4MA0 Higher January 2018 Paper 3R Q21
21 \(y = 4ax^2\)
\(z = a^2x^3\)
Given that \(y = 4a^kz^w\) find
(a)
(i) the value of \(w\), (2)
(ii) the value of \(k\). (2)
\(m = 10^3\)
\(N = m \times m^m\)
(b) Write \(N\) in the form \(10^n\) where \(n\) is an integer. (3)
| Scheme | Marks |
|---|---|
| (i) \(4a^k(a^2x^3)^w\;(= 4ax^2)\) | M1 |
| \(\dfrac{2}{3}\) | A1 |
| (ii) compare powers of \(a\) eg \(1 = k + \text{“}2w\text{”}\) | M1 |
| \(-\dfrac{1}{3}\) | A1 |
| (4) |
Notes
M1: for substitution
M1: forming and equation for \(k\)
ALT (a)
| Scheme | Marks |
|---|---|
| (i) \(x = \left(\dfrac{z}{a^2}\right)^{\frac{1}{3}}\) | M1 |
| \(\dfrac{2}{3}\) | A1 |
| (ii) \(y = 4a\left(\left(\dfrac{z}{a^2}\right)^{\frac{1}{3}}\right)^2\) | M1 |
| \(-\dfrac{1}{3}\) | A1 |
Notes
M1: for making \(x\) the subject
M1: (may be seen in a(i))
| Scheme | Marks |
|---|---|
| \(m = 1000\), so \(m \times m^m = 1000 \times 1000^{1000}\;(= 1000^{1001})\) | M1 |
| M1 | |
| \(10^{3003}\) | A1 |
| (3) |
Notes
M1: oe eg \((10^3)^{10^3 + 1}\)
M1: for method which is 1 step away from the correct answer eg \((10^3)^{1001}\)