June 2018 Paper 2 Q23
23 For each of these two function machines, when the input is \(a\) the output is \(b\).
\(k \gt 0 \quad\) and \(\quad k \ne 1 \quad\) and \(\quad a \gt 0\)
Work out an expression for \(a\) in terms of \(k\).
Give your answer in its simplest form. [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((a^2 + 3) \times k\) or \(ka^2 + 3k\) | M1 | oe eg \(b = ka^2 + 3k\) may be seen on diagram |
| \((ak)^2 + 3\) or \(a^2k^2 + 3\) | M1 | oe eg \(b = a^2k^2 + 3\) may be seen on diagram |
| \(ka^2 + 3k = a^2k^2 + 3\) | M1dep | oe equates and expands brackets correctly dep on M2 may include \(-b\) on each side |
| \(a^2(k - k^2) = 3 - 3k\) or \(ka^2(1 - k) = 3 - 3k\) or \(ka^2 - a^2k^2 = 3(1 - k)\) or \(a^2(k - k^2) = 3(1 - k)\) or \(ka^2(1 - k) = 3(1 - k)\) or \(a^2(k^2 - k) = 3k - 3\) or \(ka^2(k - 1) = 3k - 3\) or \(k^2a^2 - ka^2 = 3(k - 1)\) or \(a^2(k^2 - k) = 3(k - 1)\) or \(ka^2(k - 1) = 3(k - 1)\) | M1dep | oe eg \((a^2 =)\ \dfrac{3 - 3k}{k - k^2}\) or \((a =)\ (\pm)\sqrt{\dfrac{3 - 3k}{k - k^2}}\) or \((a^2 =)\ \dfrac{3k - 3}{k^2 - k}\) or \((a =)\ (\pm)\sqrt{\dfrac{3k - 3}{k^2 - k}}\) collects terms in \(a^2\) and factorises correctly on at least one side must use \(a^2\) as a factor if awarding mark for factorising \(\ ka^2 - a^2k^2\) dep on M3 |
| \((a^2 =)\ \dfrac{3(1 - k)}{k(1 - k)} \quad\) or \(\quad (a^2 =)\ \dfrac{3}{k}\) or \((a =)\ (\pm)\sqrt{\dfrac{3(1 - k)}{k(1 - k)}}\) | M1dep | oe eg \((a^2 =)\ \dfrac{3(k - 1)}{k(k - 1)}\) correct fraction with numerator and denominator factorised correctly dep on M4 |
| \((a =)\ \sqrt{\dfrac{3}{k}}\) or \((a =)\ \left(\dfrac{3}{k}\right)^{\frac{1}{2}}\) | A1 | oe eg \((a =)\ \dfrac{\sqrt{3}}{\sqrt{k}}\) or \((a =)\ \left(\dfrac{k}{3}\right)^{-\frac{1}{2}}\) \((a =) \pm\sqrt{\dfrac{3}{k}}\ \) M5A0 \(\qquad (a =) -\sqrt{\dfrac{3}{k}}\ \) M5A0 |
Additional guidance
| Only one machine fully correct | M1 only |
| Missing brackets must be recovered |