Higher June 2019 Paper 3 Q18
18
(a) Show that \((2x + 1)(x + 3)(3x + 7)\) can be written in the form \(ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\) and \(d\) are integers. (3)
(b) Solve \((1 - x)^2 \lt \dfrac{9}{25}\) (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(6x^3 + 35x^2 + 58x + 21\) | M1 | for a method to find the product of two linear expressions, 3 correct terms out of 4 terms e.g. \(2x^2 + x + 6x + 3\) or \(3x^2 + 7x + 9x + 21\) or \(6x^2 + 14x + 3x + 7\) |
| M1 | for a complete method to obtain all terms, at least half of which are correct (ft their first product) e.g. \(6x^3 + 32x^2 + 42x + 3x^2 + 16x + 21\) | |
| A1 | cao |
Additional guidance
Note that, for example, \(7x + 3\) is regarded as three terms in the expansion of \((2x + 1)(x + 3)\)
First product must be a 3 or 4 term quadratic but need not be simplified or may be simplified incorrectly
Accept \(a = 6\), \(b = 35\), \(c = 58\), \(d = 21\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{2}{5} \lt x \lt 1\dfrac{3}{5}\) | M1 | for first step of finding the square root of both sides eg \(1 - x \lt \pm\dfrac{3}{5}\) OR for writing in the form \(ax^2 + bx + c\ (\lt 0)\) eg \(x^2 - 2x + \dfrac{16}{25}\ (\lt 0)\) or \(25x^2 - 50x + 16\ (\lt 0)\) |
| M1 | for showing critical values \(\dfrac{2}{5}\ (= 0.4)\) and \(1\dfrac{3}{5}\ (= 1.6)\) oe | |
| A1 | for \(\dfrac{2}{5} \lt x \lt 1\dfrac{3}{5}\) oe |
Additional guidance
Condone use of an “=” sign; accept one square root (eg \(\tfrac{3}{5}\)) only shown.
Critical values can be stated, or shown in an expression (which may have incorrect inequality symbols)
Could be written as two separate expressions eg \(x \gt \dfrac{2}{5}\) and \(x \lt 1\dfrac{3}{5}\) oe