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Australian AMC 2011 - Senior

Question 1

The expression $3x(x-4)-2(5-3x)$ equals
  1. $3x^2-3x-14$
  2. $3x^2-6x-10$
  3. $3x^2-18x+10$
  4. $3x^2-18x-10$
  5. $9x^2-22x$

Question 2

A coach notices that 2 out of 5 players in his club are studying at university. If there are 12 university students in his club, how many players are there in total?
  1. 20
  2. 24
  3. 30
  4. 36
  5. 60

Question 3

The value of $14\div 0.4$ is
  1. 3.5
  2. 35
  3. 5.6
  4. 350
  5. 0.14

Question 4

In the diagram, $ABCD$ is a square. What is the value of $x$?

Question illustration
  1. 142
  2. 128
  3. 48
  4. 104
  5. 52

Question 5

Which of the following is the largest?
  1. 210
  2. $2^{10}$
  3. $10^2$
  4. $20^1$
  5. $21^0$

Question 6

If $m$ and $n$ are positive whole numbers and $mn=100$, then $m+n$ cannot be equal to
  1. 25
  2. 29
  3. 50
  4. 52
  5. 101

Question 7

$PQRS$ is a square. $T$ is a point on $RS$ such that $QT=2RT$. The value of $x$ is

Question illustration
  1. 100
  2. 110
  3. 120
  4. 150
  5. 160

Question 8

In my neighbourhood, 90% of the properties are houses and 10% are shops. Today, 10% of the houses are for sale and 30% of the shops are for sale. What percentage of the properties for sale are houses?
  1. 9%
  2. 80%
  3. $33\frac13\%$
  4. 75%
  5. 25%

Question 9

The value of $\dfrac{\frac12+\frac14+\frac18}{2+4+8}$ is
  1. 16
  2. 4
  3. 1
  4. $\frac14$
  5. $\frac1{16}$

Question 10

Anne’s morning exercise consists of walking a distance of 1 km at a rate of 5 km/h, jogging a distance of 3 km at 10 km/h and fast walking for a distance of 2 km at 6 km/h.

How long does it take her to complete her morning exercise?
  1. 30 min
  2. 35 min
  3. 40 min
  4. 45 min
  5. 50 min

Question 11

The diagram shows a square of side length 12 units divided into six triangles of equal area. What is the distance, in units, of $T$ from the side $PQ$?

Question illustration
  1. 4
  2. 3
  3. 2
  4. 1
  5. $\sqrt5$

Question 12

Each of the first six prime numbers is written on a separate card. The cards are shuffled and two cards are selected. The probability that the sum of the numbers selected is prime is
  1. $\frac15$
  2. $\frac14$
  3. $\frac13$
  4. $\frac12$
  5. $\frac16$

Question 13

Two tourists are walking 12 km apart along a flat track at a constant speed of 4 km/h. When each tourist reaches the slope of a mountain, she begins to climb with a constant speed of 3 km/h.

Question illustration

Question illustration

What is the distance, in kilometres, between the two tourists during the climb?
  1. 16
  2. 12
  3. 10
  4. 9
  5. 8

Question 14

Lines parallel to the sides of a rectangle 56 cm by 98 cm and joining its opposite edges are drawn so that they cut this rectangle into squares. The smallest number of such lines is
  1. 3
  2. 9
  3. 11
  4. 20
  5. 75

Question 15

What is the sum of the digits of the positive integer $n$ for which $n^2+2011$ is the square of an integer?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10

Question 16

Of the staff in an office, 15 rode a pushbike to work on Monday, 12 rode on Tuesday and 9 rode on Wednesday.

If 22 staff rode a pushbike to work at least once during these three days, what is the maximum number of staff who could have ridden a pushbike to work on all three days?
  1. 4
  2. 5
  3. 6
  4. 7
  5. 8

Question 17

How many integer values of $n$ make $n^2-6n+8$ a positive prime number?
  1. 1
  2. 2
  3. 3
  4. 4
  5. an infinite number

Question 18

If $x^2-9x+5=0$, then $x^4-18x^3+81x^2+42$ equals
  1. 5
  2. 25
  3. 42
  4. 67
  5. 81

Question 19

The centre of a sphere of radius 1 is one of the vertices of a cube of side 1.

Question illustration

What is the volume of the combined solid?
  1. $\frac{7\pi}{6}+1$
  2. $\frac{7\pi}{6}+\frac56$
  3. $\frac{7\pi}{6}+\frac43$
  4. $\frac{7\pi}{8}+1$
  5. $\pi+1$

Question 20

In a best of five sets tennis match (where the first player to win three sets wins the match), Chris has a probability of $\frac23$ of winning each set. What is the probability of him winning this particular match?
  1. $\frac23$
  2. $\frac{190}{243}$
  3. $\frac89$
  4. $\frac{19}{27}$
  5. $\frac{64}{81}$