| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard techniques: trapezium rule application (routine calculation), understanding concavity to determine over/underestimate (conceptual but basic), and differentiation of sec x followed by solving an equation. All parts are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply ordinates \(1, 1.2116\ldots, 2.7597\ldots\) | B1 | |
| Use correct formula, or equivalent, with \(h = 0.6\) | M1 | |
| Obtain answer 1.85 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Explain why the rule gives an overestimate | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate using quotient or chain rule | M1 | |
| Obtain correct derivative in terms of \(\sin x\) and \(\cos x\) | A1 | |
| Equate derivative to 2, use Pythagoras and obtain an equation in \(\sin x\) | M1 | |
| Obtain \(2\sin^2 x + \sin x - 2 = 0\) | A1 | OE |
| Solve a 3-term quadratic for \(x\) | M1 | |
| Obtain answer \(x = 0.896\) only | A1 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply ordinates $1, 1.2116\ldots, 2.7597\ldots$ | B1 | |
| Use correct formula, or equivalent, with $h = 0.6$ | M1 | |
| Obtain answer 1.85 | A1 | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Explain why the rule gives an overestimate | B1 | |
## Question 8(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate using quotient or chain rule | M1 | |
| Obtain correct derivative in terms of $\sin x$ and $\cos x$ | A1 | |
| Equate derivative to 2, use Pythagoras and obtain an equation in $\sin x$ | M1 | |
| Obtain $2\sin^2 x + \sin x - 2 = 0$ | A1 | OE |
| Solve a 3-term quadratic for $x$ | M1 | |
| Obtain answer $x = 0.896$ only | A1 | |8\\
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The diagram shows the graph of $y = \sec x$ for $0 \leqslant x < \frac { 1 } { 2 } \pi$.\\
(i) Use the trapezium rule with 2 intervals to estimate the value of $\int _ { 0 } ^ { 1.2 } \sec x \mathrm {~d} x$, giving your answer correct to 2 decimal places.\\
(ii) Explain, with reference to the diagram, whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).\\
(iii) $P$ is the point on the part of the curve $y = \sec x$ for $0 \leqslant x < \frac { 1 } { 2 } \pi$ at which the gradient is 2 . By first differentiating $\frac { 1 } { \cos x }$, find the $x$-coordinate of $P$, giving your answer correct to 3 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q8 [10]}}