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Functions, 2ADV EQ-Bank 10 MC

The graph of  `y = kx-3`  intersects the graph of  `y = x^2 + 8x`  at two distinct points for

  1. `k = 11`
  2. `k > 8 + 2 sqrt 3 or k < 8-2 sqrt 3`
  3. `5 <= k <= 6`
  4. `8-2 sqrt 3 <= k <= 8 + 2 sqrt 3`
Show Answers Only

`B`

Show Worked Solution

`text(Intersection occurs when:)`

`kx-3` `= x^2 + 8x`
`x^2 + (8-k)x + 3` `= 0`

 
`text(For 2 points of intersection:)`

`Delta` `> 0`
`(8-k)^2-4 (3)` `> 0`
`(8-k)^2` `>12`

 
`:. k < 8-2 sqrt 3\  uu\  k > 8 + 2 sqrt 3`

`=>   B`

Functions, 2ADV EQ-Bank 5 MC

The set of values of `k` for which  `x^2 + 2x-k = 0`  has two real solutions is

  1. `[-1, 1]`
  2. `(-1, oo)`
  3. `(-oo, -1)`
  4. `[-1]`
Show Answers Only

`B`

Show Worked Solution

`text(Two real solutions):`

`b^2-4ac` `> 0`
`4-4 ⋅ 1 ⋅ (-k)` `> 0`
`4k` `> -4`
`k` `> -1`

 
`k in (-1, oo)`

`=>   B`

Functions, 2ADV F1 EQ-Bank 20

Find the value of \(k\)  if  \(4kx^2-(3-4k) x+k=0\)  has one root.   (2 marks)

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\(k=\dfrac{3}{8}\)

Show Worked Solution

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

\(\text{1 root}\ \Rightarrow \Delta=0\)

  \(\Delta\) \(=b^2-4 a c\)
  \(0\) \(=\left[ -\left( 3-4k \right)\right]^2-4\times 4k \times k\)
  \(0\) \(=9-24k + 16k^2-16k^2\)
  \(24k\) \(=9\)
  \(k\) \(=\dfrac{9}{24}=\dfrac{3}{8}\)

Functions, 2ADV F1 EQ-Bank 13

Show that the parabola  \(2x^2-kx+k-2\)  has at least one real root.   (3 marks)

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\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Show Worked Solution

\(2x^2-kx+k-2=0\)

\(\Delta=b^2-4ac=(-k)^2-4 \times 2(k-2) = k^{2}-8k+16\)

\(\text{Real roots:}\ \ \Delta \geqslant 0\)

\(k^2-8k+16\) \(\geqslant 0\)  
\((x-4)^2\) \(\geqslant 0\)  

 
\(\therefore\ \text{At least one root exists for all}\ k\)

Functions, 2ADV 2015 HSC 12d

For what values of `k` does the quadratic equation  `x^2-8x + k = 0`  have real roots?   (2 marks)

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`k <= 16`

Show Worked Solution

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

`text(Real roots when)\ Delta >= 0:`

`b^2-4ac` `>= 0`
`(-8)^2-4 xx 1 xx k` `>= 0`
`64-4k` `>= 0`
`4k` `<= 64`
`k` `<= 16`

 
`:.\ text(Real roots exists when)\ k <= 16`

Functions, 2ADV 2004 HSC 2c

For what values of `k` does  `x^2-kx + 4 = 0`  have no real roots?   (2 marks)

Show Answers Only

`-4 < k < 4`

Show Worked Solution

`x^2-kx + 4 = 0`

`text(No real roots when)\ Delta < 0`

`b^2-4ac` `< 0`
`(text(–) k^2)-4 xx 1 xx 4` `< 0`
`k^2-16` `< 0`
`k^2` `< 16`
`:.\ -4 < k` `< 4`

 

`:.\ text(There are no real roots when)\ \ \ -4 < k < 4.`

Functions, 2ADV 2009 HSC 4b

Find the values of `k` for which the quadratic equation 

`x^2-(k + 4)x + (k + 7) = 0`

has equal roots.    (3 marks)

Show Answers Only

 `k = -6 \ \ text(or)\ \  2`

Show Worked Solution

`x^2-(k + 4)x + (k + 7) = 0`

`text(Equal roots when)\ Delta = 0:`

`[-(k + 4)]^2-4(1)(k + 7)` `= 0`
`k^2 + 8k + 16-4k\-28` `= 0`
`k^2 + 4k-12` `= 0`
`(k + 6)(k-2)` `= 0`
`k` `= -6 \ \ text(or)\ \  2`

 
`:.\ text(Equal roots when)\ \ k = -6 or 2`