If p, q and r are positive integers and p + 1 ⁄ (q + (1 / r)) = 129/31 then what is the value of p + q + r ?

15

26

31

129

160

If w = 2²³⁰ × 3²³⁴ × 5²³⁶, x = 2²³² × 3²³³ × 5²³⁵, y = 2²³⁰ × 3²³³ × 5²³⁵ and z = 2²³¹ × 3²³⁴ × 5²³⁵ then the order from smallest to largest is:

x, y, z, w

y, z, w, x

y, z, x, w

y, x, z, w

z, y, w, x

If the graphs with the following equations are drawn then which of them do not consist of entirely straight lines or half-lines?

1. y=|x+1|
2. x²-y²=0
3. x²+xy=2x+2y
4. y²=|x|

All of them

IV only

III only

II, III, and IV only

III and IV only

As n ranges over all positive integers what are the possible remainders when 2²ⁿ+3²ⁿ is divided by 5?

0 or 1

1 or 2

0

0 or 2 or 3

0 or 1 or 2 or 3 or 4

Evaluate x/yz + y/xz + z/xy given that x + y + z = 4 and xyz = ^_ 60 and xy + xz + yz = −17.

−4/17

−5/6

17/60

−33/60

33/60

Let tan x = 3/4, sin y = a, and both x and y be between 0 and π/2 radians. Then what is the value of cos(x + y)?

(5(1-a²)^0.5-4a)/3

(3(1-a²)^0.5-4a)/5

(4(1-a²)^0.5-5a)/3

(4(1-a²)^0.5-3a)/5

(5(1-a²)^0.5-3a)/4

A circle of radius r is surrounded by three circles of radius R, which touch each other externally.

What is the value of R/r?

1

3^0.5

2

1+3^0.5

3+2(3)^0.5 

Which of the following is NOT true if p² = p + 1 and p is a real number

p³ = p² + p

p⁴ = p³ + p + 1

p³ = 2p + 1

p³ + p² = p + 1

p = 1 / (p−1)

A solid metal sphere with a radius of 4 cm is melted down and 8 identical small spheres are made with the metal. What will be the radius of each small sphere?

0.5 cm

1 cm

2 cm

4 cm

8 cm

The function |x| is defined as follows: If x ≥ 0, then |x|  = x, while if x < 0, then |x|  = − x. The number of solutions of the equation 2x² − 5 |x| −3 = 0 is:

4

3

2

1

0

What is the value of the following?

(461+462+463+...+729)-(315+316+317+...+583)

36 128

38 128

39 274

39 128 

41 964

Simplifying [16^x+1+20(4^2x)] / (2^x−3•8^x+2) gives:

5•(2^4x+3)

6•(2^8x+1)

9•(2^12x+5)

9•(2⁻¹)

6•(2^4x+5)

A and B run around a circular track. They both start at the same point, but they run in opposite directions.

Both A and B run at a constant speed. 

It takes A 40 seconds to run all the way around. 

Every 15 seconds A passes B.

How many seconds does it take B to run all the way around the circular track?

23

24

25

26

27

For each real number x, let [x] be the biggest integer which is less than or equal to x.

What can you say about the following three equations?

1. [p+3]=[p]+3
2. [p+q]=[p]+[q]
3. [5p]=5[p]
   

All three equations are true for all real numbers p and q.

Equations (i) and (iii) are true for all real numbers p.

The three equations are true only when p and q are integers.

Equations (i) and (ii) are true for all real numbers p and q.

Equations (ii) and (iii) are false for some values of p and q.

There are two cubes. The points P and Q are both on the small cube: P is a point in the centre of one of the faces, and Q is a corner on the opposite face. The second cube has sides of length |PQ|. What is the surface area of the large cube divided by the surface area of the smaller cube?

(√6)/2

(√6)/4

(√3)/2

3/2

3(√6)/4

I have 100 socks. There are: 30 long ones and 70 short ones; 80 red socks and 20 blue ones; 60 wool socks and 40 cotton socks. What is the smallest possible number of short red woollen socks that I have?

0

10

20

30

40

There is a regular hexagon ABCDEF. The point B is the centre of a circle, and the points A and C are on the edge of this circle. If X is a point on the edge of the circle that is outside the hexagon then what is the angle AXC?

15°

30°

60°

90°

Cannot be determined

The inequality (x + 2)(2x + 3)² < (2x + 3)(x + 2)² is true exactly for those values of x satisfying:

x < −2

x < −2 or −2/3 < x < 1

x < −2 or −2/3 < x < −1

x < −2 or −1.5 < x < 1

x < −2 or −1.5 < x < −1

What is the side length of the largest square that will fit inside an equilateral triangle with sides of length 1.

2√3 − 3

(2 + √3) / √3

(2 − √3) / √3

2√3 + 3

(2 + √3) / 3

Find the value of cosx in the diagram below (not drawn to scale)

−527/625

−357/625

−336/625

−25/48

−7/13

If you sum the digits of each of the following numbers then you get 6: 60, 411, 123, 1050

How many positive integers are there that are less than 1000 whose digits sum to 6?

16

17

26

27

28

A sequence begins 2, 5, 7, 12, 19, … Each term (after the first two terms) is equal to the sum of the two previous terms. How many of the following statements are true?

I.         The 20th term is divisible by 2
II.        The 40th term is divisible by 2
III.       The 40th term is divisible by 3
IV.       The 60th term is divisible by 3

0

1

2

3

4

Consider the five graphs given by the following equations:

y = |sinx|
|y|=sinx 
|y|=|sinx| 
|y|=sin|x| 
|y|=|sin|x|| 

Which of the following statements about the graphs is true?

They are all different.

One graph appears twice and the other three each occur once.

There are two graphs that each appear twice and another that appears only once.

One graph appears three times and another graph appears twice.

One graph appears three times and the other two each occur once.

A circle whose radius is 1 lies on a square. If the circle and the square have the same area and their centres are at the same point, calculate the length of the line segment AB.

2 − √π

√(4 − π)

4 − √π

π − √2

None of A-D is correct.

Find the integer n that satisfies 8ⁿ + 8ⁿ + 8ⁿ + 8ⁿ = 2²⁰¹⁵

671

670

760

761

None of these.

Triangle ABC has a right angle at C. Point P on BC, point Q on AC and point R on AB are such that BP= BR and AQ = AR. Then the angle PRQ is:

22.5°

30°

45°

60°

90°

A certain function ƒ satisfies ƒ(x + y) = ƒ(x) + ƒ(y) + xy for all real numbers x and y, and it is known that ƒ(4) = 10. Then ƒ(n+1) − ƒ(n)is equal to:

n

n + 1

n+ 2

n + 3

n + 4

A sequence x₁, x₂, x₃, ... is defined as follows:

x₁=3

x_n+1=1-1/x_n

What is the value of x₂₇?

3

2/3

-1/2

∞ 

1/3

If y = (7^x − 7^-x) / (7^x + 7^−x) then x =

(log₇[(1 + y) / (1 − y)])^1/2

(1/4)[log₇(1 + y)]

(1/2)(log₇[(1 + y) / (1 − y)])

(1/4)(log₇[(y + √(y²+ 4)) / 2])

(1/2)(log₇[(y + √(y² + 4)) / 2])

The reverse of a 2-digit integer is the integer obtained by reversing the order of the 2 digits. For example the reverse of 43 is 34. How many 2-digit positive integers N exist with the property that the sum of N and the reverse of N is the square of an integer?

0

6

8

10

More than 10