IIUM Mathematics Competition
(IMC 2015)
FIRST ROUND
MULTIPLE CHOICE QUESTIONS
2 HOURS
This Question Paper Consists of 6 Printed Pages with 20 Questions
Department of Computational and Theoretical Sciences
Kulliyyah of Science
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IIUM Mathematics Competition 2015 (IMC)
First Round
Computational and Theoretical Sciences Department
Kulliyyah of Science
1. The sum of the first five terms of an arithmetic sequence is 40, and the sum of the first ten terms of
the sequence is 155. What is the sum of the first fifteen terms of the sequence
(A) 195
(B) 230
(C) 345
(D) 390
2. If
a
4
a
3
d
 and
 then the value of
is
bcd 3
bc 5
a
(A)
7
6
(B)
6
7
(C) 
12
11
(D) 
(E) 780
11
12
(E)
15
11
3. If x  0 , then | x  ( x  1) 2 | equals
(B) 1  2x
(A) 1
(C) 2x 1
(D) 1  2x
(E) 2x 1
4. Which of the following equations have the same graph?
I.
(A) I and II only
y  x2
(B) I and III only
II.
x2  4
y
x2
(C) II and III only
III.
( x  2) y  x 2  4
(D) I,II and III
(E) None
5. The diagram shows a regular heptagon, a regular hexagon and a regular pentagon all with equal
side length. Find m1.
(A)
740
21
(B)
24 0
7
(C)
2
700
21
(D)
230
7
(E)
22 0
7
IIUM Mathematics Competition 2015 (IMC)
First Round
Computational and Theoretical Sciences Department
Kulliyyah of Science
6. If a, b and c are the roots of the equation x 3  3 x  7  0 , compute the numerical value
of (a  1)(b  1)(c  1) .
(A) 9
(B) 4
(C) 4
(D) 5
(E) 11
7. The quadratic function f ( x)  ax 2  bx  c is known to pass through the points (1, 6) , (7, 6) and
(1, 6) . Find the smallest value of the function.
(A) 36
(B) 26
(C) 20
(D) 10
(E) 6
8. If A  200 and B  250 , then the value of (1 + tan A) (1 + tan B) is
(A)
3
(C) 1  2
(B) 2
(D) 4
(E) None of (A) through (D) is correct
9. In the figure, the segments of length a and b lie on perpendiculars to the diagonals of a square of
side length 2. Find a  b .
(A)
2 1
(B)
2
(C)
2
2
(D)
3
2
(E)
2 1
IIUM Mathematics Competition 2015 (IMC)
First Round
Computational and Theoretical Sciences Department
Kulliyyah of Science
10. A triangle has integer side lengths. If two of the lengths are 13 and 17, find the minimum possible
value of the perimeter, m, and the maximum possible value of the perimeter M.
(A) m = 34, M = 60
(B) m = 35, M = 60
(C) m = 34, M = 61
(D) m = 34, M = 59
(E) m = 35, M = 59
11. Let f: R R be a function such that f ( x  y )  f ( xy ) for all real numbers x and y, and f (7)  7 .
Find the value of f (49) .
(A) 1
(B) 49
(C) 7
(D) 14
(E) None of (A) through (D) is correct
12. Let x, y and z be positive real numbers such that x  y  z  1 and xy  yz  xz 
possible values of the expression
x y z
  is
y z x
(A) 1
(D) more than 3 but finitely many
(B) 2
(C) 3
1
. The number of
3
(E) infinitely many
13. The following four diagrams each depict two triangles. In which diagram can we NOT conclude
that the triangles are congruent? Note that the pictures may not be to scale angles and lengths that are
not explicitly stated to be congruent must not be assumed to be. You may assume that angles and
segments labeled with the same letters are congruent
(A)
(B)
(C)
This quadrilateral is a parallelogram
(D)
(E) None of these
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IIUM Mathematics Competition 2015 (IMC)
First Round
Computational and Theoretical Sciences Department
Kulliyyah of Science
14. In triangle ABC the segment AD bisects the angle A and the point D lies on the segment BC .
Find BD if AB = 6, AC = 4 and DC = 3.
(A)
9
2
(B) 3
(C)
5
2
(D) 8
(E) 7
4
3
(E) 0
(D) 68
(E) 76
15. If 3sin   4cos   5 , then tan  is
(A) 1
16. If x 
(B) -1
(C)
3
4
(D)
1
1
 4 , then the value of x 3  3 is
x
x
(A) 52
(B) 60
(C) 64
17. How many integers x satisfy the equation ( x2  5x  5) x
(A) 2
(B) 3
(C) 4
(D) 5
2
9 x  20
1 ?
(E) None is correct
18. Find AB given that AC = BC, AB = BD, AD = 4 and CD = 32
(A) 10
(B) 9
(C) 12
5
(D) 8
(E) 7
IIUM Mathematics Competition 2015 (IMC)
First Round
Computational and Theoretical Sciences Department
Kulliyyah of Science
19. Arrange the numbers x, y and z in increasing order, where x  mA , y  mB and z  mc.
(A) x  y  z
(B) x  z  y
(C) z  x  y
(D) y  z  x
(E) z  y  x
20. In the figure AB = 2, BC = 1 and FAE  EAD  DAC  CAB. Find AF.
(A)
5 5
2 2
(B)
5 5
4
(C)
25
2 2
(D)
25
4 2
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(E)
25
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