Higher November 2023 Paper 3 Q17

(a)

(i) Write \(x^2 - 8x + 3\) in the form \((x - a)^2 - b\) where \(a\) and \(b\) are integers. (2)

(ii) Hence, write down the coordinates of the turning point on the graph of \(y = x^2 - 8x + 3\) (1)

(b) Solve \(7x^2 + 8x - 5 = 0\)
Give your solutions correct to 3 significant figures. (3)

Alex has to find the solutions of the quadratic equation \(3k^2 + 10k - 8 = 0\)
Here is his working and answer.

\[\begin{gathered} (3k - 2)(k + 4) = 0 \\ k = 2 \text{ or } k = -4 \end{gathered}\]

Mark scheme (a)

Answer	Mark	Mark scheme
(i) \((x - 4)^2 - 13\)	M1	for start to method of completing the square, eg \((x - 4)^2\ldots\)
(i) \((x - 4)^2 - 13\)	A1	cao
(ii) \((4, -13)\)	B1	ft from answer of form \((x - a)^2 - b\)

(ii) This mark is dependent on a completed square form given in (a)(i)

Mark scheme (b)

Answer	Mark	Mark scheme
0.449 and \(-1.59\)	M1	for substitution into quadratic formula, eg \(\dfrac{-8 \pm \sqrt{8^2 - 4 \times 7 \times -5}}{2 \times 7}\) oe
	M1	for simplification to \(\dfrac{-8 \pm \sqrt{204}}{14}\) or \(\dfrac{-4 \pm \sqrt{51}}{7}\) oe or for one correct solution in decimal form.
	A1	for 0.448 to 0.449 and \(-1.59\) to \(-1.6(0)\)

Condone one sign error

One correct solution in the required form out of two solutions given scores 2 marks

If an answer is shown in the range in working and then incorrectly rounded award full marks.

Mark scheme (c)

Answer	Mark	Mark scheme
Explanation	C1	for explanation Acceptable He hasn’t considered the coefficient of 3 He forgot about the 3 The answer \(x = 2\) should be \(x = \dfrac{2}{3}\) The first bracket was not \(k - 2\) \(k = 2\) is wrong, he should have divided by 3 Not acceptable One of the answers is wrong The solutions are not correct He should have written \(3(k - 2)(k + 4)\) The \(3k\) should not be inside the bracket 2 added to \(-4\) does not give 10 \(k = 2\) is wrong