SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Functions, 2ADV EQ-Bank 2

Given \(p\) and \(q\) are rational numbers, and  \(p, q \neq 0\), show

\(px^2-(p+q) x+q=0\)

has rational roots.   (3 marks)

Show Answers Only

\(\text{Proof (See Worked Solution)}\)

Show Worked Solution
\(\Delta\) \(=b^2-4 a c\)
  \(=[-(p+q)]^2-4 \times p \times q\)
  \(=p^2+2 p q+q^2-4 p q\)
  \(=p^2-2 p q+q^2\)
  \(=(p-q)^2\)

 

\(\text{Roots of equation using quadratic formula:}\)

\(x\) \(=\dfrac{(p+q) \pm \sqrt{(p-q)^2}}{2 p}\)
  \(=\dfrac{p+q+(p-q)}{2 p} \ \ \text{or} \ \ \dfrac{p+q-(p-q)}{2 p}\)
  \(=1 \ \ \text{or} \ \ \dfrac{q}{p}\).

 

\(\text{Since \(p, q\) are rational, all roots are rational.}\)

Functions, 2ADV EQ-Bank 8 MC

The equation  `(p-1)x^2 + 4x = 5-p`  has no real roots when

  1. `p^2-6p + 6 < 0`
  2. `p^2-6p + 1 > 0`
  3. `p^2-6p-6 < 0`
  4. `p^2-6p + 1 < 0`
Show Answers Only

`B`

Show Worked Solution

`(p-1)x^2 + 4x + (p-5) = 0`

 
`text(No real solutions when)\ \ Δ<0:`

`b^2-4ac` `<0`
`4^2-4 (p-1)(p-5)` `< 0`
`16-4(p^2-6p+5)` `<0`
`−4p^2 + 24p-4` `< 0`
`p^2-6p + 1` `> 0`

 
`=> B`

Functions, 2ADV EQ-Bank 9 MC

The graphs of  `y = mx + c`  and  `y = ax^2`  will have no points of intersection for all values of `m, c` and `a` such that

  1. `a > 0 and c > 0`
  2. `m > 0 and c > 0`
  3. `a > 0 and c > -m^2/(4a)`
  4. `a < 0 and c > -m^2/(4a)`
Show Answers Only

`D`

Show Worked Solution

`text(Intersect when:)`

`mx + c` `= ax^2`
`ax^2-mx-c` `= 0`

 
`text(S)text(ince no points of intersection:)`

`Delta` `< 0`
`m^2-4a(−c)` `< 0`
`m^2 + 4ac` `< 0`

 
`text(Solve for)\ c:`

`:.\ c > (−m^2)/(4a),quada < 0`

`text(or)`

`c < (−m^2)/(4a),quada > 0`

`=>   D`

Functions, 2ADV EQ-Bank 14

Determine the value of `k` if the graph of  `y = kx-4`  intersects the graph of  `y = x^2 + 2x`  at two distinct points. Give your answer in set notation.   (3 marks)

Show Answers Only

`k \in (-oo,-2) \uu (6, oo)`

Show Worked Solution

`text(Intersection when:)\ kx-4 = x^2 + 2x`

`x^2 + (2-k)x + 4 = 0`

`text(2 solutions when)\ \ Delta>0:`

`(2-k)^2-4 xx 4` `> 0`
`k^2-4k-12` `>0`
`(k-6)(k+2)` `>0`

 
`k \in (-oo,-2) \uu (6, oo)`

Functions, 2ADV EQ-Bank 8

Solve for \(x\), giving your answers in the simplest form  \(a+b\sqrt{c}\)  where \(a, b\) and \(c\) are real:

\(5 x^2-20 x+4=0\)   (2 marks)

Show Answers Only

\(x=2 \pm \dfrac{4}{5} \sqrt{5}\)

Show Worked Solution

\(5 x^2-20 x+4=0\)

\(x\) \(=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2 a}\)
  \(=\dfrac{20 \pm \sqrt{20^2-4 \times 5 \times 4}}{2 \times 5}\)
  \(=\dfrac{20 \pm \sqrt{320}}{10}\)
  \(=2 \pm \dfrac{8 \sqrt{5}}{10}\)
  \(=2 \pm \dfrac{4}{5} \sqrt{5}\)

Functions, 2ADV EQ-Bank 7 MC

What are the solutions to  \(3x^2+2x-4=0\)?

  1. \(x=\dfrac{-1 \pm \sqrt{13}}{3}\)
  2. \(x=\dfrac{1 \pm \sqrt{13}}{3}\)
  3. \(x=\dfrac{-1 \pm \sqrt{5}}{2}\)
  4. \(x=\dfrac{1 \pm \sqrt{5}}{2}\)
Show Answers Only

\(A\)

Show Worked Solution

\(3 x^2+2 x-4=0\)

\(x\) \(=\dfrac{-b \pm \sqrt{b^2-4 a c}}{2a}\)
  \(=\dfrac{-2 \pm \sqrt{2^2-4 \times 3 \times-4}}{2 \times 3}\)
  \(=\dfrac{-2 \pm \sqrt{52}}{6}\)
  \(=\dfrac{-1 \pm \sqrt{13}}{3}\)

 

\(\Rightarrow A\)

Functions, 2ADV EQ-Bank 10

  1. Express  \(y=x^2-4 x+6\)  in the form  \(y=(x-a)^2+c\).   (1 mark)

  2. Graph the parabola, labelling its vertex and \(y\)-intercept.   (2 marks)

Show Answers Only

a.   \(y=(x-2)^2+2\)

b.   \(\text {Vertex}=(2,2)\)

\(y \text{-intercept}=(0,6)\)
 

Show Worked Solution
a.     \(y\) \(=x^2-4 x+6\)
    \(=x^2-4 x+4+2\)
    \(=(x-2)^2+2\)

 
b.    \(\text {Vertex}=(2,2)\)

\(y \text{-intercept}=(0,6)\)

Functions, 2ADV EQ-Bank 9

Using the discriminant, or otherwise, justify why the graph of  \(f(x)=-x^2+2 x-2\)  lies entirely below the \(x\)-axis.   (2 marks)

Show Answers Only

\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Show Worked Solution

\(\Delta=b^2-4 a c=2^2-4(-1)(-2)=-4\)

\(\text{Since \(\ \Delta<0, \ y=-x^2+2 x-2\ \) does not intersect the \(x\)-axis.}\)

\(\text{Since \(\ a=-1<0, f(x)\) is an upside down parabola.}\)

\(\Rightarrow f(x)\ \text{must lie entirely below} \ x\text{-axis.}\)

Functions, 2ADV F1 2025 HSC 11

The graph of a quadratic function represented by the equation  \(h=t^2-8 t+12\)  is shown.
 

  1. Find the values of \(t\) and \(h\) at the turning point of the graph.   (2 marks)

  2. The graph shows  \(h=12\)  when  \(t=0\).
  3. What is the other value of \(t\) for which  \(h=12\)?   (1 mark)

Show Answers Only

a.   \(\text{Turning point at} \ \ (4,-4)\)

b.   \(t=8\)

Show Worked Solution

a.    \(\text{Strategy 1 (no calculus)}\)

\(\text{Axis of quadratic occurs when}\ \ t= \dfrac{2+6}{2} = 4\)

\(\text{At} \ \ t=4:\)

\(h=4^2-8 \times 4+12=-4\)

\(\therefore \ \text{Turning point at} \ \ (4,-4)\)
 

\(\text{Strategy 2 (using calculus)}\)

\(h=t^2-8 t+12\)

\(h^{\prime}=2 t-8\)

\(\text{Find \(t\) when} \ \ h^{\prime}=0:\)

\(2 t-8=0 \ \Rightarrow \ t=4\)
 

b.    \(\text {When} \ \ h=12:\)

\(t^2-8 t+12\) \(=12\)
\(t(t-8)\) \(=0\)

 
\(\therefore \ \text{Other value:} \ \ t=8\)

Functions, 2ADV F1 EQ-Bank 1 MC

The graph of a quadratic function  \(f(x)=a x^2+b x+c\)  is drawn below.
 

Which of the following are true?

  1. \(a<0, c=0\)  and  \(b^2-4 a c=0\)
  2. \(a>0, c=0\)  and  \(b^2-4 a c=0\)
  3. \(a>0, c>0\)  and  \(b^2-4 a c>0\)
  4. \(a<0, c>0\)  and  \(b^2-4 a c=0\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Quadratic touches } x \text{-axis once only} \ \ \Rightarrow b^2-4 a c=0\ \ \text{(eliminate C)}\)

\(\text{Quadratic is inverted} \Rightarrow a<0 \ \ \text{(eliminate B)}\)

\(\text{If} \ \ c=0, f(x)=a x^2+b x+0=x(a x+b) \Rightarrow \text{cuts twice (Eliminate A)}\)

\(\Rightarrow D\)

Functions, 2ADV F1 EQ-Bank 19

\(R\left(r, r^2\right), S\left(s, s^2\right)\) and \(T\left(t, t^2\right)\) are points on the parabola  \(y=x^2\).

Given \(RT\) is parallel to \(SO\), show  \(r+t=s\)   (2 marks)
 

Show Answers Only

\(R\left(r, r^2\right), S\left(s, s^2\right), T\left(t, t^2\right)\)

\(m_{S O}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{s^2-0}{s-0}=s\)

\(m_{R T}=\dfrac{t^2-r^2}{t-r}=\dfrac{(t-r)(t+r)}{(t-r)}=t+r\)

\(\text{Given}\ R T \ \| \  SO \ \Rightarrow \ m_{SO}=m_{R T}\)

\(\therefore s=r+t\ \ …\ \text{as required} \)

Show Worked Solution

\(R\left(r, r^2\right), S\left(s, s^2\right), T\left(t, t^2\right)\)

\(m_{S O}=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{s^2-0}{s-0}=s\)

\(m_{R T}=\dfrac{t^2-r^2}{t-r}=\dfrac{(t-r)(t+r)}{(t-r)}=t+r\)

\(\text{Given}\ R T \ \| \  SO \ \Rightarrow \ m_{SO}=m_{R T}\)

\(\therefore s=r+t\ \ …\ \text{as required} \)

Functions, 2ADV F1 EQ-Bank 17

The tangent to the parabola  \(y=x^2+2 x-4\)  is  \(y=px-5\)  where  \(p>0\).

Find the value of \(p\).   (2 marks)

Show Answers Only

\(p=4\)

Show Worked Solution

\(\text{Intersection occurs when:}\)

\(x^2+2x-4\) \(=px-5\)  
\(x^2+(2-p)x+1\) \(=0\)  

 
\(\text{Tangent touches once}\ \Rightarrow\ \text{Discriminant}\ \Delta=0\)

\((2-p)^2-4 \times 1 \times 1\) \(=0\)  
\(4-4p+p^2-4\) \(=0\)  
\(p(p-4)\) \(=0\)  
\(p\) \(=4\ \ \ (p\gt 0)\)  
COMMENT: Key is to recognise this is a discriminant question, not a calculus application.

Functions, 2ADV F1 EQ-Bank 45

Determine whether the function  \(f(x)=2x^3-5x\)  is even, odd or neither. Show all working.   (2 marks)

Show Answers Only

\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Show Worked Solution

\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Functions, 2ADV F1 2023 HSC 10 MC

The graph  \(y = x^2\)  meets the line  \(y = k\)  (where \(k>0\)) at points \(P\) and \(Q\) as shown in the diagram. The length of the interval \(PQ\) is \(L\).
 

Let \(a\) be a positive number. The graph  \(y=\dfrac{x^2}{a^2}\)  meets the line  \(y=k\)  at points \(S\) and \(T\).

What is the length of \(ST\)?

  1. \(\dfrac{L}{a}\)
  2. \(\dfrac{L}{a^2}\)
  3. \(aL\)
  4. \(a^2L\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Intersection of}\ \ y=x^2\ \ \text{and}\ \ y=k:\)

\(x^2=k\ \ \Rightarrow\ \ x=\pm \sqrt k\)

\(\therefore L=2\sqrt k\)

\(\text{Intersection of}\ \ y=\dfrac{x^2}{a^2}\ \ \text{and}\ \ y=k:\)

\(\dfrac{x^2}{a^2} \) \(=k\)  
\(x^2\) \(=a^2k\)  
\(x\) \(=\pm a\sqrt k\)  

\(\therefore ST=a \times 2\sqrt k = aL \)

\(\Rightarrow C\)

♦♦♦ Mean mark 24%.

Functions, 2ADV F1 2023 HSC 9 MC

Let \(f(x)\) be any function with domain all real numbers.

Which of the following is an even function, regardless of the choice of \(f(x)\)?

  1. \(2 f(x)\)
  2. \(f(f(x))\)
  3. \((f(-x))^2\)
  4. \(f(x) f(-x)\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Even function}\ \rightarrow \ f(x)=f(-x)\)

\(\text{Consider the function}\ \ f(x) = x-2\)

\( 2f(1)=-2,\ \ 2f(-1)=-6\ \ \text{(not even)}\)

\( f(f(1))=f(-1)=-3,\ \ f(f(-1))=f(-3)=-5\ \ \text{(not even)}\)

\( (f(-1))^2=(-3)^2=9,\ \ (f(1))^2=(-1)^2=1\ \ \text{(not even)}\)

\( f(1)f(-1)=-1 \times -3=3,\ \ f(-1)f(1)=-3 \times -1=3 \ \text{(possibly even)}\)

\(=>D\)

♦♦♦ Mean mark 24%.

Functions, 2ADV F1 2023 HSC 4 MC

The graph of a polynomial is shown.
 


  

Which row of the table is correct for this polynomial?
 

Show Answers Only

`B`

Show Worked Solution

`text{By elimination:}`

`text{Double root where graph SP touches}\ xtext{-axis}`

`:.\ C and D\ text{incorrect}`

`(x-c)^2\ text{occurs where}\ x>0`

`:. c>0\ (A\ text{is incorrect})`

`=>B`

Functions, EXT1 F1 EQ-Bank 17

Solve the inequality  `x^2<=|\ 2x-1\ |`  for `x`.

Express your answer in interval notation.   (3 marks)

Show Answers Only

`:. x in [-1-sqrt2, -1 + sqrt2]\ \ ∪\ \ x in [1]`

Show Worked Solution

`text(Case 1:)`

`x^2` `<=2x-1`
`x^2-2x+1` `<=0`
`(x-1)^2` `<=0`

 
`x=1`
 

`text(Case 2:)`

`x^2` `<=-(2x-1)`
`x^2+2x-1` `<=0`

 

`x` `=(-2+-sqrt(4+4*1*1))/2`  
  `=-1+-sqrt2`  

 
`text(Test)\ \ x=0\ \ =>\ \ 0<=-(-1)\ \ text{(correct)}`

`:. x in [-1-sqrt2, -1 + sqrt2]\ \ ∪\ \ x in [1]`

Functions, 2ADV F1 2021 HSC 8 MC

The graph of  `y = f(x)`  is shown.

Which of the following could be the equation of this graph?

  1. `y = (1 - x)(2 + x)^3`
  2. `y = (x + 1)(x - 2)^3`
  3. `y = (x + 1)(2 - x)^3`
  4. `y = (x - 1)(2 + x)^3`
Show Answers Only

`C`

Show Worked Solution

`text(By elimination:)`

`text(A single negative root occurs when)\ \ x =–1`

`->\ text(Eliminate A and D)`

`text(When)\ \ x = 0, \ y > 0`

`->\ text(Eliminate B)`

`=> C`

Functions, 2ADV F1 SM-Bank 23

Find the values of `k` for which the expression  `x^2-3x + (4-2k)`  is always positive.  (3 marks)

Show Answers Only

`k < 7/8`

Show Worked Solution

`x^2-3x + (4-2k) > 0`

`x^2-3x + (4-2k) = 0\ \ text(is a concave up parabola)`

`=>\ text{Always positive (no roots) if}\ \ Delta < 0`
 

`b^2-4ac < 0`

`(−3)^2-4 · 1 · (4-2k)` `< 0`
`9-16 + 8k` `< 0`
`8k` `< 7`
`k` `< 7/8`

Functions, EXT1 F1 EQ-Bank 1

Find the values of `x` that satisfy the equation

  `x^2 + 8x + 3 <= 0`.   (3 marks)

Show Answers Only

`{x:\  -4-sqrt{13} <=x <= -4 + sqrt{13}}`

Show Worked Solution
`x` `= (-8 ± sqrt(8^2-4 · 1 · 3))/2`
  `= (-8 ± sqrt(52))/2`
  `= -4 ± sqrt13`

 


 

`{x:\  -4-sqrt13 <=x <= -4 + sqrt13}`

Functions, 2ADV F1 2017 HSC 2 MC

Which expression is equal to  `3x^2-x-2`?

  1. `(3x-1) (x + 2)`
  2. `(3x + 1) (x-2)`
  3. `(3x-2) (x + 1)`
  4. `(3x + 2) (x-1)`
Show Answers Only

`D`

Show Worked Solution

`3x^2-x-2= (3x + 2) (x-1)`

`=>  D`

Functions, 2ADV F1 2016 HSC 11e

Find the points of intersection of  `y=-5-4x`  and  `y=3-2x-x^2.`  (3 marks)

Show Answers Only

`(4, – 21) and (– 2, 3)`

Show Worked Solution

`y = 3 – 2x – x^2`

`text(Substitute)\ \ y = -5 – 4x\ \ text(into equation)`

`-5 – 4x` `= 3 – 2x – x^2`
`x^2 – 2x – 8` `= 0`
`(x – 4) (x + 2)` `= 0`

  
`:. x = 4 or -2`
 

`text(When)\ \ x = 4,\ \ y = -5 – 4(4) = -21`

`text(When)\ \ x = -2,\ \ y = -5 – 4 (-2) = 3`  
 

`:.\ text(Intersection at)\ \ (4, – 21) and (– 2, 3)`

Functions, 2ADV F1 2016 HSC 4 MC

Which diagram shows the graph of an odd function?
 

hsc-2016-4mci

hsc-2016-4mcii

Show Answers Only

`A`

Show Worked Solution

`text(Odd functions occur when:)`

♦ Mean mark 38%.

`f(x) = – f(x)`

`text(Graphically, this occurs when a function has)`

`text(symmetry when rotated 180° about the origin.)`

`=>  A`

Functions, 2ADV 2006 HSC 7c

  1. Write down the discriminant of  `2x^2 + (k-2)x + 8`  where  `k`  is a constant.   (1 mark)
  2. Hence, or otherwise, find the values of `k` for which the parabola  `y = 2x^2 + kx + 9` does not intersect the line  `y = 2x + 1`.   (2 marks)

Show Answers Only

i.    `\Delta= k^2-4k-60`

ii.  `-6 < k < 10`

Show Worked Solution

i.   `2x^2 + (k-2)x + 8`

`Delta` `= b^2-4ac`
  `= (k-2)^2-4 xx 2 xx 8`
  `= k^2-4k + 4-64`
  `= k^2-4k-60`

 

ii.   `y` `= 2x^2 + kx + 9` `\ \ text{…  (1)}`
`y` `= 2x + 1` `\ \ text{…  (2)}`

 
`text(Substitute)\ y = 2x + 1\ text{into (1)}`

`2x + 1 = 2x^2 + kx + 9`

`2x^2 + kx-2x + 8 = 0`

`2x^2 + (k-2)x + 8 = 0\ …\ text{(∗)}`
 

`text{The graphs will not intercept if (∗) has no roots, i.e.)\ \ Delta <0`

`k^2-4k-60` `< 0`
`(k-10) (k + 6)` `< 0`

 

HSC quadratic

`text(From the graph, no intersection when)`

`-6 < k < 10`

Functions, 2ADV F1 2014 HSC 6 MC

Which expression is a factorisation of  `8x^3 + 27`? 

  1. `(2x - 3)(4x^2 + 12x - 9)`
  2. `(2x + 3)(4x^2 - 12x + 9)`
  3. `(2x - 3)(4x^2 + 6x - 9)`
  4. `(2x + 3)(4x^2 - 6x + 9)`
Show Answers Only

`D`

Show Worked Solution

`8x^3 + 27`

COMMENT: Factorising a cubic is only examinable with scaffolding, as provided here by expanding the answer options.

`= (2x)^3 + 3^3`

`= (2x + 3)(4x^2 – 6x + 9)`

 
`=>  D`

Functions, EXT1* F1 2010 HSC 2b

Solve the inequality  `x^2-x-12 < 0`.   (2 marks) 

Show Answers Only

 `-3 < x < 4`

Show Worked Solution
 MARKER’S COMMENT: Drawing the parabola proved the most efficient way of answering this question for students.

`x^2-x-12<0`

`text(Solve)\ \ \ ` `x^2-x-12` `=0`
  `(x-4)(x + 3)` `=0`
`=>x = 4\ text(or)\ -3`

 

 

`x^2-x-12 < 0\ \ \text(when)\ \ -3 < x < 4`

Functions, 2ADV F1 2013 HSC 1 MC

What are the solutions of   `2x^2-5x-1 = 0`? 

  1. `x = (-5 +-sqrt17)/4` 
  2. `x = (5 +-sqrt17)/4`
  3. `x = (-5 +-sqrt33)/4`
  4. `x = (5 +-sqrt33)/4`
Show Answers Only

`D`

Show Worked Solution

`2x^2-5x-1 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`x` `= (5 +- sqrt{\ \ (-5)^2-4 xx 2 xx(-1) })/ (2 xx 2)`
  `= (5 +- sqrt(25 + 8) )/4`
  `= (5 +- sqrt(33) )/4`

 
`=>  D`