Aussie Maths & Science Teachers: Save your time with SmarterEd
The lengths of the sides of a triangle are 3 cm, 6 cm and 5 cm, as shown below.
The angle, `x`, can be found using
A. `cos(x) = (3^2 + 6^2 - 5^2)/(2 xx 3 xx 6)`
B. `cos(x) = (3^2 + 6^2 + 5^2)/(2 xx 3 xx 6)`
C. `cos(x) = (3^2 + 5^2 - 6^2)/(2 xx 3 xx 6)`
D. `cos(x) = 3/5`
E. `cos(x) = 3/6`
The top of a table is in the shape of a trapezium, as shown below.
The area of the tabletop, in square centimetres, is
A. `200`
B. `260`
C. `4200`
D. `4800`
E. `288\ 000`
The closing price of a share on Wednesday was $160.
The closing price of the same share on Thursday was 3% less than its closing price on Wednesday.
The closing price of the same share on Friday was 4.5% more than its closing price on Thursday.
The closing price of the share on Friday is closest to
A. $157.38
B. $161.98
C. $162.18
D. $162.40
E. $172.22
Fong’s gas bill is $368.40. If he pays this bill on time, it will be reduced by 5%.
In this case, the bill would be reduced by
A family bought a country property.
At the end of the first year, there were two thistles per hectare on the property.
At the end of the second year, there were six thistles per hectare on the property.
At the end of the third year, there were 18 thistles per hectare on the property.
Assume the number of thistles per hectare continues to follow a geometric pattern of growth.
At the end of the seventh year, the number of thistles per hectare is expected to be
A. `972`
B. `1458`
C. `2916`
D. `4374`
E. `8748`
Let `(dy)/(dx) = x^3 - xy` and `y = 2` when `x = 1`.
Using Euler’s method with a step size of 0.1, the approximation to `y` when `x = 1.1` is
A. 0.9
B. 1.0
C. 1.1
D. 1.9
E. 2.1
An agent charged $20 commission for selling a rare book for $500.
What percentage of the selling price is the commission?
A. 4%
B. 5%
C. 20%
D. 25%
E. 40%
The price of a one-way airfare between two cities varies each day according to demand.
• On Monday the price is $160.
• The price on Tuesday is 25% greater than the price on Monday.
• The price on Wednesday is 10% less than the price on Tuesday.
• The price on Thursday is 25% less than the price on Wednesday.
• The price on Friday is 20% greater than the price on Thursday.
• The Saturday price is the same as the Friday price.
The price on Saturday is
A. $150
B. $162
C. $176
D. $288
E. $330
Peter received a quote from the Artificial Grass Company for his new front lawn.
The quote is for $1880 plus a Goods and Services Tax (GST) of 10%.
The final amount that Peter pays for the new front lawn is
A. $188
B. $1880
C. $1890
D. $1899
E. $2068
$6000 is invested in an account that earns simple interest at the rate of 3.5% per annum.
The total interest earned in the first four years is
A. `$70`
B. `$84`
C. `$210`
D. `$840`
E. `$885`
Sally purchased an electronic game machine on hire purchase. She paid $140 deposit and then $25.50 per month for two years.
The total amount that Sally paid is
A. $191
B. $446
C. $612
D. $740
E. $752
The second term of a Fibonacci-related sequence is 36 and the third term is 72.
The first term of this sequence is
A log transformation is used to linearise the relationship between the weight of a mouse, in grams, and its age, in weeks.
When a least squares regression line is fitted to the transformed data, its equation is
`text(weight) = – 7 + 30 log_10(text(age))`
This equation predicts that a mouse aged five weeks has a weight, in grams, that is closest to
A. `14`
B. `21`
C. `23`
D. `41`
E. `143`
For an ordered set of data containing an odd number of values, the middle value is always
A. the mean.
B. the median.
C. the mode.
D. the mean and the median.
E. the mean, the median and the mode.
The following information relates to Parts 1 and 2.
A farmer plans to breed sheep to sell.
In the first year she starts with 50 breeding sheep.
During the first year, the sheep numbers increase by 84%.
At the end of the first year, the farmer sells 40 sheep.
Part 1
How many sheep does she have at the start of the second year?
A. 2
B. 42
C. 52
D. 84
E. 92
Part 2
If `S_n` is the number of sheep at the start of year `n`, a difference equation that can be used to model the growth in sheep numbers over time is
| A. `S_(n+1) = 1.84S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` | |
| B. `S_(n+1) = 0.84S_n - 50` | `\ \ \ \ \ text(where)\ \ S_1 = 40` | |
| C. `S_(n+1) = 0.84S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` | |
| D. `S_(n+1) = 0.16S_n - 50` | `\ \ \ \ \ text(where)\ \ S_1 = 40` | |
| E. `S_(n+1) = 0.16S_n - 40` | `\ \ \ \ \ text(where)\ \ S_1 = 50` |
The first three terms of a geometric sequence are `6, x, 54.`
A possible value of `x` is
A. `9`
B. `15`
C. `18`
D. `24`
E. `30`
The distribution of test marks obtained by a large group of students is displayed in the percentage frequency histogram below.
Part 1
The pass mark on the test was 30 marks.
The percentage of students who passed the test is
A. `7text(%)`
B. `22text(%)`
C. `50text(%)`
D. `78text(%)`
E. `87text(%)`
Part 2
The median mark lies between
A. `35 and 40`
B. `40 and 45`
C. `45 and 50`
D. `50 and 55`
E. `55 and 60`
The back-to-back ordered stemplot below shows the distribution of maximum temperatures (in °Celsius) of two towns, Beachside and Flattown, over 21 days in January.
Part 1
The variables
temperature (°Celsius), and
town (Beachside or Flattown), are
A. both categorical variables.
B. both numerical variables.
C. categorical and numerical variables respectively.
D. numerical and categorical variables respectively.
E. neither categorical nor numerical variables.
Part 2
For Beachside, the range of maximum temperatures is
A. `3°text(C)`
B. `23°text(C)`
C. `32°text(C)`
D. `33°text(C)`
E. `38°text(C)`
Part 3
The distribution of maximum temperatures for Flattown is best described as
A. negatively skewed.
B. positively skewed.
C. positively skewed with outliers.
D. approximately symmetric.
E. approximately symmetric with outliers.
In the first three layers of a stack of soup cans there are 20 cans in the first layer, 19 cans in the second layer and 18 cans in the third layer.
This pattern of stacking cans in layers continues.
The maximum number of cans that can be stacked in this way is
A. `190`
B. `210`
C. `220`
D. `380`
E. `590`
The difference equation
`t_(n+1) = at_n + 6 quad text (where) quad t_1 = 5`
generates the sequence
`5, 21, 69, 213\ …`
The value of `a` is
A. – 1
B. 3
C. 4
D. 15
E. 16
The yearly membership of a club follows an arithmetic sequence.
In the club’s first year it had 15 members.
In its third year it had 29 members.
How many members will the club have in the fourth year?
A. `8`
B. `22`
C. `36`
D. `43`
E. `57`
The dot plot below shows the distribution of the number of bedrooms in each of 21 apartments advertised for sale in a new high-rise apartment block.
Part 1
The mode of this distribution is
A. `1`
B. `2`
C. `3`
D. `7`
E. `8`
Part 2
The median of this distribution is
A. `1`
B. `2`
C. `3`
D. `4`
E. `5`
A sequence is generated by a first-order linear difference equation.
The first four terms of this sequence are 1, 3, 7, 15.
The next term in the sequence is
A. 17
B. 19
C. 22
D. 23
E. 31
`1,3,7,15\ ...`
`text(Sequence has the pattern,)`
`Τ_(n+1)=2T_n+1`
| `:. T_5` | `= 2 xx T_4 +1` |
| `=2 xx 15 +1` | |
| `=31` |
`=>E`
The sequence `12, 15, 27, 42, 69, 111 …` can best be described as
A. fibonacci-related
B. arithmetic with `d > 1`
C. arithmetic with `d < 1`
D. geometric with `r > 1`
E. geometric with `r < 1`
The values of the first seven terms of a geometric sequence are plotted on the graph above.
Values of `a` and `r` that could apply to this sequence are respectively
| (A) `a=90` | `\ \ \ \ r= – 0.9` |
| (B) `a=100` | `\ \ \ \ r= – 0.9` |
| (C) `a=100` | `\ \ \ \ r= – 0.8` |
| (D) `a=100` | `\ \ \ \ r=0.8` |
| (E) `a=90` | `\ \ \ \ r=0.9` |
Consider the following discrete probability distribution for the random variable `X.`
The mean of this distribution is
Consider the tangent to the graph of `y = x^2` at the point `(2, 4).`
Which of the following points lies on this tangent?
A. `text{(1, −4)}`
B. `(3, 8)`
C. `text{(−2, 6)}`
D. `(1, 8)`
E. `text{(4, −4)}`
The inverse of the function `f: R^+ -> R,\ f(x) = 1/sqrt x + 4` is
| A. | `f^-1: (4, oo) -> R` | `f^-1(x) = 1/(x - 4)^2` |
| B. | `f^-1: R^+ -> R` | `f^-1(x) = 1/x^2 + 4` |
| C. | `f^-1: R^+ -> R` | `f^-1(x) = (x + 4)^2` |
| D. | `f^-1:\ text{(−4, ∞)} -> R` | `f^-1(x) = 1/(x + 4)^2` |
| E. | `f^-1:\ text{(−∞, 4)} -> R` | `f^-1(x) = 1/(x - 4)^2` |
The linear function `f: D -> R, f (x) = 4 - x` has range `text{[−2, 6)}`.
The domain `D` of the function is
`:. D = (−2,6)`
`=> D`
The point `P\ text{(4, −3)}` lies on the graph of a function `f`. The graph of `f` is translated four units vertically up and then reflected in the `y`-axis.
The coordinates of the final image of `P` are
| `y - z` | `= 8` |
| `5x - y` | `= 0` |
| `x + z` | `= 4` |
The system of three simultaneous linear equations above can be written in matrix form as
| A. | `[[0,1,-1],[0,5,-1],[1,0,1]][[x],[y],[z]]=[[8],[0],[4]]` | B. | `[[0,1,-1],[5,-1,0],[1,0,1]][[x],[y],[z]]=[[8],[0],[4]]` |
| C. | `[[1,-1],[5,-1],[1,1]][[x],[y],[z]]=[[8],[0],[4]]` | D. | `[[0,5,1],[1,-1,0],[-1,0,1]][[x],[y],[z]]=[[8],[0],[4]]` |
| E. | `[[0,5,0],[-1,-1,0],[1,1,0]][[x],[y],[z]]=[[8],[0],[4]]` |
A turkey is taken from the refrigerator. Its temperature is 5°C when it is placed in an oven preheated to 190°C.
Its temperature, `T`° C, after `t` hours in the oven satisfies the equation
`(dT)/(dt) = -k(T − 190)`.
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At what time (to the nearest minute) will it be cooked? (3 marks)
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At the beginning of 1991 Australia’s population was 17 million. At the beginning of 2004 the population was 20 million.
Assume that the population `P` is increasing exponentially and satisfies an equation of the form `P = Ae^(kt)`, where `A` and `k` are constants, and `t` is measured in years from the beginning of 1991.
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The diagram shows a right-angled triangle `ABC` with `∠ABC = 90^@`. The point `M` is the midpoint of `AC`, and `Y` is the point where the perpendicular to `AC` at `M` meets `BC`.
| (i) |  |
`text(In)\ ΔAYM\ text(and)\ ΔCYM`
| `∠AMY` | `= ∠CMY = 90^@\ \ \ (MY ⊥ AC)` |
| `AM` | `=CM\ \ \ text{(given)}` |
| `YM\ text(is common)` | |
`:.ΔAYM ≡ ΔCYM\ \ text{(SAS)}`
(ii) `text(Let)\ \ /_BAC = 2 theta`
`∠YAB = ∠YAM = theta\ \ \ \ (AY\ text(bisects)\ ∠BAC)`
| `∠YAM` | `= ∠YCM=theta` | `\ \ \ text{(corresponding angles of}` |
| `\ \ \ text{congruent triangles)}` | ||
| `:.∠YAB= ∠ YAM = ∠YCM=theta` | ||
`text(In)\ \ ΔABC,`
| `∠ABC + ∠BAC + ∠YCM` | `= 180^@` |
| `:.90^@ +2theta+theta` | `= 180^@` |
| `3theta` | `= 90^@` |
| `:.theta` | `= 30^@` |
`text(In)\ \ Delta MYC,`
| `:.tan 30^@` | `= (MY)/(MC)` |
| `(MY)/(MC)` | `= 1/sqrt3` |
| `(MY)/(2MC)` | `=1/(2 sqrt3)` |
| `(MY)/(AC)` | `= 1/(2sqrt3)\ \ \ text{(given}\ \ AC=2MC text{)}` |
| `:.MY : AC` | `= 1 : 2sqrt3.` |
A particle moves along a straight line so that its displacement, `x` metres, from a fixed point `O` is given by `x = 1 + 3 cos 2t`, where `t` is measured in seconds.
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i. `x = 1 + 3 cos 2t`
`text(When)\ \ t = 0,`
| `x` | `= 1 + 3 cos 0` |
| `= 1 + 3` | |
| `= 4` |
`:.\ text(Initial displacement is 4 m to the right of)\ O.`
ii. `text(Period)\ = (2pi)/n = (2pi)/2 = pi`
`text(Considering the range)`
| `-1` | `<=cos 2t<=1` |
| `-3` | `<=3cos 2t<=3` |
| `-2` | `<=1 + 3 cos 2t<=4` |
| iii. | `x` | `= 1 + 3 cos 2t` |
| `:.v` | `= −6 sin 2t` |
`text(The particle comes to rest when)\ \ v=0`
| `-6 sin 2t` | `= 0` |
| `sin 2t` | `= 0` |
| `2t` | `= 0, pi, 2pi…` |
| `t` | `= 0, pi/2, pi…` |
`:.\ text(After)\ \ t=0, text(particle first comes to rest when)`
`t = pi/2\ text(seconds.)`
`text(When)\ t = pi/2,`
| `x` | `= 1 + 3 cos 2(pi/2)` |
| `= 1 + 3 cos pi` | |
| `= 1 + 3(−1)` | |
| `= −2` |
`:.\ text(Particle first comes to rest at 2 m to the left of)\ O.`
iv. `x = 1 + 3 cos 2t`
| `v` | `= -6 sin 2t` |
| `a` | `= -12 cos 2t` |
`text(MAX occurs when)\ \ a=0`
| `−12 cos 2t` | `= 0` |
| `cos 2t` | `= 0` |
| `2t` | `= pi/2, (3pi)/2, …` |
| `t` | `= pi/4, (3pi)/4, …` |
`:.\ text(Maximum at)\ \ t=pi/4,\ \ (3pi)/4, …\ text(seconds,)`
`text(When)\ \ t = pi/4\ text(seconds,)`
| `v` | `= -6 sin 2(pi/4)` |
| `= -6 sin(pi/2)` | |
| `= −6` |
`:.\ text(Maximum is 6 m s)^(−1).`
Clare is learning to drive. Her first lesson is 30 minutes long. Her second lesson is 35 minutes long. Each subsequent lesson is 5 minutes longer than the lesson before.
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If `f prime (x) = 3x^2 - 4`, which one of the following graphs could represent the graph of `y = f (x)`?
Demelza is a badminton player. If she wins a game, the probability that she will win the next game is 0.7. If she loses a game, the probability that she will lose the next game is 0.6. Demelza has just won a game.
The probability that she will win exactly one of her next two games is
A. `0.33`
B. `0.35`
C. `0.42`
D. `0.49`
E. `0.82`
`text(After a win in game 1,)`
`text{Pr(exactly one of next two)}`
`= text(Pr)(WWL) + text(Pr)(WLW)`
`= 1 xx 0.7 xx 0.3 + 1 xx 0.3 xx 0.4`
`= 0.33`
`=> A`
The adult membership fee for a cricket club is $150. Junior members are offered a discount of $30 off the adult membership fee. --- 2 WORK AREA LINES (style=lined) --- Adult members of the cricket club pay $15 per match in addition to the membership fee of $150. --- 3 WORK AREA LINES (style=lined) --- If a member does not pay the membership fee by the due date, the club will charge simple interest at the rate of 5% per month until the fee is paid. Michael paid the $150 membership fee exactly two months after the due date. --- 2 WORK AREA LINES (style=lined) --- The cricket club received a statement of the transactions in its savings account for the month of January 2014. The statement is shown below. --- 2 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
Interest for this account is calculated on the minimum balance for the month and added to the account on the last day of the month.
What is the annual rate of interest for this account? Write your answer, correct to one decimal place. (1 mark)
A section of the graph of `f` is shown below.
The rule of `f` could be
If `A = P(1 + r)^n`, find `A` given `P = $300`, `r = 0.12` and `n = 3` (give your answer to the nearest cent). (2 marks)
What is the value of `5a^2-b`, if `a =-4` and `b = 3`. (2 marks)
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If `f: text{(−∞, 1)} -> R,\ \ f(x) = 2 log_e (1 - x)\ \ text(and)\ \ g: text{[−1, ∞)} -> R, g(x) = 3 sqrt (x + 1),` then the maximal domain of the function `f + g` is
Part of the graph of `y = f(x)`, where `f: R -> R,\ \ f(x) = 3-e^x,` is shown below.
Which one of the following could be the graph of `y = f^{-1}(x),` where `f^{-1}` is the inverse of `f?`
The function with rule `f(x) = -3 tan(2 pi x)` has period
The cubic equation `x^3 – px + q = 0` has roots `alpha, beta` and `gamma`.
It is given that `alpha^2 + beta^2 + gamma^2 = 16` and `a^3 + beta^3 + gamma^3 = -9`.
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A small spherical balloon is released and rises into the air. At time `t` seconds, it has radius `r` cm, surface area `S = 4 pi r^2` and volume `V = 4/3 pi r^3`.
As the balloon rises it expands, causing its surface area to increase at a rate of `((4 pi)/3)^(1/3)\ \text(cm)^2 text(s)^-1`. As the balloon expands it maintains a spherical shape.
The hyperbolas `H_1:\ \ x^2/a^2 - y^2/b^2 = 1` and `H_2:\ \ x^2/a^2 - y^2/b^2 = -1` are shown in the diagram.
Let `P(a sec theta, b tan theta)` lie on `H_1` as shown on the diagram.
Let `Q` be the point `(a tan theta, b sec theta)`.
| i. `((x – 2) (x – 5))/(x – 1)` | `= (x^2 – 7x + 10)/(x – 1)` |
| `= (x(x – 1) – 6 (x – 1) + 4)/(x – 1)` | |
| `= x – 6 + 4/(x – 1)` |
`:.text(Equation of oblique asymptote is)\ \ y = x – 6`
ii. `text(Vertical asymptote at)\ \ x = 1`
`x text(-intercepts)\ \ 2 and 5,\ \ y text(-intercept) = -10`
The polynomial `P(x) = x^4 - 4x^3 + 11x^2 - 14x + 10` has roots `a + ib` and `a + 2ib` where `a` and `b` are real and `b != 0.`
The complex number `z` is such that `|\ z\ |=2` and `text(arg)(z) = pi/4.`
Plot each of the following complex numbers on the same half-page Argand diagram.
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Find the value of `(dy)/(dx)` at the point `(2, text(−1))` on the curve `x + x^2 y^3 = -2.` (3 marks)
Consider the complex numbers `z = -sqrt 3 + i` and `w = 3 (cos\ pi/7 + i sin\ pi/7).`
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A particle, `P`, of mass `m` is attached by two strings, each of length `l`, to two fixed points, `A` and `B`, which lie on a vertical line as shown in the diagram.
The system revolves with constant angular velocity `omega` about `AB`. The string `AP` makes an angle `alpha` with the vertical. The tension in the string `AP` is `T_1` and the tension in the string `BP` is `T_2` where `T_1 >= 0` and `T_2 >= 0`. The particle is also subject to a downward force, `mg`, due to gravity.
| (i) |  |
`text(Resolving the forces vertically)`
`T_1 cos alpha = mg + T_2 cos alpha\ \ \ …\ (1)`
`text(Resolving the forces horizontally)`
` T_1 sin alpha+ T_2 sin alpha = mr omega^2\ \ \ …\ (2)`
`r = l sin alpha`
`:. (T_1 + T_2) sin alpha = m l sin alpha omega^2`
`T_1 + T_2 = m l omega^2`
(ii) `text(Substitute)\ \ T_2 = 0\ \ text{into (1) and (2)}`
| `T_1` | `= (mg)/cos alpha\ \ \ …\ (1)` |
| `T_1 sin alpha` | `= m r omega^2` |
| `=ml sin alpha omega^2\ \ \ \ text{(since}\ \ r=l sin alpha text{)}` | |
| `T_1` | `=ml omega^2\ \ \ …\ (2)` |
| `:. m l omega^2` | `= (mg)/(cos alpha)` |
| `omega^2` | `= g/(l cos alpha)` |
| `:.omega` | `= sqrt(g/(l cos alpha))` |
Let `P(p, 1/p), Q(q, 1/q)` and `R(r, 1/r)` be three distinct points on the hyperbola `xy = 1.`
The polynomial `p(x) = ax^3 + bx + c` has a multiple zero at 1 and has remainder 4 when divided by `x + 1`. Find `a, b` and `c`. (3 marks)
The diagram shows the graph of the hyperbola
`x^2/144 - y^2/25 = 1.`
The diagram shows the graph of `y =f(x)`. The graph has a horizontal asymptote at `y =2`.
Draw separate one-third page sketches of the graphs of the following:
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i.
ii.
iii.
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