cylinder is cm^{3}. = 30^{0}, the perimeter of the traingle ABC is _____________ and the curved surface area of the cone with vertex C (having the same base) is ______________
iv. If the sum of the intercepts of the line with coordinate is 5 and the area of the traingle formed by the line and coordinate axes is 3 then intercepts are ______________________
v. In the figure given below
= = 90^{0} AP = 7.5 cm, AB = 10 cm, BC = 26 cm, Then CD is ________
vi. Then a, b are __________________
III. This section contains (8) statements in GROUPA and (4) questions in GROUPB. Answer questions in GROUPB on the basis of the statements made in GROUPA. (4 x 1 = 4 MARKS)
GROUPA
A,B and C are three distinct points in a plane .
i . There is no point D equidistant from A,B,C
ii. B lies on the perpendicular bisector of AC
iii. Among the three lengths AB, BC, AC the largest is the sum of the other two
iv. There is a point D such that ABCD is a rhombus
v. There is a point D such that ABCD is a rectangle
vi. In triangle ABC, the angular bisector B is perpendicular to the side AC
vii. AB is a tangent to the circle through B with centre C
viii. AC is a diameter of the circle passing through A, B, C.
GROUPB
1. Which of the above statements mean that A, B, C are collinear
2. Which of the above statements mean AB = BC
3. Which of the above statements mean that AB is perpendicular to BC
4. Which of the above statements infer ABCD is cyclic quadrilateral
PARTB
IV. The numerator of a fraction is less than its denominator by 2. When 1 is added to both the numerator and denominator we get another fraction. The sum of these two fractions is . Find the first fraction . (5)
V. Let be the roots of the quadratic equation x^{2}+ax+b=0 and , be the roots of the equation
x^{2}ax+b2=0. Given that and = 24. Find the value of the coefficient a. (5)
VI. Find the number of points with integral coordinates in the cartesian plane satisfying the inequalities
x 100; y 100; xy 100 (5)
VII. Given an ABC of magnitude less than 180^{0} and O is a point inside it. Draw a straight line through O which forms a traingle with minimum perimeter. (Describe the construction and give proof). (5)
VIII. Perpendiculars are drawn from the vertex of the obtuse angle of a rhombus to its sides. The length of each perpendicular is equal to `a’ units. The distance between their feet being equal to `b’ units. Determine the area of the rhombus. (5)
IX. Let the sequence be defined by
(a) u_{1} = 5 and the relation u_{n+1}  u_{n} = 3 + 4(n1) n N
Express u_{n} as a polynomial in n.
(b) The remainder R, when x^{100} is divided by x^{2}3x+2, is a polynomial. Find x.(3)