4MA0 Higher January 2017 Paper 4 Q16
16 \(g = 2^3 \times 3 \times 7^2 \qquad\qquad h = 2 \times 3 \times 7^3\)
(a) Express \(gh\) as a product of powers of its prime factors.
Simplify your answer. (2)
Simplify your answer. (2)
\(\dfrac{g}{h} = 2^a \times 3^b \times 7^c\)
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\). (2)
(c) Show that \(\;\left(7 - 2\sqrt{5}\right)\left(7 + 2\sqrt{5}\right) = 29\)
Show your working clearly. (2)
Show your working clearly. (2)
\(\dfrac{1}{\sqrt[3]{9^4}} = 3^n\)
(d) Work out the exact value of \(n\). (3)
| Scheme | Marks |
|---|---|
| \(2^{3+1} \times 3^{1+1} \times 7^{2+3}\) | M1 |
| \(2^4 \times 3^2 \times 7^5\) | A1 |
| (2) |
Notes
M1: or for a product of powers of 2, 3 and 7 with two powers correct,
or for an attempt to find prime factors of 2420208 (allow one arithmetical error) or
for \(2^4\), \(3^2\), \(7^5\)
| Scheme | Marks |
|---|---|
| \(2^{3-1} \times 3^{1-1} \times 7^{2-3}\) | M1 |
| 2, 0, −1 | A1 |
| (2) |
Notes
M1: or for any two correct.
A1: Accept \(2^2 \times 3^0 \times 7^{-1}\)
| Scheme | Marks |
|---|---|
| Eg \(7^2 - (2\sqrt{5})^2\) or \(7^2 - 14\sqrt{5} + 14\sqrt{5} - (2\sqrt{5})^2\) | M1 |
| Show that | A1 |
| (2) |
Notes
M1: For a correct unsimplified exact expansion
\(7^2\) may be simplified to 49 and \((2\sqrt{5})^2\) as far as 20
A1: Correct solution (simplified correctly) dep on M1
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{9^{\frac{4}{3}}}\) or \(9^{-\frac{4}{3}}\) or \(\dfrac{1}{\sqrt[3]{(3^2)^4}}\) or \(\dfrac{1}{\sqrt[3]{3^8}}\) oe | M1 |
| \((3^2)^{-\frac{4}{3}}\) or \(3^{-\frac{8}{3}}\) or \(\dfrac{1}{3^{\frac{8}{3}}}\) | M1 |
| \(-\dfrac{8}{3}\) | A1 |
| (3) | |
| (9 marks) |
Notes
M1: Or for \(9^4 = 3^8\)
A1: oe
Eg \(-2\dfrac{2}{3}\) or \(-2.\dot{6}\) but not a decimal approximation.