Cambridge Pre-U Mathematics (Principal Subject) Syllabus

For arc length and sector area with angle $\theta$ in radians and radius $r$, give the formulas.

Arc length $s=r\theta$; sector area $A=\tfrac{1}{2}r^{2}\theta$.

What is the derivative of $x^{n}$ with respect to $x$?

$$\frac{d}{dx}\,x^{n}=nx^{n-1}$$

State the derivatives of $e^{x}$, $\ln x$, $\sin x$ and $\cos x$.

$\frac{d}{dx}e^{x}=e^{x}$; $\frac{d}{dx}\ln x=\frac{1}{x}$; $\frac{d}{dx}\sin x=\cos x$; $\frac{d}{dx}\cos x=-\sin x$.

How do you classify a stationary point of $y=f(x)$ using the second derivative?

At a stationary point where $f'(x)=0$: if $f''(x)>0$ it is a local minimum; if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive (examine further).

State the power rule for indefinite integration of $x^{n}$.

$$\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq -1)$$

What does the definite integral $\int_a^b f(x)\,dx$ represent geometrically, and how is it evaluated?

It gives the signed area between $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. Evaluated as $[F(x)]_a^b=F(b)-F(a)$ where $F'=f$.

State the chain rule for differentiating a composite function $y=f(g(x))$.

$$\frac{dy}{dx}=f'(g(x))\cdot g'(x),\quad\text{or}\quad \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

State the product rule and the quotient rule for differentiation.

Product: $\frac{d}{dx}(uv)=u'v+uv'$. Quotient: $\frac{d}{dx}\!\left(\frac{u}{v}\right)=\dfrac{u'v-uv'}{v^{2}}$.

What is the integration by parts formula?

$$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$$

What is the value of $\int \frac{1}{x}\,dx$?

$$\int\frac{1}{x}\,dx=\ln|x|+C$$

How do you split $\frac{px+q}{(x-a)(x-b)}$ into partial fractions?

Write it as $\dfrac{A}{x-a}+\dfrac{B}{x-b}$, then find $A$ and $B$ by equating numerators (or substituting $x=a$ and $x=b$).

What is the general method for solving a first-order separable differential equation $\frac{dy}{dx}=f(x)g(y)$?

Separate variables: $\int\frac{1}{g(y)}\,dy=\int f(x)\,dx$, integrate both sides, then add a constant and apply boundary conditions.

Write the general solution of the differential equation $\frac{dy}{dx}=ky$ (exponential growth/decay).

$$y=Ae^{kx}$$ where $A$ is a constant determined by initial conditions.

State the trapezium rule for approximating $\int_a^b y\,dx$ with $n$ strips of width $h$.

$$\int_a^b y\,dx\approx \frac{h}{2}\big(y_0+y_n+2(y_1+y_2+\dots+y_{n-1})\big),\quad h=\frac{b-a}{n}$$

Describe the Newton-Raphson iterative method for solving $f(x)=0$.

Iterate $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ from an initial estimate $x_0$, converging to a root.

How does a fixed-point iteration of the form $x_{n+1}=g(x_n)$ converge to a root?

Rearrange $f(x)=0$ into $x=g(x)$. The iteration converges to a root near $x=\alpha$ if $|g'(\alpha)|<1$.

Compare the mean, median and mode as measures of central tendency, noting which is robust to outliers.

Mean = arithmetic average (sensitive to outliers); median = middle value when ordered (robust to outliers); mode = most frequent value (usable for categorical data).

Give the formula for the variance and standard deviation of a data set of $n$ values.

Variance $\sigma^{2}=\dfrac{\sum x^{2}}{n}-\bar{x}^{2}$ (mean of squares minus square of mean); standard deviation $\sigma=\sqrt{\sigma^{2}}$.

How is the interquartile range (IQR) defined, and what is it used for?

$\text{IQR}=Q_3-Q_1$ (upper quartile minus lower quartile). It measures the spread of the middle 50% of data and is used to identify outliers (typically beyond $Q_1-1.5\,\text{IQR}$ or $Q_3+1.5\,\text{IQR}$).

State the addition rule of probability for two events $A$ and $B$.

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ For mutually exclusive events $P(A\cap B)=0$.

Define conditional probability and state the multiplication rule.

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(A\cap B)=P(A\mid B)\,P(B)$$ Independent events: $P(A\cap B)=P(A)P(B)$.

For a discrete random variable $X$, give the formulas for the expected value $E(X)$ and variance $\operatorname{Var}(X)$.

$$E(X)=\sum x\,P(X=x),\qquad \operatorname{Var}(X)=E(X^{2})-[E(X)]^{2}=\sum x^{2}P(X=x)-\mu^{2}$$

State the mean and variance of a binomial distribution $X\sim B(n,p)$, and its probability function.

$P(X=r)=\binom{n}{r}p^{r}(1-p)^{n-r}$; mean $E(X)=np$; variance $\operatorname{Var}(X)=np(1-p)$.

What are the key properties of the normal distribution $N(\mu,\sigma^{2})$ regarding shape and the 68-95-99.7 rule?

Bell-shaped, symmetric about $\mu$. Approximately 68% of data lies within $\mu\pm\sigma$, 95% within $\mu\pm 2\sigma$, and 99.7% within $\mu\pm 3\sigma$.

How do you standardise a normal variable $X\sim N(\mu,\sigma^{2})$ to the standard normal $Z$?

$$Z=\frac{X-\mu}{\sigma},\qquad Z\sim N(0,1)$$

In hypothesis testing, define the null hypothesis $H_0$, the alternative $H_1$, and the significance level.

$H_0$: the default/no-effect claim being tested. $H_1$: the alternative claim. The significance level (e.g. 5%) is the probability of rejecting $H_0$ when it is true (Type I error), defining the critical region.

Distinguish a Type I error from a Type II error in hypothesis testing.

Type I error: rejecting $H_0$ when it is actually true (false positive). Type II error: failing to reject $H_0$ when it is actually false (false negative).

Define the distribution of the sample mean $\bar{X}$ from a population $N(\mu,\sigma^{2})$ of sample size $n$.

$$\bar{X}\sim N\!\left(\mu,\frac{\sigma^{2}}{n}\right)$$ so the standard error is $\dfrac{\sigma}{\sqrt{n}}$.

Define displacement, velocity and acceleration in terms of derivatives with respect to time.

Velocity $v=\dfrac{dx}{dt}$ (rate of change of displacement); acceleration $a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{dt^{2}}$.

State the four SUVAT equations of motion for constant acceleration.

$$v=u+at,\quad s=ut+\tfrac{1}{2}at^{2},\quad v^{2}=u^{2}+2as,\quad s=\tfrac{1}{2}(u+v)t$$

State Newton's three laws of motion.

1st: a body remains at rest or moves with constant velocity unless acted on by a resultant force. 2nd: $F=ma$ (resultant force = mass × acceleration). 3rd: every action has an equal and opposite reaction.

What is the limiting friction model for a body on the point of moving?

Friction $F\leq\mu R$, where $\mu$ is the coefficient of friction and $R$ is the normal reaction. At the point of slipping (limiting), $F=\mu R$.

For two particles connected by a light inextensible string over a smooth pulley, what is true of their accelerations and the tension?

Both particles have the same magnitude of acceleration, and the tension is the same throughout the string. Apply $F=ma$ to each particle separately and solve simultaneously.

Define linear momentum and state the impulse-momentum relationship.

Momentum $p=mv$. Impulse $=Ft=$ change in momentum: $$Ft=mv-mu$$

State the principle of conservation of momentum for a collision between two bodies.

Total momentum before = total momentum after (no external forces): $$m_1u_1+m_2u_2=m_1v_1+m_2v_2$$

Give the formulas for work done, kinetic energy and gravitational potential energy.

Work $W=Fs\cos\theta$; kinetic energy $E_k=\tfrac{1}{2}mv^{2}$; gravitational PE $E_p=mgh$.

State the work-energy principle and define power.

Work-energy principle: the work done by the resultant force equals the change in kinetic energy. Power = rate of doing work: $P=\dfrac{W}{t}=Fv$.

For a projectile launched with speed $u$ at angle $\theta$, give the horizontal and vertical components of velocity and the equations of motion.

Horizontal: $u_x=u\cos\theta$ (constant), $x=u\cos\theta\,t$. Vertical: $u_y=u\sin\theta$, $y=u\sin\theta\,t-\tfrac{1}{2}gt^{2}$, with acceleration $-g$.

Derive the formulas for the time of flight and range of a projectile launched at angle $\theta$ on level ground with speed $u$.

Time of flight $T=\dfrac{2u\sin\theta}{g}$; range $R=\dfrac{u^{2}\sin 2\theta}{g}$ (maximum when $\theta=45^{\circ}$).

What is the maximum height reached by a projectile launched with speed $u$ at angle $\theta$?

$$H=\frac{u^{2}\sin^{2}\theta}{2g}$$ reached when the vertical velocity is zero.

How can you tell whether a function $f$ has an inverse, and how do their graphs relate?

$f$ has an inverse if and only if it is one-to-one (injective). The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$, and the domain and range are swapped.

Define the modulus function $|x|$ and describe how to solve $|f(x)|=k$.

$|x|=x$ if $x\geq 0$ and $-x$ if $x<0$. To solve $|f(x)|=k$ ($k\geq0$): solve $f(x)=k$ and $f(x)=-k$, then check solutions.

Give the exact derivative results for $\tan x$ and $\sqrt{x}$.

$\dfrac{d}{dx}\tan x=\sec^{2}x$; $\dfrac{d}{dx}\sqrt{x}=\dfrac{d}{dx}x^{1/2}=\dfrac{1}{2\sqrt{x}}$.