Pure Mathematics 1

A suspension bridge: the main cable hangs in a parabola.

A quadratic 二次式 is an expression of the form $ax^2 + bx + c$, where $a \neq 0$. The letters $a$, $b$, $c$ are the coefficients 系数 (the fixed numbers). Much of this section is about solving the equation $ax^2 + bx + c = 0$.

Completing the square

To complete the square 配方 means to write the quadratic in the form

This form is useful: it shows the vertex 顶点 (turning point) of the curve at $(-p,\ q)$, and it gives a quick way to solve the equation.

Worked example. Write $9x^2 - 36x + 8$ in the form $p(x + q)^2 + r$.

Take the factor $9$ out of the first two terms, then complete the square inside:

So $p = 9$, $q = -2$, $r = -28$.

The completed square hands you the vertex: least value -28 at x equals 2

The discriminant

The discriminant 判别式 of $ax^2 + bx + c$ is

It tells you how many real roots 实根 (real solutions) the equation $ax^2 + bx + c = 0$ has:

Discriminant	Roots
$b^2 - 4ac > 0$	two distinct 相异 real roots
$b^2 - 4ac = 0$	one repeated real root
$b^2 - 4ac < 0$	no real roots

The sign of $b^2-4ac$ decides how many times the parabola meets the $x$-axis.

Worked example. Find the values of the constant $k$ for which $3kx^2 + (k + 8)x + 3 = 0$ has two distinct real roots.

Here $a = 3k$, $b = k + 8$, $c = 3$. For two distinct real roots you need $b^2 - 4ac > 0$:

Factorise: $(k - 4)(k - 16) > 0$, so $k < 4$ or $k > 16$. You also need $a \neq 0$, so $k \neq 0$.

Quadratic equations and inequalities

To solve a quadratic equation 二次方程, factorise, complete the square, or use the formula

To solve a quadratic inequality 二次不等式 such as $(k - 4)(k - 16) > 0$, find the two roots, then decide which side of each root makes the statement true. A sketch of the parabola 抛物线 helps: the curve is above the $x$-axis (positive) outside the roots and below it (negative) between them.

Simultaneous equations

To solve a pair of simultaneous equations 联立方程 where one is linear and one is quadratic, use substitution 代入: rearrange the linear equation for one letter, then put that into the quadratic. This gives a single quadratic to solve.

Equations that are quadratic in disguise

Some equations are quadratic in some function of $x$. For example $x^4 - 5x^2 + 4 = 0$ is quadratic in $x^2$: let $u = x^2$, solve $u^2 - 5u + 4 = 0$, then go back to $x$. You will use this idea again in trigonometry.

A function 函数 is a rule that sends each input to exactly one output. Write it as $f(x)$. The composition of functions 复合函数 $fg(x)$ means apply $g$ first, then apply $f$ to the result.

• The domain 定义域 is the set of allowed inputs $x$.
• The range 值域 is the set of outputs the function actually produces.

A function is one-one 一一对应 if different inputs always give different outputs. (No output is repeated.) Only a one-one function has an inverse function 反函数 $f^{-1}$, which reverses the rule.

The composition 复合 of two functions means doing one after the other. $fg(x)$ means "do $g$ first, then $f$": $fg(x) = f(g(x))$. The composite function $fg$ exists only when the range of $g$ lies inside the domain of $f$.

Finding an inverse

To find $f^{-1}$: write $y = f(x)$, make $x$ the subject, then swap letters.

Worked example. The function $f(x) = (x + 3)^2 - 12$ is defined for $x \geqslant 0$. Find $f^{-1}(x)$.

Write $y = (x + 3)^2 - 12$ and solve for $x$:

You take the positive square root because $x \geqslant 0$ means $x + 3 \geqslant 3 > 0$. So

Graphs of inverses and transformations

The graph of $y = f^{-1}(x)$ is the reflection 反射 of $y = f(x)$ in the line $y = x$.

Reflecting $y=f(x)$ in the line $y=x$ gives its inverse; a point $(a,b)$ becomes $(b,a)$.

You should know these transformations 变换 of $y = f(x)$:

New equation	Effect on the graph
$y = f(x) + a$	translation 平移 up by $a$
$y = f(x + a)$	translation left by $a$
$y = a\,f(x)$	stretch 伸缩 in the $y$-direction, scale factor $a$
$y = f(ax)$	stretch in the $x$-direction, scale factor $\tfrac{1}{a}$

When two transformations are combined, the order can matter. State each one fully (type, direction, and amount).

Adding to the output or input slides the curve; a multiplier stretches it.

Coordinate geometry 坐标几何 studies lines and circles using their equations.

Straight lines

The gradient 斜率 (steepness) of the line joining $(x_1, y_1)$ and $(x_2, y_2)$ is

You can write the equation of a straight line 直线方程 in any of these forms: Two lines are parallel 平行 when their gradients are equal, and perpendicular 垂直 (at right angles) when the product of their gradients is $-1$.

Circles

The circle 圆 with centre 圆心 $(a, b)$ and radius 半径 $r$ has equation

An expanded form like $x^2 + y^2 - 6x + 10y - 27 = 0$ is the same circle: complete the square in $x$ and in $y$ to find the centre and radius.

A tangent 切线 to a circle touches it at one point and is perpendicular to the radius at that point. This right-angle fact solves most circle problems.

A tangent touches the circle once and meets the radius at a right angle.

Worked example. The points $P(1, 1)$ and $Q(7, 11)$ are the ends of a diameter 直径 of a circle. Find the equation of the circle.

The centre is the midpoint of $PQ$:

The radius is half the length of $PQ$: So the circle is $(x - 4)^2 + (y - 6)^2 = 34$.

Radians

A radian 弧度 is another way to measure angles. One radian is the angle at the centre of a circle that cuts off an arc 弧 equal in length to the radius. The link between radians and degrees 度 is

So to change degrees to radians, multiply by $\dfrac{\pi}{180}$; to change radians to degrees, multiply by $\dfrac{180}{\pi}$.

One radian: the angle whose arc equals the radius

Arc length and sector area

For a sector 扇形 with radius $r$ and angle $\theta$ in radians:

A chord 弦 cuts the sector into a triangle and a segment 弓形. The segment area is the sector minus the triangle:

The shaded segment is the part of the sector between the chord and the arc.

Worked example. A sector has centre $O$ and the angle at $O$ is $\tfrac{2}{3}\pi$ radians. Show that the segment cut off by the chord has area about $0.614 r^2$.

A Ferris wheel: a point on the rim rises and falls like a sine curve.

Graphs and exact values

You must know the shape of the graphs of the sine 正弦, cosine 余弦 and tangent function 正切 (written $\sin$, $\cos$, $\tan$). The sine and cosine graphs wave between $-1$ and $1$ and repeat every $360^\circ$ ($2\pi$). Learn these exact values:

$\theta$	$30^\circ$	$45^\circ$	$60^\circ$
$\sin\theta$	$\tfrac12$	$\tfrac{1}{\sqrt2}$	$\tfrac{\sqrt3}{2}$
$\cos\theta$	$\tfrac{\sqrt3}{2}$	$\tfrac{1}{\sqrt2}$	$\tfrac12$
$\tan\theta$	$\tfrac{1}{\sqrt3}$	$1$	$\sqrt3$

Over one turn $\sin$ and $\cos$ stay between $-1$ and $1$; $\tan$ shoots off at $90^\circ$ and $270^\circ$.

The notations $\sin^{-1}x$, $\cos^{-1}x$, $\tan^{-1}x$ mean the inverse angle (the principal value 主值).

Identities

An identity 恒等式 is true for every value of the angle. The two you must know are

Use them to rewrite an equation so it contains only one trig function.

Solving trigonometric equations

To solve a trigonometric equation 三角方程, first reduce it to one function, then find every solution in the given interval.

Worked example. Solve $6\sin\theta = 1 + \dfrac{2}{\sin\theta}$ for $-180^\circ < \theta < 180^\circ$.

Multiply through by $\sin\theta$ to clear the fraction. This makes a quadratic in $\sin\theta$:

So $\sin\theta = \tfrac23$ or $\sin\theta = -\tfrac12$.

• $\sin\theta = \tfrac23$: $\theta = 41.8^\circ$ or $\theta = 180^\circ - 41.8^\circ = 138.2^\circ$.
• $\sin\theta = -\tfrac12$: $\theta = -30^\circ$ or $\theta = -150^\circ$.

The four solutions are $\theta = -150^\circ,\ -30^\circ,\ 41.8^\circ,\ 138.2^\circ$.

The binomial expansion

For a positive integer $n$, the binomial expansion 二项展开式 is

where $\binom{n}{r} = \dfrac{n!}{r!\,(n - r)!}$ is a binomial coefficient 二项式系数.

Worked example. Find the first three terms, in ascending powers of $x$, of $(2 - px)^5$.

Arithmetic and geometric progressions

A progression 数列 (sequence) is a list of terms following a rule.

• An arithmetic progression 等差数列 (AP) adds a fixed common difference 公差 $d$ each step. The $n$th term 项 is $u_n = a + (n - 1)d$, and the sum of the first $n$ terms is $S_n = \tfrac{n}{2}\big(2a + (n - 1)d\big)$.
• A geometric progression 等比数列 (GP) multiplies by a fixed common ratio 公比 $r$ each step. The $n$th term is $u_n = ar^{\,n-1}$, and $S_n = \dfrac{a(1 - r^n)}{1 - r}$.

An AP climbs in equal steps; a GP's steps grow by the same ratio

A GP is convergent — it converges 收敛 (settles to a limit) — when $|r| < 1$. Then it has a sum to infinity 无穷和

Worked example. The third term of a GP is $18$ and the sum of the first three terms is $26$. The common ratio is negative. Find the sum to infinity.

From $ar^2 = 18$ you get $a = \dfrac{18}{r^2}$. Put this into $a(1 + r + r^2) = 26$:

The ratio is negative, so $r = -\tfrac34$ and $a = \dfrac{18}{(3/4)^2} = 32$. Then

Differentiation 微分 finds the gradient of a curve 曲线斜率 at each point. The gradient is the limit 极限 of the gradients of shorter and shorter chords, called the derivative 导数.

The rules

Write the derivative as $f'(x)$ or $\dfrac{dy}{dx}$. The basic rule is

Differentiate sums term by term, and use the chain rule 链式法则 for a function inside a function: Differentiating again gives the second derivative 二阶导数 $f''(x)$ or $\dfrac{d^2y}{dx^2}$.

Using the derivative

• Tangent and normal. The gradient of the curve at a point is the gradient of the tangent there. The normal 法线 is perpendicular to the tangent, so its gradient is $-\dfrac{1}{\text{(tangent gradient)}}$.
• Increasing or decreasing. The function is an increasing function 增函数 where $\dfrac{dy}{dx} > 0$, and a decreasing function 减函数 where $\dfrac{dy}{dx} < 0$.
• Rate of change. A derivative is a rate of change 变化率. Linked rates use the chain rule, e.g. $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$.

Stationary points

A stationary point 驻点 is where $\dfrac{dy}{dx} = 0$. Test its nature with the second derivative: $f''(x) > 0$ gives a minimum point 极小值点, and $f''(x) < 0$ gives a maximum point 极大值点.

At a maximum or a minimum the tangent is flat, so $\frac{dy}{dx}=0$.

Worked example. The curve $y = 4x^{1/2} - x$ has a maximum point at $x = a$. Find $a$.

So $a = 4$.

Integration 积分 is the reverse of differentiation. Reversing the power rule gives

The $+\,C$ is the constant of integration 积分常数. If you know one point on the curve, substitute it to find $C$.

Definite integrals and area

A definite integral 定积分 has limits and gives a number:

The area of the region 区域 between a curve and the $x$-axis, from $x = p$ to $x = q$, is $\displaystyle\int_p^q y \, dx$. For the area between two curves, integrate (top curve $-$ bottom curve).

The definite integral $\int_0^4 y\,dx$ is the shaded area under the curve.

Worked example. The curve $y = 4x^{1/2} - x$ meets the $x$-axis again at $x = 16$. Find the area between the curve and the $x$-axis from $x = 0$ to $x = 4$.

Volume of revolution

When a region is turned all the way around an axis it sweeps out a solid. The volume of revolution 旋转体体积 about the $x$-axis is

and about the $y$-axis it is $V = \pi \displaystyle\int x^2 \, dy$. For example, the region under $y = \sqrt{x}$ from $x = 0$ to $x = 4$, turned about the $x$-axis, has volume $\pi\displaystyle\int_0^4 x\, dx = \pi\big[\tfrac{x^2}{2}\big]_0^4 = 8\pi$.

Rotating the region under a curve around the $x$-axis sweeps out a solid; each thin slice is a disc of area $\pi y^2$, so $V = \pi\int y^2\,dx$

• Show every line of algebra — method marks are lost by jumping steps; use the discriminant $b^2 - 4ac$ to decide the number of real roots.
• Work in radians for circular measure (arc $= r\theta$, sector area $= \frac{1}{2}r^2\theta$) and for calculus of trig functions.
• For differentiation, set $\frac{dy}{dx} = 0$ for stationary points and use the second derivative to classify them.
• For integration, add $+ c$ to an indefinite integral and use limits for area; an area below the axis gives a negative integral.