CAIE P2 2007 November — Question 4 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2007
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicTrigonometric equations in context
TypeConvert sin/cos ratio to tan
DifficultyStandard +0.3 This is a straightforward calculus problem requiring differentiation of tan x, solving sec²x = 2 (a standard trig equation), and finding corresponding y-values. While it involves multiple steps, each is routine for A-level: the derivative is immediate, the trig equation has a well-known solution method, and the range restriction is clearly stated. Slightly above average difficulty due to combining calculus with trig equations, but no novel insight required.
Spec1.05a Sine, cosine, tangent: definitions for all arguments1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives
4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
AnswerMarks Guidance
State derivative \(2 - \sec^2 x\), or equivalentB1
Equate derivative to zero and solve for \(x\)M1
Obtain \(x = \frac{1}{4}\pi\), or 0.785 (± 45° gains A1)A1
Obtain \(x = -\frac{1}{4}\pi\) (allow negative of first solution)A1√
Obtain corresponding \(y\)-values \(\frac{1}{2}\pi - 1\) and \(-\frac{1}{2}\pi + 1\), ± 0.571A1 [5] 
State derivative $2 - \sec^2 x$, or equivalent | B1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain $x = \frac{1}{4}\pi$, or 0.785 (± 45° gains A1) | A1 |
Obtain $x = -\frac{1}{4}\pi$ (allow negative of first solution) | A1√ |
Obtain corresponding $y$-values $\frac{1}{2}\pi - 1$ and $-\frac{1}{2}\pi + 1$, ± 0.571 | A1 | [5]
4 The equation of a curve is $y = 2 x - \tan x$, where $x$ is in radians. Find the coordinates of the stationary points of the curve for which $- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi$.

\hfill \mbox{\textit{CAIE P2 2007 Q4 [5]}}
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