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EXAMCOPY Algebra, STD2 A4 2019 HSC 33

The time taken for a car to travel between two towns at a constant speed varies inversely with its speed.

It takes 1.5 hours for the car to travel between the two towns at a constant speed of 80 km/h.

  1. Calculate the distance between the two towns.  (1 mark)

  2. By first plotting four points, draw the curve that shows the time taken to travel between the two towns at different constant speeds.  (3 marks)

Show Answers Only
  1. `120\ text(km)`
  2.  
Show Worked Solution
a.    `D` `= S xx T`
    `= 80 xx 1.5`
    `= 120\ text(km)`

 
b.   

\begin{array} {|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ s\ \  \rule[-1ex]{0pt}{0pt} & 20 & 40 & 60 & 80 \\
\hline
\rule{0pt}{2.5ex} t \rule[-1ex]{0pt}{0pt} & 6 & 3 & 2 & 1.5 \\
\hline
\end{array}

EXAMCOPY Algebra, STD2 A4 2011 HSC 28a

The air pressure, `P`, in a bubble varies inversely with the volume, `V`, of the bubble. 

  1. Write an equation relating `P`, `V` and `a`, where `a` is a constant.    (1 mark)

  2. It is known that `P = 3` when `V = 2`.

     

    By finding the value of the constant, `a`, find the value of `P` when `V = 4`.    (2 marks)

  3. Sketch a graph to show how `P` varies for different values of `V`.

     

    Use the horizontal axis to represent volume and the vertical axis to represent air pressure.   (2 marks)


Show Answers Only
  1. `P = a/V`
  2. `P = 1 1/2`
  3.  
     
Show Worked Solution
♦ Mean mark (i) 39%
COMMENT: Expressing the proportional relationship `P prop 1/V` as the equation `P=k/V` is a core skill here.
i. `P` `prop 1/V`
    `= a/V`

 

ii. `text(When)\ P=3,\ V = 2`
`3` `= a/2`
`a` `=6`

 

`text(Need to find)\ P\ text(when)\ V = 4`  

♦ Mean mark (ii) 47%
`P` `=6/4`
  `= 1 1/2`

  

♦♦ Mean mark (iii) 26%
COMMENT: An inverse relationship is reflected by a hyperbola on the graph.
iii.

EXAMCOPY Algebra, STD2 A4 2007 HSC 15 MC

If pressure (`p`) varies inversely with volume (`V`), which formula correctly expresses  `p`  in terms of  `V`  and  `k`, where  `k`  is a constant?

  1. `p = k/V`
  2. `p = V/k`
  3. `p = kV`
  4. `p = k + V`
Show Answers Only

`A`

Show Worked Solution

`p prop 1/V`

`p = k/V`

`=>  A`

EXAMCOPY Algebra, STD2 A4 2010 HSC 13 MC

The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.

How long will it take the same size block of ice to melt at 12°C?  

  1. 3.2 hours
  2. 20 hours
  3. 26  hours
  4. 45 hours
Show Answers Only

`B`

Show Worked Solution
 
♦ Mean mark 50% 

`text{Time to melt}\ (T) prop1/text(Temp) \ => \ T=k/text(Temp)`

`text(When) \ T=8, text(Temp = 30)`

`8` `=k/30`
`k` `=240`

  

`text{Find}\ T\ text{when  Temp = 12:}`

`T` `=240/12`
  `=20\ text(hours)`

 
`=>  B`

EXAMCOPY Algebra, STD2 A1 2009 HSC 16 MC

The time for a car to travel a certain distance varies inversely with its speed.

Which of the following graphs shows this relationship?
 

Show Answers Only

`A`

Show Worked Solution
`T` `prop 1/S`
`T` `= k/S`

 
`text{By elimination:}`

`text(As   S) uarr text(, T) darr => text(cannot be B or D)`

♦ Mean mark 38%

`text(C  is incorrect because it graphs a linear relationship)`

`=>  A`

Circle Geometry, SMB-008

In the diagram, \(AC\) is a diameter of the circle centred at \(O\), and \(OA = AB\).
 

Find the value of \(\theta\).   (3 marks)   

Show Answers Only

\(\theta = 30^{\circ}\)

Show Worked Solution

\(\angle ABC=90^{\circ}\ \ \text{(angle in semi-circle)}\)

\(OA=OB\ \ \text{(radii)} \)

\( \angle OAB=60^{\circ}\ \ ( \Delta OAB\ \text{is equilateral}) \)

\(\theta\) \(= 180-(90+60)\ \ (180^{\circ}\ \text{in}\ \Delta) \)  
  \(= 30^{\circ} \)  

Cicle Geometry, SMB-007

In the diagram, a line from the centre of the circle meets a chord at its midpoint.

Find the value of \(\theta\).  (2 marks) 
  

Show Answers Only

\(\theta = 47^{\circ}\)

Show Worked Solution

\(\text{Line from centre bisects chord}\ \ \Rightarrow\ \ \text{Line is ⊥ to chord}\)

\(\theta\) \(= 180-(90+43)\ \ (180^{\circ}\ \text{in}\ \Delta) \)  
  \(= 47^{\circ} \)  

Circle Geometry, SMB-006

In the circle centred at \(O\), the chord \(AC\) has length 15 and \(OB\) meets the chord \(AC\) at right angles.

Find the length of \(BC\).  (1 mark) 
  

Show Answers Only

\(BC = 7.5\)

Show Worked Solution
\(BC\) \(= \dfrac{1}{2} \times 15\ \ \text{(perpendicular from centre to chord bisects chord)}\)  
  \(= 7.5 \)  

Circle Geometry, SMB-003

In the diagram, the vertices of  \(\Delta ABC\)  lie on the circle with centre \(O\). The point \(D\) lies on \(BC\) such that \(\Delta ABD\) is isosceles and \(\angle ABC = x\).

Explain why \(\angle AOC =2x\).   (2 marks)

Show Answers Only

\(\text{See Worked Solutions}\)

Show Worked Solution

\(\text{Angles at circumference and centre are both on arc}\ AC\)

\(\therefore \angle AOC = 2x\)

Circle Geometry, SMB-002

The line \(AT\) is the tangent to the circle at \(A\), and \(BT\) is a secant meeting the circle at \(B\) and \(C\).
  

Given that  \(AT = 12\),  \(BC = 7\)  and  \(CT = x\), find the value of \(x\).  (2 marks)

Show Answers Only

\(x = 9\)

Show Worked Solution

\(\text{Property: square of tangent = product of secant intercepts}\)

\(AT^2\) \(= CT \times BT\)
\(12^2\) \(= x(x + 7)\)
\(144\) \(= x^2 + 7x\)
\(x^2 + 7x-144\) \(= 0\)
\((x + 16)(x-9)\) \(= 0\)

 

\(\therefore x = 9,\  (x \gt 0) \)

Circle Geometry, SMB-001

In the circle centred at \(O\) the chord \(AB\) has length 7. The point \(E\) lies on \(AB\) and \(AE\) has length 4. The chord \(CD\) passes through \(E\).

Let the length of \(CD\) be \(\ell\) and the length of \(DE\) be \(x\).

Show that  \(x^2-\ell x + 12 = 0\).  (2 marks)

Show Answers Only

\(\text{See Worked Solutions}\)

Show Worked Solution

\(\text{Show}\ \ x^2-\ell x + 12 = 0\)
 


 

\(AB = 7 \ \ \Rightarrow \ EB=7-4=3\)

\(AE \times EB = ED \times CE\ \ \text{(intercepts of intersecting chords)}\)

\(4 \times 3\) \(= x(\ell-x)\)
\(12\) \(= x\ell-x^2\)
\(\therefore x^2-\ell x+12\) \(=0\)

Rates of Change, SMB-010

Moses finds that for a Froghead eel, its mass is directly proportional to the square of its length.

An eel of this species has a length of 72 cm and a mass of 8250 grams.

What is the expected length of a Froghead eel with a mass of 10.2 kg? Give your answer to one decimal place.  (3 marks)

Show Answers Only

`80.1\ text{cm}`

Show Worked Solution

`text(Mass) prop text(length)^2`

`m = kl^2`
  

`text(Find)\ k:`

`8250` `= k xx 72^2`
`k` `= 8250/72^2`
  `= 1.591…`

 
`text(When)\ \ l\ \ text(when)\ \ m = 10\ 200:`

`10\ 200` `= 1.591… xx l^2`
`l^2` `= (10\ 200)/(1.591…)`
`:. l` `= 80.058…`
  `= 80.1\ text{cm  (to 1 d.p.)}`

Rates of Change, SMB-009

The number of trees that can be planted along the fence line of a paddock varies inversely with the distance between each tree.

There will be 108 trees if the distance between them is 5 metres.

  1. How many trees can be planted if the distance between them is 6 metres?  (2 marks)

  2. What is the distance between the trees if 120 trees are planted?  (1 mark)

Show Answers Only
  1. `90`
  2. `4.5\ text(metres)`
Show Worked Solution

i.   `t prop 1/d`

`t` `= k/d`
`108` `= k/5`
`k` `= 540`

 
`text(Find)\ t\ text(when)\ d = 6:`

`t` `= 540/6`
  `= 90`

 

ii.   `text(Find)\ d\ text(when)\ t = 120:`

`120` `= 540/d`
`d` `= 540/120`
  `= 4.5\ text(metres)`

Rates of Change, SMB-008

It is known that the quantity of steel produced in tonnes `(S)`, is directly proportional to the tonnes of iron ore used in the process `(I)`.

If 16 tonnes or iron ore produces 10 tonnes of steel, calculate the tonnes of iron ore required to produce 48 tonnes of steel.  (3 marks)

Show Answers Only

`76.8\ text{tonnes}`

Show Worked Solutions

`S prop I\ \ =>\ \ S=kI`

`text(Find)\ k\ text{given}\ S=10\ text{when}\ I=16:`

`10` `=k xx 16`
`k` `=10/16`
  `=0.625`

 
`text{Find}\ I\ text{when}\ S=48:`

`48` `=0.625 xx I`
`:. I` `=48/0.625`
  `=76.8\ text{tonnes}`

Rates of Change, SMB-007

It is known that a quantity `P` kgs is proportional to the reciprocal of another quantity `Q` kgs such that `P prop 1/Q`.

If  `P=12` when `Q=20`, calculate the estimated quantity of `Q` when `P=45` kgs, to the nearest gram.  (3 marks)

Show Answers Only

`5333\ text{g}`

Show Worked Solutions

`P prop 1/Q\ \ =>\ \ P=k/Q`

`text(Find)\ k\ text{given}\ P=12\ text{when}\ Q=20:`

`12` `=k/20`
`:. k` `=12 xx 20`
  `=240`

 
`text{Find}\ Q\ text{when}\ P=45:`

`45` `=240/Q`
`:. Q` `=240/45`
  `=5.3333\ text{kg}`
  `=5333\ text{g}`

Rates of Change, SMB-006

The stopping distance of a car on a certain road, once the brakes are applied, is directly proportional to the square of the speed of the car when the brakes are first applied.

A car travelling at 70 km/h takes 58.8 metres to stop.

How far does it take to stop if it is travelling at 105 km/h?  (3 marks)

Show Answers Only

`132.3\ text(metres)`

Show Worked Solution

`text(Let)\ \ d\ text(= stopping distance)`

`d prop s^2`

`d = ks^2`
 

`text(Find)\ k,`

`58.8` `= k xx 70^2`
`k` `= 58.8/(70^2)`
  `= 0.012`

 
`text(Find)\ d\ \ text(when)\ s = 105:`

`d` `= 0.012 xx 105^2`
  `= 132.3\ text(metres)`

Rates of Change, SMB-005

Fuifui finds that for Giant moray eels, the mass of an eel `(M)` is directly proportional to the cube of its length `(l)`.

An eel of this species has a length of 15 cm and a mass of 675 grams.

What is the expected length of a Giant moray eel with a mass of 3.125 kg? Give your answer to one decimal place.  (3 marks)

Show Answers Only

`25\ text{cm}`

Show Worked Solution

`M prop l^3`

`M = kl^3`
 

`text(Find)\ k:`

`675` `= k xx 15^3`
`k` `= 675/15^3`
  `= 0.2`

 
`text(Find)\ \ l\ \ text(when)\ \ M = 3125:`

`3125` `= 0.2 xx l^3`
`l^3` `= 3125/0.2`
`:. l` `= root3(15\ 625)`
  `= 25\ text{cm}`

Rates of Change, SMB-004

Jacques is a marine biologist and finds that the mass of a crab `(M)` is directly proportional to the cube of the diameter of its shell `(d)`.

If a crab with a shell diameter of 15 cm weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place.  (3 marks)

Show Answers Only

`17.6\ text(cm)`

Show Worked Solution
`M` `prop d^3`  
`M` `= kd^3`  

 
`text(When)\ \ M=680, \ d=15`

`680` `=k xx 15^3`  
`k` `=0.201481…`  

 
`text(Find)\ \ d\ \ text(when)\ \ M=1100:`

`1100` `=0.20148… xx d^3`  
`d` `=root3(1100/(0.20148…))`  
  `=17.608…`  
  `=17.6\ text{cm  (to 1 d.p.)}`  

Rates of Change, SMB-003

The current of an electrical circuit, measured in amps (A), varies inversely with its resistance, measured in ohms (R).

When the resistance of a circuit is 28 ohms, the current is 3 amps.

What is the current when the resistance is 8 ohms? (2 marks)

Show Answers Only

`10.5`

Show Worked Solution
`A` `prop 1/R`  
`A` `= k/R`  

 
`text(When)\ \ A=3, \ R=28`

`3` `=k/28`  
`k` `=84`  

 
`text(Find)\ \ A\ \ text(when)\ \ R=8:`

`A` `=84/8`  
  `=10.5`  

Rates of Change, SMB-002

It is known that at a constant speed, the distance travelled in kilometres `(d)` is directly proportional to the time of travel in hours `(t)`, or  `d prop t`.

  1. If `d=75` when `t=5`, calculate the constant of variation `k`.  (2 marks)

  2. In the context of this question, what does the value of `k` represent?  (1 mark)

Show Answers Only

i.    `k=15`

ii.   `text{Speed}`

Show Worked Solutions

i.    `d prop t`

`d=kt`

`text(Find)\ k\ text{given}\ d=75\ text{when}\ t=5:`

`75` `=k xx 5`
`:. k` `=75/5`
  `=15`

 
ii.   `k\ text{represents the speed.}`

Rates of Change, SMB-001

It is known that a quantity `y` is inversely proportional to another quantity `x`.

If  `y=3` when `x=1.8`, calculate the constant of variation `k`.  (2 marks)

Show Answers Only

`k=5.4`

Show Worked Solutions

`y prop 1/x`

`y=k/x`

`text(Find)\ k\ text{given}\ y=3\ text{when}\ x=1.8:`

`3` `=k/1.8`
`:. k` `=3xx1.8`
  `=5.4`

Functions and Graphs, SMB-019

Shade the region defined by  `y+3x>3`  on the graph below and verify your result.  (3 marks)

Show Answers Only

`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(Incorrect – not in shaded area)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`0+3x=3\ \ =>\ \ x=1`

`ytext{-intercept occurs when}\ x=0:`

`y+3(0)=3\ \ =>\ \ y=3`

`text{Test}\ (0,0):`

`0+3(0)>3\ \ =>\ \ 0>3\ \ text{(Incorrect – not in shaded area)}`

Functions and Graphs, SMB-018

Shade the region defined by  `x/2-y>0`  on the graph below and verify your result.  (2 marks)

Show Answers Only

`text{Test}\ (1,0):`

`1/2-0>0\ \ text{(correct)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`x/2-0=0\ \ =>\ \ x=0`

`text{Graph cuts at}\ (0,0)`

`text{Also passes through}\ (2,1)`
 

`text{Test}\ (1,0):`

`1/2-0>0\ \ text{(Correct – in shaded area)}`

Functions and Graphs, SMB-017

  1. Express the function  `5y-3x=15`  in the form  `y=mx+b`.  (1 mark)

  2. Shade the region defined by  `5y-3x>15`  on the graph below and verify your result.  (3 marks)
      

Show Answers Only

i.    `y=3/5x+3`

ii.  

`text{Test}\ (0,0):`

`5(0)-3(0)>15\ \ =>\ \ 0<15\ \ text{(incorrect)}`

Show Worked Solution
i.     `5y-3x` `=15`
  `5y` `=3x+15`
  `y` `=3/5x+3`

 

ii.   `xtext{-intercept occurs when}\ y=0:`

`5(0)-3x=15\ \ =>\ \ x=-5`

`ytext{-intercept at}\ \ y=3`

`text{Test}\ (0,0):`

`5(0)-3(0)>15\ \ =>\ \ 0>15\ \ text{(Incorrect – not in shaded area.)}`

Functions and Graphs, SMB-016

Shade the region defined by  `3x-4y<12`  on the graph below and verify your result.  (3 marks)
 


Show Answers Only

`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(correct)}`

Show Worked Solution

`xtext{-intercept occurs when}\ y=0:`

`3x-4(0)=12\ \ =>\ \ x=4`

`ytext{-intercept occurs when}\ x=0:`

`3(0)-4y=12\ \ =>\ \ y=-3`

`text{Test}\ (0,0):`

`3(0)-4(0)<12\ \ =>\ \ 0<12\ \ text{(Correct – is in shaded area)}`

Functions and Graphs, SMB-013

  1. Factorise the expression  `x^2-x-6`.  (1 mark)

  2. What is the domain of the function  `f(x) = log_2(x^2-x-6)`?  (2 marks)

Show Answers Only

i.    `x^2-x-6=(x-3)(x+2)`

ii.   `text{Domain:}\ x< -2 \ ∪ \ x>3`

Show Worked Solution

i.    `x^2-x-6=(x-3)(x+2)`
 

ii.    `f(x) = log_2(x^2-x-6)=log_2(x-3)(x+2)`

`f(x)\ text{exists when:}`

`(x-3)(x+2)>0`

`x< -2 and x>3`

`text{Domain:}\ x< -2 \ ∪ \ x>3`

Functions and Graphs, SMB-010

A function has the equation  `h(x)=-1-(x-3)^2`.

State the domain and range of `h(x)`.   (2 marks)

Show Answers Only

`text{Domain}\ h(x):\ text{all}\ x`

`text{Range}\ h(x):\ y<=-1`

Show Worked Solution

`h(x)\ text{exists for all}\ x`

`text{Domain}\ h(x):\ text{all}\ x`
 

`text{Consider the function transformation:}`

`y=x^2\ text{translated 3 units right}\ \ =>\ \ y=(x-3)^2`

`y=(x-3)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x-3)^2`

`y=-(x-3)^2\ text{translated 1 unit down}\ =>\ \ y=-1-(x-3)^2`

`:.\ text{Range}\ h(x): \ y<=-1`

Functions and Graphs, SMB-008

A function has the equation  `f(x)=4-(x+1)^2`.

State the domain and range of `f(x)`.   (3 marks)

Show Answers Only

`text{Domain}\ f(x): \ text{all}\ x`

`text{Range}\ f(x): \ y<=4`

Show Worked Solution

`f(x)\ text{exists for all}\ x`

`text{Domain}\ f(x): \ text{all}\ x`
 

`text{Consider the function transformation:}`

`y=x^2\ text{translated 1 unit left}\ =>\ \ y=(x+1)^2`

`y=(x+1)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x+1)^2`

`y=-(x+1)^2\ text{translated 4 units up}\ \ =>\ \ y=4-(x+1)^2`

`:.\ text{Range}\ f(x): \ y<=4`

Functions and Graphs, SMB-003

State the domain and range of  `y = -sqrt(12-x^2)`.   (2 marks)

Show Answers Only

`text(Domain:)\ -sqrt12<=x<= sqrt12`

`text(Range:)\ -sqrt12<=y<= 0`

Show Worked Solution

`y = -sqrt(12-x^2)`

`12-x^2>=0\ \ =>\ \ x^2<=12`

`:.\ text(Domain:)\ -sqrt12<=x<= sqrt12`
 

`y_max =0`

`y_min = -sqrt12\ \ (text{when}\ x=0)`

`:.\ text(Range:)\ -sqrt12<=y<= 0`

Functions and Graphs, SMB-002

A function has the equation  `f(x)=(x-2)^2-5`.

State the range of `f(x)`.   (2 marks)

Show Answers Only

`text{Range}\ f(x): \ y>=-5`

Show Worked Solution

`text{Consider the function transformation:}`

`y=x^2\ \ text{translated 2 units to the right}\ \ =>\ \ y=(x-2)^2`

`y=(x-2)^2\ text{translated 5 units down}\ \ =>\ \ y=(x-2)^2-5`

`:.\ text{Range}\ f(x): \ y>=-5`

Functions and Graphs, SMB-001

`f(x)` is defined by the equation  `f(x)=3-x^2`.

  1. Find the coordinates of the `x`-intercepts of `f(x)`.  (1 mark)

  2. State the domain and range of `f(x)`.   (2 marks)

Show Answers Only

i.    `(sqrt3,0) and (-sqrt3, 0)`

ii.    `text{Domain: all}\ x`

`text{Range}\ f(x): \ y<=3`

Show Worked Solution

i.    `xtext{-intercepts occur when}\ y=0`

`3-x^2` `=0`  
`x^2` `=3`  
`x` `=+-sqrt3`  

 
`:. xtext{-intercepts at} (sqrt3,0) and (-sqrt3, 0)`
 

ii.   `text{Domain: all}\ x`

`text{Find range}\ f(x):`

`x^2>=0\ text{for all}\ x \ \ => \ \ 3-x^2<=3\ text{for all}\ x`

`:.\ text{Range}\ f(x): \ y<=3`

Polynomials, SMB-017

Sketch  `y=-(x+2)(x-1)^2` on the graph below, clearly showing all intercepts.  (3 marks)
 

Show Answers Only

Show Worked Solution

`xtext{-axis intercepts at}\ \ x=-2 and 1\ text{(touches only)}`

`ytext{-intercept when}\ \ x=0\ \ =>\ \ y=-2xx(-1)^2=-2`
 

Polynomials, SMB-016

Sketch  `y=(x+2)^2(2x-1)` on the graph below, clearly showing all intercepts .  (3 marks)
 

Show Answers Only

Show Worked Solution

`xtext{-axis intercepts at}\ \ x=-2\ text{(touches only) and}\ 1/2`

`ytext{-intercept when}\ x=0\ \ =>\ \ y=2^2(-1)=-4`
 

Polynomials, SMB-015

Sketch  `y=(x+1)^2(x-3)` on the graph below, clearly showing all intercepts .  (3 marks)
 

 

Show Answers Only

Show Worked Solution

`xtext{-axis intercepts at}\ \ x=-1\ text{(touches only) and}\ 3`

`ytext{-intercept when}\ x=0\ \ =>\ \ y=1^2(-3)=-3`
 

Polynomials, SMB-014

Sketch  `y=x(x-2)(x+3)` on the graph below, clearly showing all intercepts .  (3 marks)

 

Show Answers Only

Show Worked Solution

`xtext{-axis intercepts at}\ \ x=-3, 0 and 2`

`ytext{-intercept at}\ (0,0)`

`text{At}\ \ x=1\ \ =>\ \ y=-4`
  

Polynomials, SMB-013

Sketch  `y=(x+1)(x+2)(x-1)` on the graph below, clearly showing all intercepts .  (3 marks)

Show Answers Only

Show Worked Solution

`xtext{-axis intercepts at}\ \ x=-2, -1 and 1`

`ytext{-intercept when}\ \ x=0\ \ =>\ \  y=1xx2xx-1=-2`
  

Polynomials, SMB-012

`h(x)=x^3+3x^2+x-5`.

  1. Show  `h(1)=0`  (1 mark)

  2. Express `h(x)` in the form `h(x)=(x-1)*g(x)` where `g(x)` is a quadratic factor.  (2 marks)

  3. Justify that `h(x)` only has one zero.  (2 marks)

Show Answers Only

i.    `text{Proof (See worked solutions)}`

ii.    `h(x)=(x-1)(x^2+4x+5)`

iii.   `text{Proof (See worked solutions)}`

Show Worked Solution

i.   `h(x)=x^3+3x^2+x-5`.

`h(1) = 1+3+1-5=0`
 

ii.   `h(x)=(x-1)*g(x)`

`text{By long division:}`
 

`h(x)=(x-1)(x^2+4x+5)`
 

iii.   `text{Consider the roots of}\ \ y=x^2+4x+5`

`Δ = b^2-4ac=4^2-4*1*5=-4<0`

`text{Since}\ \ Δ<0\ \ =>\ \ text{No zeros (roots)}`

`:. h(x)\ text{only has 1 zero at}\ x=1\ (h(1)=0)`

Polynomials, SMB-011

`g(x)=(x-1)(x^2-2x+8)`.

Justify that `g(x)` only has one zero.  (2 marks)

Show Answers Only

`text{Proof (See worked solutions)}`

Show Worked Solution

`g(x)=(x-1)(x^2-2x+8)`.

`g(1)=0\ \ =>\ \ text{one zero at}\ \ x=1`

`text{Consider the roots of}\ \ y=x^2-2x+8`

`Δ = b^2-4ac=(-2)^2-4*1*8=-28<0`

`text{Since}\ \ Δ<0\ \ =>\ \ text{No zeros (roots)}`

`:. g(x)\ text{only has 1 zero}`

Polynomials, SMB-010 MC

`P(x)` is a monic polynomial of degree 4.

The maximum number of zeros that `P(x)` can have is

  1. `0`
  2. `1`
  3. `3`
  4. `4`
Show Answers Only

`D`

Show Worked Solution

`text{A polynomial of degree 4 has a leading term}\ ax^4`

`text{A monic polynomial of degree 4 has a leading term}\ x^4`

`:.\ text{Maximum number of zeroes}\ = 4`

`=>D`

Polynomials, SMB-009

Let  `P(x) = x^3+5x^2+2x-8`.

  1. Show that  `P(-2) = 0`.  (1 mark)

  2. Hence, factor the polynomial  `P(x)`  as  `A(x)B(x)`, where  `B(x)`  is a quadratic polynomial.  (2 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `P(x)=(x+2)(x^2+3x-4)`
Show Worked Solution
i.    `P(-2)` `= (-2)^3+ 5(-2)^2+2(-2)-8`
    `=-8+20-4-8`
    `= 0`

 

ii.  `text{Since}\ \ P(-2)=0\ \ =>\ \ (x+2)\ text{is a factor of}\ P(x)`

`P(x)=A(x)B(x)=(x+2)*B(x)`

`text{Using long division:}\ P(x)-:(x+2)=B(x)`
 

`:.P(x)=(x+2)(x^2+3x-4)`

Polynomials, SMB-008

Consider the polynomial  `P(x) = 2x^4+3x^3-12x^2-7x+6`.

Fully factorised, `P(x) = (2x-1)(x+3)(x+a)(x-b)`

Find the value of `a` and `b` where `a,b>0`.  (3 marks)

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`a=1, b=2`

Show Worked Solution

`text{Test for factors (by trial and error):}`

`P(1) = 2+3-12-7+6 = -8`

`P(-1) = 2-3-12+7+6 = 0\ \ =>\ \ (x+1)\ \ text{is a factor}`

`P(2) = 32+24-48-14+6 = 0\ \ =>\ \ (x-2)\ \ text{is a factor}`

`:. a=1, b=2`

Polynomials, SMB-007

Consider the polynomial  `P(x) = 3x^3+x^2-10x-8`.

Is `(x+2)` a factor of `P(x)`? Justify your answer.  (2 marks)

Show Answers Only

`P(-2) = -24+4+20-8=-8`

`:. (x+2)\ \ text(is not a factor of)\ P(x)`

Show Worked Solution

`P(-2) = -24+4+20-8=-8`

`:. (x+2)\ \ text(is not a factor of)\ P(x)`

Polynomials, SMB-006

Consider the polynomial  `P(x) = 2x^3-7x^2-7x+12`.

  1. Show that  `(x-1)`  is a factor of  `P(x)`.  (1 mark)

  2. Fully factorise `P(x)`.  (2 marks)

Show Answers Only
  1. `text(See Worked Solution)`
  2. `P(x)=(x-1)(2x+3)(x-4)`
Show Worked Solution

i.   `P(1) = 2-7-7+12=0`

`:. (x-1)\ \ text(is a factor of)\ P(x)`

 

ii.   `text{Using part (i)} \ => (x-1)\ text{is a factor of}\ P(x)`

`P(x) = (x-1)*Q(x)`
 

`text(By long division:)`
 

`P(x)` `= (x-1) (2x^2-5x-12)`
  `= (x-1)(2x+3)(x-4)`

Polynomials, SMB-005

Consider the polynomial  `P(x) = x^3-4x^2+x+6`.

  1. Show that  `x = -1`  is a zero of  `P(x)`.  (1 mark)

  2. Find the other zeros.  (2 marks)

Show Answers Only
  1. `text(See Worked Solution)`
  2. `x = 2 and 3`
Show Worked Solution

i.   `P(-1) = -1-4-1+6 = 0`

`:. x=-1\ \ text(is a zero)`

 

ii.   `text{Using part (i)} \ => (x+1)\ text{is a factor of}\ P(x)`

`P(x) = (x+1)*Q(x)`
 

`text(By long division:)`

`P(x)` `= (x+1) (x^2-5x+6)`
  `= (x+1)(x-2)(x-3)`

 
`:.\ text(Other zeroes are:)`

`x = 2 and x = 3`

Polynomials, SMB-004

Let  `p(x)=x^{3}-2 a x^{2}+x-1`. When `p(x)` is divided by `(x+2)`, the remainder is 5.

Find the value of `a`.  (2 marks)

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`-2`

Show Worked Solution

`text{Since}\ \ p(x) -: (x+2)\ \ text{has a remainder of 5:}`

`P(-2)` `=5`  
`5` `=(-2)^3-2a(-2)^2-2-1`  
`5` `=-8-8a-2-1`  
`8a` `=-16`  
`:.a` `=-2`