Question 3 - A-Level Maths

CAIE P3 2015 November — Question 3 6 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2015
Session	November
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Product & Quotient Rules
Type	Find equation of tangent
Difficulty	Standard +0.3 This is a straightforward application of the quotient rule followed by finding a tangent line equation. While it involves trigonometric functions and requires careful algebraic manipulation, it's a standard textbook exercise with clear steps: differentiate using quotient rule, evaluate at the given point, then use point-slope form. The main challenge is arithmetic accuracy rather than conceptual difficulty.
Spec	1.07m Tangents and normals: gradient and equations 1.07q Product and quotient rules: differentiation

3 A curve has equation $$y = \frac { 2 - \tan x } { 1 + \tan x }$$ Find the equation of the tangent to the curve at the point for which $x = \frac { 1 } { 4 } \pi$, giving the answer in the form $y = m x + c$ where $c$ is correct to 3 significant figures.

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Answer	Marks	Guidance
Use correct quotient rule or equivalent to find first derivative	M1*
Obtain $\frac{-(1 + \tan x)\sec^2 x - \sec^2 x(2 - \tan x)}{(1 + \tan x)^2}$ or equivalent	A1
Substitute $x = \frac{1}{4}\pi$ to find gradient	dep M1*
Obtain $-\frac{3}{2}$	A1
Form equation of tangent at $x = \frac{1}{4}\pi$	M1
Obtain $y = -\frac{3}{2}x + 1.68$ or equivalent	A1	[6]

Use correct quotient rule or equivalent to find first derivative | M1* |
Obtain $\frac{-(1 + \tan x)\sec^2 x - \sec^2 x(2 - \tan x)}{(1 + \tan x)^2}$ or equivalent | A1 |
Substitute $x = \frac{1}{4}\pi$ to find gradient | dep M1* |
Obtain $-\frac{3}{2}$ | A1 |
Form equation of tangent at $x = \frac{1}{4}\pi$ | M1 |
Obtain $y = -\frac{3}{2}x + 1.68$ or equivalent | A1 | [6]

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3 A curve has equation

$$y = \frac { 2 - \tan x } { 1 + \tan x }$$

Find the equation of the tangent to the curve at the point for which $x = \frac { 1 } { 4 } \pi$, giving the answer in the form $y = m x + c$ where $c$ is correct to 3 significant figures.

\hfill \mbox{\textit{CAIE P3 2015 Q3 [6]}}

This paper (10 questions)

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Q1 2 Q2 3 Q3 6 Q4 8 Q5 8 Q6 8 Q7 10 Q8 10 Q9 10 Q10 10

Answer	Marks	Guidance
Use correct quotient rule or equivalent to find first derivative	M1*
Obtain \(\frac{-(1 + \tan x)\sec^2 x - \sec^2 x(2 - \tan x)}{(1 + \tan x)^2}\) or equivalent	A1
Substitute \(x = \frac{1}{4}\pi\) to find gradient	dep M1*
Obtain \(-\frac{3}{2}\)	A1
Form equation of tangent at \(x = \frac{1}{4}\pi\)	M1
Obtain \(y = -\frac{3}{2}x + 1.68\) or equivalent	A1	[6]