A-Level Edexcel Maths Pure Graphs of Functions: The straight line with equation

The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2px^2 - 6px + 4p$, where $p$ is a constant - Edexcel - A-Level Maths Pure - Question 10 - 2016 - Paper 1

Question 10

The-straight-line-with-equation-$y-=-3x---7$-does-not-cross-or-touch-the-curve-with-equation-$y-=-2px^2---6px-+-4p$,-where-$p$-is-a-constant-Edexcel-A-Level Maths Pure-Question 10-2016-Paper 1.png

The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2px^2 - 6px + 4p$, where $p$ is a constant. (a) Show that $4p^2 - ... show full transcript

Worked Solution & Example Answer:The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2px^2 - 6px + 4p$, where $p$ is a constant - Edexcel - A-Level Maths Pure - Question 10 - 2016 - Paper 1

Step 1

(a) Show that $4p^2 - 20p + 9 < 0$

96%

114 rated

Answer

To determine whether the straight line intersects the curve, we can set the two equations equal:

$3x - 7 = 2px^2 - 6px + 4p$

Rearranging gives:

$2px^2 - (6p + 3)x + (4p + 7) = 0$

For the line not to intersect the curve, the discriminant of the quadratic must be less than zero. The discriminant $D$ for the equation $ax^2 + bx + c = 0$ is given by:

$D = b^2 - 4ac$

Here, we have:

$a = 2p$
$b = -(6p + 3)$
$c = 4p + 7$

Calculating the discriminant, we get:

$D = [-(6p + 3)]^2 - 4(2p)(4p + 7)$

Simplifying,

$D = (6p + 3)^2 - 8p(4p + 7)$

Expanding both sides yields:

$(36p^2 + 36p + 9) - (32p^2 + 56p)$

This simplifies to:

$4p^2 - 20p + 9$

Thus, we need to show that:

$4p^2 - 20p + 9 < 0$

Step 2

(b) Hence find the set of possible values of $p$

99%

104 rated

Answer

To solve the inequality $4p^2 - 20p + 9 < 0$ , we can first find the roots of the equation $4p^2 - 20p + 9 = 0$ using the quadratic formula:

$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting in our values ( $a = 4, b = -20, c = 9$ ):

$p = \frac{20 \pm \sqrt{(-20)^2 - 4(4)(9)}}{2(4)}$

Calculating the discriminant:

$400 - 144 = 256$

Thus, we have:

$p = \frac{20 \pm 16}{8}$

Calculating the two roots:

$p = \frac{36}{8} = 4.5$
$p = \frac{4}{8} = 0.5$

The quadratic $4p^2 - 20p + 9$ opens upwards (as $a > 0$ ), therefore it is negative between the two roots. Thus, the set of possible values for $p$ is:

$0.5 < p < 4.5$

The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2px^2 - 6px + 4p$, where $p$ is a constant - Edexcel - A-Level Maths Pure - Question 10 - 2016 - Paper 1

Worked Solution & Example Answer:The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2px^2 - 6px + 4p$, where $p$ is a constant - Edexcel - A-Level Maths Pure - Question 10 - 2016 - Paper 1

(a) Show that $4p^2 - 20p + 9 < 0$

(b) Hence find the set of possible values of $p$

Join the A-Level students using SimpleStudy...