| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Sketch reciprocal function graphs |
| Difficulty | Moderate -0.3 This is a straightforward C3 question testing standard knowledge of reciprocal trig functions. Part (i) is routine graph sketching from memory, part (ii) is direct calculator work (cos x = 1/3), and part (iii) requires the standard identity manipulation sec θ = 5 cosec θ → sin θ = 5 cos θ → tan θ = 5, then calculator work. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show correct general shape (alternating above and below x-axis) | M1 | with no branch reaching x-axis |
| Draw (more or less correct) sketch | A1 | 2 with at least one of 1 and –1 indicated or clearly implied |
| (ii) Attempt solution of \(\cos x = \frac{1}{4}\) | M1 | maybe implied; or equiv |
| Obtain \(1.23\) or \(0.392\pi\) | A1 | or greater accuracy |
| Obtain \(5.05\) or \(1.61\pi\) | A1 | 3 or greater accuracy and no others within \(0 \le x \le 2\pi\); penalise answer(s) to 2sf only once |
| (iii) Either: Obtain equation of form \(\tan \theta = k\) | M1 | any constant \(k\); maybe implied |
| Obtain \(\tan \theta = 5\) | A1 | |
| Obtain two values only of form \(\theta, \theta + \pi\) | M1 | within \(0 \le x \le 2\pi\); allow degrees at this stage |
| Obtain \(1.37\) and \(4.51\) (or \(0.437\pi\) and \(1.44\pi\)) | A1 | 4 allow ±1 in third sig fig; or greater accuracy |
| Or: (for methods which involve squaring etc.) | ||
| Attempt to obtain eqn in one trig ratio | M1 | |
| Obtain correct value | A1 | \(\tan^2 \theta = 25, \cos^2 \theta = \frac{1}{26}, \ldots\) |
| Attempt solution at least to find one value in first quadrant and one value in third | M1 | |
| Obtain \(1.37\) and \(4.51\) (or equivs as above) | A1 | ignoring values in second and fourth quadrants |
(i) Show correct general shape (alternating above and below x-axis) | M1 | with no branch reaching x-axis
Draw (more or less correct) sketch | A1 | 2 with at least one of 1 and –1 indicated or clearly implied
(ii) Attempt solution of $\cos x = \frac{1}{4}$ | M1 | maybe implied; or equiv
Obtain $1.23$ or $0.392\pi$ | A1 | or greater accuracy
Obtain $5.05$ or $1.61\pi$ | A1 | 3 or greater accuracy and no others within $0 \le x \le 2\pi$; penalise answer(s) to 2sf only once
(iii) Either: Obtain equation of form $\tan \theta = k$ | M1 | any constant $k$; maybe implied
Obtain $\tan \theta = 5$ | A1 |
Obtain two values only of form $\theta, \theta + \pi$ | M1 | within $0 \le x \le 2\pi$; allow degrees at this stage
Obtain $1.37$ and $4.51$ (or $0.437\pi$ and $1.44\pi$) | A1 | 4 allow ±1 in third sig fig; or greater accuracy
Or: (for methods which involve squaring etc.) | |
Attempt to obtain eqn in one trig ratio | M1 |
Obtain correct value | A1 | $\tan^2 \theta = 25, \cos^2 \theta = \frac{1}{26}, \ldots$
Attempt solution at least to find one value in first quadrant and one value in third | M1 |
Obtain $1.37$ and $4.51$ (or equivs as above) | A1 | ignoring values in second and fourth quadrants
---7 (i) Sketch the graph of $y = \sec x$ for $0 \leqslant x \leqslant 2 \pi$.\\
(ii) Solve the equation $\sec x = 3$ for $0 \leqslant x \leqslant 2 \pi$, giving the roots correct to 3 significant figures.\\
(iii) Solve the equation $\sec \theta = 5 \operatorname { cosec } \theta$ for $0 \leqslant \theta \leqslant 2 \pi$, giving the roots correct to 3 significant figures.
\hfill \mbox{\textit{OCR C3 2007 Q7 [9]}}