| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with given constant |
| Difficulty | Moderate -0.3 This is a straightforward AS-level integration question requiring standard techniques: integrating powers of x, applying limits, and solving a quadratic (via substitution u=√k). Part (a) is routine algebraic manipulation after integration, and part (b) involves standard quadratic solving. Slightly easier than average due to the guided 'show that' structure and use of familiar integration rules. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^n \to x^{n+1}\) | M1 | Correct indices; implied by sight of \(x^{\frac{1}{2}}\) or \(x\) |
| \(\int\left(\frac{5}{2\sqrt{x}}+3\right)dx = 5\sqrt{x}+3x\) | A1 | \(5\sqrt{x}+3x\) or \(5x^{\frac{1}{2}}+3x\); allow \(+c\); condone spurious notation |
| \(\left[5\sqrt{x}+3x\right]_1^k = 4 \Rightarrow 5\sqrt{k}+3k-8=4\) | dM1 | Uses both limits, subtracts, sets equal to \(4\) |
| \(3k+5\sqrt{k}-12=0\) | A1* | Fully correct proof with no errors leading to given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3k+5\sqrt{k}-12=0 \Rightarrow (3\sqrt{k}-4)(\sqrt{k}+3)=0\) | M1 | Correct method of solving (quadratic in \(\sqrt{k}\), factorisation, formula etc.) |
| \(\sqrt{k} = \frac{4}{3}, (-3)\) | A1 | Both values; allow \(\sqrt{k}=\pm\frac{4}{3},(\pm3)\) |
| \(\sqrt{k}=\ldots \Rightarrow k=\ldots\) | dM1 | Squares their value(s) of \(\sqrt{k}\) to find \(k\); dependent on first M |
| \(k = \frac{16}{9}\), reject \(\cancel{k}\) (negative) | A1 | \(k=\frac{16}{9}\) only; negative solution must be rejected |
## Question 7:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^n \to x^{n+1}$ | M1 | Correct indices; implied by sight of $x^{\frac{1}{2}}$ or $x$ |
| $\int\left(\frac{5}{2\sqrt{x}}+3\right)dx = 5\sqrt{x}+3x$ | A1 | $5\sqrt{x}+3x$ or $5x^{\frac{1}{2}}+3x$; allow $+c$; condone spurious notation |
| $\left[5\sqrt{x}+3x\right]_1^k = 4 \Rightarrow 5\sqrt{k}+3k-8=4$ | dM1 | Uses both limits, subtracts, sets equal to $4$ |
| $3k+5\sqrt{k}-12=0$ | A1* | Fully correct proof with no errors leading to given answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3k+5\sqrt{k}-12=0 \Rightarrow (3\sqrt{k}-4)(\sqrt{k}+3)=0$ | M1 | Correct method of solving (quadratic in $\sqrt{k}$, factorisation, formula etc.) |
| $\sqrt{k} = \frac{4}{3}, (-3)$ | A1 | Both values; allow $\sqrt{k}=\pm\frac{4}{3},(\pm3)$ |
| $\sqrt{k}=\ldots \Rightarrow k=\ldots$ | dM1 | Squares their value(s) of $\sqrt{k}$ to find $k$; dependent on first M |
| $k = \frac{16}{9}$, reject $\cancel{k}$ (negative) | A1 | $k=\frac{16}{9}$ only; negative solution must be rejected |
---\begin{enumerate}
\item Given that $k$ is a positive constant and $\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$\\
(a) show that $3 k + 5 \sqrt { k } - 12 = 0$\\
(b) Hence, using algebra, find any values of $k$ such that
\end{enumerate}
$$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$
\hfill \mbox{\textit{Edexcel AS Paper 1 2020 Q7 [8]}}