| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to approximate integral |
| Difficulty | Standard +0.3 This is a straightforward application of standard Maclaurin series techniques. Part (a) requires substituting x/3 into the standard cos x series and using cos²θ = (1+cos2θ)/2, which is routine for CP1. Parts (b)-(d) involve basic integration of the polynomial approximation and calculator comparison—mechanical steps with no conceptual challenges. Slightly above average only due to the multi-part nature and requiring careful algebraic manipulation. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks |
|---|---|
| 2 (a) | ( x ) 2 ( x ) 4 2 x 2 x 4 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 3 2 2 3 4 ! 3 | M1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| 9 243 | A1 | 1.1b |
| Answer | Marks |
|---|---|
| (b) | x 2 1 |
| Answer | Marks | Guidance |
|---|---|---|
| where A, B and C 0 | M1 | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| 18 972 | A1ft | 1.1b |
| = awrt 0.98295 | A1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | Calculator = awrt 0.98280 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | E.g. the approximation is correct to 3 d.p. | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Scheme | Marks |
Question 2:
--- 2 (a) ---
2 (a) | ( x ) 2 ( x ) 4 2 x 2 x 4 2
c o s 2 x = 1 − 32 + 34 − ... = ... or 1 − + − ... = ... or
1 8 1 9 4 4
3 2
2 4
1 2 x 1 1 2 x 1 2 x
1 c o s = 1 1 − + − ...
2 3 2 2 3 4 ! 3 | M1 | 2.2a
x2 1
=1− + x4
9 243 | A1 | 1.1b
(2)
(b) | x 2 1
1 − + x 4
9 2 4 3 = 1 − x + 1 x 3 = A l n x + B x 2 + C x 4
x x 9 2 4 3
where A, B and C 0 | M1 | 3.1a
x2 1
lnx− + x4
18 972 | A1ft | 1.1b
= awrt 0.98295 | A1 | 2.2a
(3)
(c) | Calculator = awrt 0.98280 | B1 | 1.1b
(1)
(d) | E.g. the approximation is correct to 3 d.p. | B1 | 3.2b
(1)
(7 marks)
Notes:
(a)
x
M1: Deduces the required series by using the Maclaurin series for cos x, replacing x with and
3
squares, or first applying the double angle identity (allow sign error) and then applying the series for
2 x
cos x with . Attempts at finding from differentiation score M0 as the cosine series is required.
3
A1: Correct series
(b)
M1: Divides their series in part (a) by x and integrates to the form A l n x + B x 2 + C x 4
A1ft: Correct integration, follow through on their coefficients and need not be simplified.
A1: Deduces the definite integral awrt 0.98295
(c)
B1: Correct value.
(d)
B1: Makes a quantitative statement about the accuracy, so e.g. how many decimal places or
significant figures it is correct to, or calculates a percentage accuracy to deduce it is reasonable. Do
not accept just “underestimate” or similar without quantitative evidence. Allow for a reasonable
comment as long as (b) is correct to at least 2 s.f. but (c) must be the correct value.
Question | Scheme | Marks | AOs\begin{enumerate}[label=(\alph*)]
\item Use the Maclaurin series expansion for $\cos x$ to determine the series expansion of $\cos^2\left(\frac{x}{3}\right)$ in ascending powers of $x$, up to and including the term in $x^4$
Give each term in simplest form.
[2]
\item Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for
$$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$
[3]
\item Use the integration function on your calculator to evaluate
$$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$
Give your answer to 5 decimal places.
[1]
\item Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
[1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP1 2021 Q2 [7]}}