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Functions, MET2 2023 VCAA 19 MC

Find all the values of \(k\), such that the equation \(x^2+(4k+3)x+4k^2-\dfrac{9}{4}=0\) has two real solutions for \(x\), one positive and one negative.

  1. \(k>-\dfrac{3}{4}\)
  2. \(k\geq-\dfrac{3}{4}\)
  3. \(k>\dfrac{3}{4}\)
  4. \(-\dfrac{3}{4}<k<\dfrac{3}{4}\)
  5. \(k<-\dfrac{3}{4}\ \text{or}\ k>\dfrac{3}{4}\)
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\(D\)

Show Worked Solution

\(\text{Use the quadratic formula and solve for }k\text{ using CAS.}\)

\(x^2+(4k+3)x+4k^2-\dfrac{9}{4}=0\)

\(\text{Need one positive and one negative solution.}\)

\(\therefore\ \text{Solve:}\)

\(\ \dfrac{-b+\sqrt{b^2-4ac}}{2a}\) \(>0\)      
\(\ \dfrac{-4k-3+\sqrt{(4k+3)^2-4(4k^2-\dfrac{9}{4}}}{2}\) \(>0\)      

 
\(\to\ \ -\dfrac{3}{4}<k<\dfrac{3}{4}\)

 \(\dfrac{-b-\sqrt{b^2-4ac}}{2a}\) \(<0\)
\(\ \dfrac{-4k-3-\sqrt{(4k+3)^2-4(4k^2-\dfrac{9}{4}}}{2}\) \(<0\)

 
\(\to\ \ \ k>-\dfrac{3}{4}\)
 
\(\text{Must be the intersection of the values of }k\)

\(\therefore\ \ -\dfrac{3}{4}<k<\dfrac{3}{4}\ \text{is the correct range of values of }k\)

\(\Rightarrow D\)


♦♦ Mean mark 32%.
MARKER’S COMMENT: 27% incorrectly chose A.

Algebra, MET2 2019 VCAA 2 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}`
  5. `[-1, oo)`
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`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`

Algebra, MET2 2017 VCAA 7 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`
  5. `p^2 - 6p + 6 > 0`
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`B`

Show Worked Solution

`(p – 1)x^2 + 4x + (p – 5) = 0`

♦♦ Mean mark 32%.

 

`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`

Graphs, MET2 2007 VCAA 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`
  5. `k = 5`
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`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`

 

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

`=>   B`

Algebra, MET2 2015 VCAA 21 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. `a > 0 and c < 0`
  3. `a > 0 and c > -m^2/(4a)`
  4. `a < 0 and c > -m^2/(4a)`
  5. `m > 0 and c > 0`
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`D`

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`text(Intersect when:)`

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

 

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

♦ Mean mark 37%.
`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`

Algebra, MET2 2014 VCAA 18 MC

The graph of  `y = kx - 4`  intersects the graph of  `y = x^2 + 2x`  at two distinct points for

  1. `k = 6`
  2. `k > 6 or k < -2`
  3. `-2 <= k <= 6`
  4. `6 - 2 sqrt 3 <= k <= 6 + 2 sqrt 3`
  5. `k = -2`
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`B`

Show Worked Solution

`text(Intersect 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 < − 2 \ \ ∪ \ \ k>6`

`=>   B`

Calculus, MET2 2013 VCAA 21 MC

The cubic function  `f: R -> R, f(x) = ax^3-bx^2 + cx`, where `a, b` and `c` are positive constants, has no stationary points when

  1. `c > b^2/(4a)`
  2. `c < b^2/(4a)`
  3. `c < 4b^2a`
  4. `c > b^2/(3a)`
  5. `c < b^2/(3a)`
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`D`

Show Worked Solution

`text(If no stationary points,)`

♦♦ Mean mark 29%.

`=>\ text(No solution to)\ \ f{′}(x) = 0`

`f^{′}(x) = 3ax^2 -2bx +c`
 

`text(No solution when,)`

`Delta` `< 0`
`(−2b)^2-4(3ac)` `< 0`
`3ac` `> b^2`
`:. c` `> (b^2)/(3a)`

`=>   D`