| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 Part (i) is a routine differentiation using the chain rule. Part (ii) is a standard algebraic manipulation of a trigonometric identity that can be verified by working from one side or multiplying by the conjugate. Part (iii) requires recognizing that the identity from (ii) allows direct integration using the result from (i), but this connection is fairly straightforward once the identity is established. Overall, this is a well-structured multi-part question slightly easier than average due to its guided nature and use of standard techniques. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07l Derivative of ln(x): and related functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | Use quotient or chain rule | M1 |
| Obtain given answer correctly | A1 | |
| Total: | 2 |
| Answer | Marks |
|---|---|
| 7(ii) | EITHER: |
| Multiply numerator and denominator of LHS by 1+sinθ | (M1 |
| Use Pythagoras and express LHS in terms of sec θ and tanθ | M1 |
| Complete the proof | A1) |
| Answer | Marks |
|---|---|
| Express RHS in terms of cos θ and sin θ | (M1 |
| Use Pythagoras and express RHS in terms of sin θ | M1 |
| Complete the proof | A1) |
| Answer | Marks | Guidance |
|---|---|---|
| Express LHS in terms of secθ and tanθ | (M1 | |
| Multiply numerator and denominator by secθ + tanθ and use Pythagoras | M1 | |
| Complete the proof | A1) | |
| Total: | 3 | |
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 7(iii) | Use the identity and obtain integral2tanθ+2secθ−θ | B2 |
| Use correct limits correctly in an integral containing terms a tanθ and b secθ | M1 |
| Answer | Marks |
|---|---|
| 4 | A1 |
| Total: | 4 |
Question 7:
--- 7(i) ---
7(i) | Use quotient or chain rule | M1
Obtain given answer correctly | A1
Total: | 2
--- 7(ii) ---
7(ii) | EITHER:
Multiply numerator and denominator of LHS by 1+sinθ | (M1
Use Pythagoras and express LHS in terms of sec θ and tanθ | M1
Complete the proof | A1)
OR1:
Express RHS in terms of cos θ and sin θ | (M1
Use Pythagoras and express RHS in terms of sin θ | M1
Complete the proof | A1)
OR2:
Express LHS in terms of secθ and tanθ | (M1
Multiply numerator and denominator by secθ + tanθ and use Pythagoras | M1
Complete the proof | A1)
Total: | 3
Question | Answer | Marks
--- 7(iii) ---
7(iii) | Use the identity and obtain integral2tanθ+2secθ−θ | B2
Use correct limits correctly in an integral containing terms a tanθ and b secθ | M1
1
Obtain answer 2 2− π
4 | A1
Total: | 4\begin{enumerate}[label=(\roman*)]
\item Prove that if $y = \frac{1}{\cos \theta}$ then $\frac{dy}{d\theta} = \sec \theta \tan \theta$. [2]
\item Prove the identity $\frac{1 + \sin \theta}{1 - \sin \theta} = 2 \sec^2 \theta + 2 \sec \theta \tan \theta - 1$. [3]
\item Hence find the exact value of $\int_0^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} d\theta$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q7 [9]}}