Cambridge A‑Level Mathematics 9709 – Pure Mathematics 3 (P3)
Coordinate Geometry, Conics, Parametric Equations & Related Algebra
1. Algebra (P3 3.1)
- Polynomial division
- Long division – works for any degree.
- Synthetic division – quicker when the divisor is of the form x − a.
- Factor & remainder theorems
If \(f(x)=0\) at \(x=a\) then \((x-a)\) is a factor of \(f(x)\).
The remainder on division of \(f(x)\) by \((x-a)\) is \(f(a)\).
- Partial‑fraction decomposition (required for integration of rational functions)
Any proper rational function \(\displaystyle\frac{P(x)}{Q(x)}\) can be written as a sum of simpler fractions. Three cases are required by the syllabus:
- Distinct linear factors
\[
\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}.
\]
Example: \(\displaystyle\frac{3x+5}{(x-1)(x+2)}\)
\[
\frac{3x+5}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}
\Longrightarrow 3x+5=A(x+2)+B(x-1).
\]
Solving gives \(A=2,\;B=1\); hence \(\displaystyle\frac{3x+5}{(x-1)(x+2)}=\frac{2}{x-1}+\frac{1}{x+2}\).
- Repeated linear factors
\[
\frac{P(x)}{(x-a)^{2}}=\frac{A}{x-a}+\frac{B}{(x-a)^{2}}.
\]
Example: \(\displaystyle\frac{4x-1}{(x-3)^{2}}\)
\[
\frac{4x-1}{(x-3)^{2}}=\frac{A}{x-3}+\frac{B}{(x-3)^{2}}
\Longrightarrow 4x-1=A(x-3)+B.
\]
Comparing coefficients gives \(A=4,\;B=11\); thus \(\displaystyle\frac{4x-1}{(x-3)^{2}}=\frac{4}{x-3}+\frac{11}{(x-3)^{2}}\).
- Irreducible quadratic factors
\[
\frac{P(x)}{x^{2}+px+q}= \frac{Ax+B}{x^{2}+px+q}.
\]
Example: \(\displaystyle\frac{2x^{2}+3x+5}{x^{2}+4}\)
\[
\frac{2x^{2}+3x+5}{x^{2}+4}= \frac{Ax+B}{x^{2}+4}.
\]
Multiply through: \(2x^{2}+3x+5=Ax+B\).
Equating coefficients gives \(A=2,\;B=3\); hence \(\displaystyle\frac{2x^{2}+3x+5}{x^{2}+4}=2+\frac{3}{x^{2}+4}\).
- Binomial expansion with rational exponents (valid for \(|x|<1\))
\[
(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+\cdots
\]
Example (non‑integer exponent): \((1+x)^{3/2}\)
\[
(1+x)^{3/2}=1+\frac{3}{2}x-\frac{3}{8}x^{2}+\frac{1}{16}x^{3}+\cdots\qquad(|x|<1).
\]
Example 1 – Polynomial division (synthetic)
- Divide \(f(x)=2x^{3}-3x^{2}+5x-7\) by \(g(x)=x-2\).
- Synthetic division:
2 | 2 -3 5 -7
4 2 14
----------------
2 1 7 7
Quotient \(=2x^{2}+x+7\), remainder \(=7\).
\[
\frac{2x^{3}-3x^{2}+5x-7}{x-2}=2x^{2}+x+7+\frac{7}{x-2}.
\]
2. Logarithmic & Exponential Functions (P3 3.2)
- Law of logarithms (no change‑of‑base required)
\[
\log(ab)=\log a+\log b,\qquad
\log\frac{a}{b}=\log a-\log b,\qquad
\log(a^{n})=n\log a.
\]
Note: The Cambridge exam never asks for a change‑of‑base; you should work directly with the given base.
- Solving exponential equations – take logarithms of both sides.
Example: \(\displaystyle5^{2x-1}=3^{x+2}\)
\[
(2x-1)\log5=(x+2)\log3\;\Longrightarrow\;x=\frac{2\log5-\log3}{2\log5-\log3}.
\] (solve algebraically after expanding).
- Linearisation of power laws – converting \(y=kx^{n}\) to a straight line on a log–log plot:
\[
\log y = \log k + n\log x.
\]
Gradient = \(n\), intercept = \(\log k\).
Example 2 – Linearising data
Given \(y=4x^{1.5}\), take logs: \(\log y = \log4 + 1.5\log x\).
Plotting \(\log y\) against \(\log x\) yields a straight line of gradient \(1.5\) and intercept \(\log4\).
3. Trigonometry (P3 3.3)
- Fundamental identities: \(\sin^{2}\theta+\cos^{2}\theta=1\), \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
- Sum & difference formulae:
\[
\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta,\qquad
\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta.
\]
- Double‑, half‑ and triple‑angle formulae (as required for exam questions).
- R‑formula – expressing \(a\sin\theta+b\cos\theta\) as a single sinusoid.
Derivation:
- Assume \(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)\).
- Expand RHS: \(R\sin\theta\cos\alpha+R\cos\theta\sin\alpha\).
- Match coefficients:
\[
a=R\cos\alpha,\qquad b=R\sin\alpha.
\]
Hence \(R=\sqrt{a^{2}+b^{2}}\) and \(\displaystyle\tan\alpha=\frac{b}{a}\) (choose \(\alpha\) in the correct quadrant).
Common pitfalls
- If \(a<0\) the angle \(\alpha\) lies in the second or third quadrant; adjust \(\alpha\) by adding \(\pi\) to keep \(\tan\alpha=b/a\) correct.
- Remember that \(R\ge0\); the sign is absorbed into \(\alpha\).
Example 3 – Solving a trig equation with the R‑formula
Solve \(2\sin\theta-\sqrt3\cos\theta=1\) for \(0\le\theta<2\pi\).
- Write \(2\sin\theta-\sqrt3\cos\theta=R\sin(\theta+\alpha)\).
\(R=\sqrt{2^{2}+(\!-\sqrt3)^{2}}=\sqrt7\),
\(\tan\alpha=\dfrac{-\sqrt3}{2}\) ⇒ \(\alpha\approx-0.713\) rad (or \(5.570\) rad).
- Equation becomes \(\sqrt7\,\sin(\theta+\alpha)=1\) ⇒ \(\sin(\theta+\alpha)=\dfrac1{\sqrt7}\).
- Solutions for \(\theta+\alpha\):
\(\theta+\alpha =\sin^{-1}\!\left(\frac1{\sqrt7}\right)\) or \(\pi-\sin^{-1}\!\left(\frac1{\sqrt7}\right)\).
Subtract \(\alpha\) to obtain the two values of \(\theta\) in \([0,2\pi)\).
4. Differentiation (P3 3.4)
| Function | Derivative |
|---|
| \(e^{ax}\) | \(ae^{ax}\) |
| \(\ln x\) | \(\dfrac1x\) |
| \(\tan^{-1}x\) | \(\dfrac1{1+x^{2}}\) |
| \(u\,v\) (product) | \(u'v+uv'\) |
| \(\dfrac{u}{v}\) (quotient) | \(\dfrac{u'v-uv'}{v^{2}}\) |
- Parametric differentiation – if \(x=x(t),\;y=y(t)\) and \(\dfrac{dx}{dt}eq0\), then
\[
\frac{dy}{dx}= \frac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}.
\]
- Implicit differentiation – differentiate both sides of an equation involving \(x\) and \(y\) and solve for \(\dfrac{dy}{dx}\).
Example 4 – Tangent to a parametric curve
Given \(x=t^{2},\;y=t^{3}\), find \(\dfrac{dy}{dx}\) at \(t=2\).
- \(\displaystyle\frac{dx}{dt}=2t,\;\frac{dy}{dt}=3t^{2}\).
- \(\displaystyle\frac{dy}{dx}= \frac{3t^{2}}{2t}= \frac{3t}{2}\).
- At \(t=2\), \(\displaystyle\frac{dy}{dx}=3.\)
5. Integration (P3 3.5)
| Integrand | Integral + C |
|---|
| \(e^{ax}\) | \(\dfrac{1}{a}e^{ax}\) |
| \(\sin(ax)\) | \(-\dfrac{1}{a}\cos(ax)\) |
| \(\cos(ax)\) | \(\dfrac{1}{a}\sin(ax)\) |
| \(\sec^{2}(ax)\) | \(\dfrac{1}{a}\tan(ax)\) |
| \(\dfrac{1}{x}\) | \(\ln|x|\) |
| \(\dfrac{1}{x^{2}+a^{2}}\) | \(\dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right)\) |
- Integration by parts – \(\displaystyle\int u\,dv = uv-\int v\,du\).
- Trigonometric substitution – used for integrands containing \(\sqrt{a^{2}\!-\!x^{2}},\;\sqrt{a^{2}\!+\!x^{2}},\;\sqrt{x^{2}\!-\!a^{2}}\).
- Partial‑fraction integration – decompose first (see Section 1) and then integrate each term using the table above.
Example 5 – \(\displaystyle\int\frac{x}{x^{2}+4}\,dx\)
Let \(u=x^{2}+4\), then \(du=2x\,dx\) ⇒ \(\dfrac{x\,dx}{x^{2}+4}=\dfrac12\frac{du}{u}\). Hence
\[
\int\frac{x}{x^{2}+4}\,dx = \frac12\ln|x^{2}+4|+C.
\]
6. Numerical Solution of Equations (P3 3.6)
- Graphical method – plot \(y=f(x)\) and read off the x‑intercepts.
- Fixed‑point iteration – rewrite \(f(x)=0\) as \(x=g(x)\) and iterate
\[
x_{n+1}=g(x_{n})\quad\text{until }|x_{n+1}-x_{n}|<\text{required tolerance}.
\]
- Newton–Raphson method – rapid convergence when the initial guess is close:
\[
x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}.
\]
Example 6 – Newton–Raphson for \(x^{3}-x-2=0\)
- \(f(x)=x^{3}-x-2,\;f'(x)=3x^{2}-1.\)
- Start with \(x_{0}=1.5\):
\[
x_{1}=1.5-\frac{1.5^{3}-1.5-2}{3(1.5)^{2}-1}=1.5-\frac{-0.125}{5.75}=1.5217.
\]
- Repeating gives \(x\approx1.52138\) (root to 5 dp).
7. Vectors (P3 3.7)
- Notation – \(\mathbf{a}= \langle a_{1},a_{2}\rangle\) (2‑D) or \(\langle a_{1},a_{2},a_{3}\rangle\) (3‑D).
- Operations
- Addition: \(\mathbf{a}+\mathbf{b}= \langle a_{1}+b_{1},\,a_{2}+b_{2}\rangle\).
- Scalar multiplication: \(k\mathbf{a}= \langle ka_{1},ka_{2}\rangle\).
- Dot product: \(\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}\); magnitude \(|\mathbf{a}|=\sqrt{\mathbf{a}\cdot\mathbf{a}}\).
- Equation of a line (vector form)
\[
\mathbf{r}= \mathbf{r}_{0}+t\mathbf{d},
\]
where \(\mathbf{r}_{0}\) is a known point and \(\mathbf{d}\) a direction vector.
- Parallel & perpendicular conditions
- Parallel: \(\mathbf{d}_{1}=k\mathbf{d}_{2}\) (or slopes equal).
- Perpendicular: \(\mathbf{d}_{1}\cdot\mathbf{d}_{2}=0\).
- Angle between two vectors
\[
\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}.
\]
Example 7 – Distance from a point to a line (vector method)
Line through \(A(1,2)\) with direction \(\mathbf{d}= \langle 3,-4\rangle\); point \(P(5,0)\).
- \(\mathbf{AP}= \langle4,-2\rangle\).
- Perpendicular distance:
\[
d=\frac{|\mathbf{d}\times\mathbf{AP}|}{|\mathbf{d}|}
=\frac{|3(-2)-(-4)4|}{\sqrt{3^{2}+(-4)^{2}}}
=\frac{2}{5}=0.4.
\]
8. Coordinate Geometry – Straight Lines (P3 4.1)
- Gradient (slope) form – from two points \((x_{1},y_{1}), (x_{2},y_{2})\):
\[
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}},\qquad
y-y_{0}=m(x-x_{0}).
\]
- Intercept form – line cutting the axes at \((a,0)\) and \((0,b)\):
\[
\frac{x}{a}+\frac{y}{b}=1.
\]
- General (standard) form – \(Ax+By+C=0\).
Gradient \(m=-\dfrac{A}{B}\) (if \(Beq0\)).
- Distance from a point \((x_{0},y_{0})\) to the line
\[
d=\frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}.
\]
- Angle between two lines with slopes \(m_{1},m_{2}\):
\[
\tan\theta=\left|\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}\right|.
\]
If either line is vertical, use the direction‑vector form \(\tan\theta=|(\mathbf{d}_{1}\times\mathbf{d}_{2})|/(\mathbf{d}_{1}\cdot\mathbf{d}_{2})\).
Example 8 – Finding the equation of a line
Through points \(P(2,‑1)\) and \(Q(5,3)\).
- Gradient: \(m=\dfrac{3-(-1)}{5-2}= \dfrac{4}{3}\).
- Using point‑slope with \(P\): \(y+1=\dfrac{4}{3}(x-2)\) ⇒ \(3y+3=4x-8\) ⇒ \(4x-3y-11=0.\)
9. Coordinate Geometry – Conics (P3 4.2 & 4.3)
- Circle – centre \((h,k)\), radius \(r\): \((x-h)^{2}+(y-k)^{2}=r^{2}\).
- Parabola – standard forms
\[
y^{2}=4ax\;( \text{opens right}),\qquad
x^{2}=4ay\;( \text{opens up}).
\]
General form \(Ax^{2}+By^{2}+2Gx+2Fy+C=0\) with exactly one of \(A,B\) zero.
- Ellipse – \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) (a ≥ b). Foci at \((\pm c,0)\) where \(c^{2}=a^{2}-b^{2}\).
- Hyperbola – \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) (opens left‑right) or \(\dfrac{y^{2}}{b^{2}}-\dfrac{x^{2}}{a^{2}}=1\) (opens up‑down). Asymptotes \(y=\pm\frac{b}{a}x\).
- Key properties required for the exam: centre, vertices, foci, directrices, eccentricity, latus‑rectum, and the method of completing the square to put a quadratic equation into standard form.
Example 9 – Putting a conic into standard form
Given \(9x^{2}+16y^{2}-54x+64y+71=0\).
- Group x‑terms and y‑terms: \((9x^{2}-54x)+(16y^{2}+64y)=-71\).
- Factor coefficients: \(9(x^{2}-6x)+16(y^{2}+4y)=-71\).
- Complete squares:
\(x^{2}-6x+9=(x-3)^{2}\) (add \(9\) inside),
\(y^{2}+4y+4=(y+2)^{2}\) (add \(4\) inside).
Adjust RHS: \(-71+9\cdot9+16\cdot4=-71+81+64=74.\)
- Result: \(\displaystyle\frac{(x-3)^{2}}{ \frac{74}{9}}+\frac{(y+2)^{2}}{ \frac{74}{16}}=1\) ⇒ ellipse with centre \((3,-2)\).
10. Parametric Equations (P3 4.4)
- General form: \(x=x(t),\;y=y(t)\) where \(t\) is the parameter.
- Eliminating the parameter – solve one equation for \(t\) (or a function of \(t\)) and substitute into the other to obtain a Cartesian equation.
- Differentiation – slope of the curve: \(\displaystyle\frac{dy}{dx}= \frac{dy/dt}{dx/dt}\).
Second derivative: \(\displaystyle\frac{d^{2}y}{dx^{2}}=\frac{d}{dt}\!\left(\frac{dy}{dx}\right)\Big/ \frac{dx}{dt}\).
- Arc length – for \(t\in[a,b]\):
\[
s=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\;dt.
\]
- Area under a parametric curve – between \(t_{1}\) and \(t_{2}\):
\[
A=\int_{t_{1}}^{t_{2}} y\,\frac{dx}{dt}\;dt.
\]
Example 10 – Eliminating the parameter
Given \(x=2\cos t,\;y=3\sin t\).
- Square both: \(\displaystyle\frac{x^{2}}{4}+\frac{y^{2}}{9}= \cos^{2}t+\sin^{2}t=1.\)
- Cartesian equation: \(\displaystyle\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) – an ellipse.
Example 11 – Tangent to a parametric curve
Curve: \(x=1+t^{2},\;y=2t\). Find the equation of the tangent at \(t=1\).
- \(\displaystyle\frac{dx}{dt}=2t,\;\frac{dy}{dt}=2.\)
- Slope at \(t=1\): \(\displaystyle m=\frac{dy/dt}{dx/dt}\Big|_{t=1}= \frac{2}{2}=1.\)
- Point on the curve: \((x,y)=(1+1^{2},\,2\cdot1)=(2,2).\)
- Using point‑slope: \(y-2=1(x-2)\) ⇒ \(y=x.\)
11. Summary of Key Formulae
| Topic | Key formulae |
|---|
| Gradient form | \(y-y_{0}=m(x-x_{0}),\; m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\) |
| Intercept form | \(\dfrac{x}{a}+\dfrac{y}{b}=1\) |
| Distance point‑line | \(d=\dfrac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}\) |
| Angle between lines | \(\tan\theta=\Bigl|\dfrac{m_{2}-m_{1}}{1+m_{1}m_{2}}\Bigr|\) |
| R‑formula | \(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha),\;R=\sqrt{a^{2}+b^{2}},\;\tan\alpha=\dfrac{b}{a}\) |
| Parametric slope | \(\displaystyle\frac{dy}{dx}= \frac{dy/dt}{dx/dt}\) |
| Arc length (parametric) | \(\displaystyle s=\int\sqrt{\bigl(dx/dt\bigr)^{2}+\bigl(dy/dt\bigr)^{2}}\,dt\) |
| Area (parametric) | \(\displaystyle A=\int y\,\frac{dx}{dt}\,dt\) |
12. Quick Revision Checklist
- Can you perform synthetic division and state the remainder theorem?
- Are you comfortable decomposing any proper rational function into partial fractions (three cases)?
- Do you remember the three logarithm laws and that no change‑of‑base is needed?
- Can you derive and apply the R‑formula, paying attention to the quadrant of \(\alpha\)?
- Are you fluent with parametric differentiation, second derivative, arc length and area formulas?
- Can you quickly write the distance‑from‑point‑to‑line formula and the angle‑between‑lines formula?
- Do you know how to complete the square to convert a general quadratic into the standard form of a conic?