Scholastic Aptitude Test previous year question papers

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Scholastic Aptitude Test previous year question papers
SECTION - A
This section has Six Questions. Each question in provided with five alternative
answers, Only one of them is the correct answer. Indicate the correct answer by A, B,
C, D, E.
(6×2 = 12 MARKS)
1. C is a circle and P is a point exterior to it. Several lines are drawn through P such that each
line has nonempty intersection with C. If 2005 points of intersection are formed, then
among the lines [ ]
A) there need not be even one tangent
B) there may be two tangents
C) there may be three tangents
D) there has to be precisely one tangents
E) none of these
2. For i = 1, 2, …., 2005, ai is a positive integer an, an + 1, an + 2 is a G.P. if n odd. an, an + 1, an + 2
is an A.P. if n is even. [ ]
A) there is no such sequence
B) ai = 1 for all i {1, 2, …., 2005}
C) there are only a finite number of such sequences
D) 1 a an is the square of a rational number, if n is odd
E) none of these
3. P is an interior point of a circle C whose centre is O and O P. C1 is the circle with OP as
diameter. Q is a point on the circumference of C1 such that Q {O, P}. PQ intersects C in
A and B. Then QB AQ / is
[ ]
A) independent of P and Q B) independent neither of P nor of Q
C) independent of P but not of Q D) independent of Q but not of P
E) dependent on the radius of C
4. n is a positive integer. The number of its positive integral divisors is 2005. Then [ ]
A) it cannot be the 4-th power of a positive integer
B) it cannot be the 3-rd power of a positive integer
C) it cannot be the 12-th power of a positive integer
D) it cannot be the 6-th power a positive integer
E) none of these
5. D, G are points on the side AB of ABC. E and F are points on the sides AC and BC
respectively such that DE // BC , EF // AB and FG // CA . Then D, E, F, G are the
consecutive vertices of a quadrilateral [ ]
A) always B) only if AB
AD > 2
1
C) only if AB
AD = 2
1 D) only if AB
AD < 2
1
E) none of these
6. Let S be the set of points common to the lines ax + by = p, cx + dy = q. Let S' be the set of
points common to the lines dx + by = p' and cx + ay = q'. Then : [ ]
A) if S is empty then S' is empty
B) if S is infinite so is S'
C) if S consists of only one element, then S' is empty
D) if S consists of only one element, then S' is infinite
E) if S consists of only one element, so does S'.
SECTION - B
This section has Six Questions. In each question a blank is left, Fill in the blank.
(6×2 = 12 MARKS)
1. The number of positive integers n such that 2005 is a divisor of n2 + n + 1 is __________
2. I1, I2, …., I2005 are arcs of circles. Ik is part of a circle with radius rk and subtends an angle
k radians at the centre. C is the circle with radius r1 + r2 + …. + r2005. The arc of C whose
length is the sum of the lengths of I1, I2, …., I2005 subtends at the centre of C an angle of
___________________ radians.
3. The greatest positive integer k such that xk – 1 is a divisor both of x2005 – 1 and x1203 – 1 is
______________
4. The distance between the parallel sides AB and CD of a trapezium is 12 units. AB = 24
units, CD = 15 units. E is the midpoint of AB . O is the point of intersection of DE with
AC . The area of the quadrilateral EBCO is __________________
5. The set of all natural numbers n such that n 2 and 2005 is the sum of n consecutive
natural numbers is _______________
6. Consider the smallest multiple of 2005 such that its digits are the lengths of the sides of a
pentagon. The number of unit sides in the pentagon is ___________________
SECTION - C
(6×2 = 12 MARKS)
1. C is in the interior of AOB. Locate a point P on 
OA and Q on 
OB such that C is the
midpoint of PQ.
2. P(x) is a polynomial with integral coefficients. P(x) = 4010 for 5 different integral values
of x. Prove that there is no integer x such that P(x) = 2005.
3. Determine the radius of the circle inscribed in a rhombus whose diagonals measure 10
units and 24 units.
4. There are 15 sets of lines in a plane, one consisting of 77 parallel lines, another 5 parallel
lines, another 4 parallel lines and another 3 parallel lines. The remaining sets consist of
one line each. If no two lines are coincident, no three of them are concurrent and lines
belonging to different sets intersect, determine the points of intersection.
5. In two triangles ABC and DEF, AB = DE, AC = DF, ACB, DEF are of equal
measure. Is it necessary that ABC and DEF are of equal measure? Discuss.
6. In the adjoining figure OAB OPQ in the indicated order of correspondence. 
QP
meets AB in X. Prove that 
OX bisects AXP.
SECTION - D
(6×4 = 24 MARKS)
1. Factorize 2(a – b)2 + (b – c) ( c – a).
2. In a ABC, ABC = 70. Explain how you locate all points P in the plane of the triangle
such that APC = 1120 .
3. x 0. If u1 = 0 and if un + 1 = ( 1 – x) un + nx for all natural numbers n, then prove by
induction that un = x
1 {nx – 1 + (1 – x)n} for all natural numbers n.
4. A, B, C are three non collinear points. Explain how you will draw a circle with centre at C
such that parallel tangents can be drawn from A and B.
5. Given positive integers a0, a1, a2, ….., a2005. It is known that a1 > a0, a2 = 3a1 – 2a0, a3
= 3a2 – 2a1, …….., a2005 = 3a2004 -2a2003. Prove that a2005 > 22004
6. AB is a diameter of a circle, 1 , 2 the two half – planes determined by 
AB . P1, P2, ….,
P2004 are points on AB such that AP1 = P1 P2 = P2 P3 = …. = P2003P2004 = P2004 B. For every r
{1, 2, 3, …., 2004} draw a hook, which is the figure formed by the semicircle in 1 on
r AP as diameter and the semicircle in 2 on B Pr as diameter. Prove that these hooks
divide the circle into 2005 regions of equal area.