Easy -1.2 This is a straightforward C1 question requiring direct application of standard power rule for differentiation and integration. Students need only rewrite 1/x³ as x⁻³ and apply memorized formulas—no problem-solving, multi-step reasoning, or conceptual insight required. Easier than average A-level question.
Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form, (a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\int y \mathrm {~d} x\).
(a) \(\frac{dy}{dx} = 10x^4 - 3x^{-1}\) or \(10x^4 - \frac{3}{x^4}\)
M1 A1 A1
M1: Attempt to differentiate \(x^n \to x^{n-1}\) (for any of the 3 terms) i.e. \(ax^3\) or \(ax^{-4}\), where \(a\) is any non-zero constant or the 7 differentiated to give 0 is sufficient evidence for M1. 1st A1: One correct (non-zero) term, possibly unsimplified. 2nd A1: Fully correct simplified answer.
M1: Attempt to integrate \(x^n \to x^{n+1}\) (i.e. \(ax^5\) or \(ax\) or \(ax^{-2}\), where \(a\) is any non-zero constant). 1st A1: Two correct terms, possibly unsimplified. 2nd A1: All three terms correct and simplified. B1: +C appearing at any stage in part (b) (independent of previous work). Allow correct equivalents to printed answer, e.g. \(\frac{x^6}{3} + 7x - \frac{1}{2x^2}\) or \(\frac{1}{3}x^6 + 7x - \frac{1}{2}x^{-2}\). Allow \(\frac{1x^6}{3}\) or \(7x^1\)
(a) $\frac{dy}{dx} = 10x^4 - 3x^{-1}$ or $10x^4 - \frac{3}{x^4}$ | M1 A1 A1 | M1: Attempt to differentiate $x^n \to x^{n-1}$ (for any of the 3 terms) i.e. $ax^3$ or $ax^{-4}$, where $a$ is any non-zero constant or the 7 differentiated to give 0 is sufficient evidence for M1. 1st A1: One correct (non-zero) term, possibly unsimplified. 2nd A1: Fully correct simplified answer.
(b) $(\int=) \frac{2x^6}{6} + 7x + \frac{x^{-2}}{-2} = \frac{x^6}{3} + 7x - \frac{x^{-2}}{2} + C$ | M1 A1 A1 B1 | M1: Attempt to integrate $x^n \to x^{n+1}$ (i.e. $ax^5$ or $ax$ or $ax^{-2}$, where $a$ is any non-zero constant). 1st A1: Two correct terms, possibly unsimplified. 2nd A1: All three terms correct and simplified. B1: +C appearing at any stage in part (b) (independent of previous work). Allow correct equivalents to printed answer, e.g. $\frac{x^6}{3} + 7x - \frac{1}{2x^2}$ or $\frac{1}{3}x^6 + 7x - \frac{1}{2}x^{-2}$. Allow $\frac{1x^6}{3}$ or $7x^1$
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