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

The value of (log_a(b²))×(log_b(a³)) is    

6

1/6 

-1 

1/5 

5 

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

If 4^x - 2^x+1 = 3 then 4^x= 

1

9

1 or 9

6

1 or 6   

If 3x² - 6x +1 is written in the form a(x+b)² + c then the value of a+b+c is

2

0

-3

-9

-26

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.

The graph with equation x² + y² + 10 = 7x + 3y is a circle. 

The radius of this circle is

(√3)/2

(9√2)/2

(3√3)/2

19

(3√2)/2

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

If a, b and c are positive integers and

a|b-x| > c

then

x < (ab-c)/a

x > (ab+c)/a

-(ab+c)/a < x  < (ab+c)/a

(ab-c)/a < x < (ab+c)/a

x > (ab+c)/a or x < (ab-c)/a

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

If log_a8 - 3log_at = 3 where a > 1 then t =

2/a

(3^a)/a

[∛(3^a)]/a

[∛(3a)]/a

3/a

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

If t > 1 then simplifying the expression (t^2log_t4 - t^−log_t3)/2 gives

13/2

19/2

94/3

49/6

47/6

Let arccos(x) denote the inverse function of cos(x).

If 0 < θ < 2π and cos(θ) = – 0.2 then all the possible values of θ are

arccos(– 0.2) or π + arccos (– 0.2)

---

---

π/2 + arccos(0.2) or (3π)/2 - arccos(0.2)

arccos(–0.2) or π - arccos(0.2)

π - arccos(0.2) or π+ arccos(0.2)

arccos(–0.2) or (3π)/2 - arccos(–0.2)

If n is a positive integer and the expression 1/[(n+1)!] – 1/[(n+2)!] is simplified to a single fraction, then the result is:

[If m is any positive integer then m! denotes the product of all the positive integers less than or equal to m.  ]

n/[(n+2)!]

(n+1)/(n³+3n²+2n)

n/(n²+3n+2)

1/[(n+1)!(n+2)!]

(n+1)/[(n+2)!]

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

671

670

760

761

None of these.

If 3 sec²θ − 5 tanθ − 4 = 0 then the value of tanθ is

[5±√(37)] /6

[−5±√(13)] /6

[−5±√(13)] /2

[5±√(13)] /6

[−5±√(37)] /2

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