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Functions, 2ADV EQ-Bank 28

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)

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

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 25

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)

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`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 17

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)

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

\(\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 EQ-Bank 7 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 29

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)

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\(p=4\)

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 23

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

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`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, 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`