Cambridge Pre-U Mathematics (Principal Subject) Syllabus

Mathematics (Principal Subject) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (Principal Subject) in Cambridge Pre-U, not a summary of it.

1. 1

   Pure Mathematics 1

   6 topics

   • Algebra and Functions
     • Polynomials, factor and remainder theorems
     • Quadratic functions, discriminant and completing the square
     • Indices, surds and rationalising denominators
     • Composite and inverse functions
     • Modulus function and graphs
   • Coordinate Geometry
     • Equations of straight lines, gradients and intersections
     • Equation of a circle and tangent/normal lines
     • Parametric representation of curves
   • Sequences and Series
     • Arithmetic and geometric progressions
     • Sum to infinity of a convergent geometric series
     • Binomial expansion for positive integer powers
   • Trigonometry
     • Radian measure, arc length and sector area
     • Sine, cosine and tangent graphs and transformations
     • Identities and solution of trigonometric equations
   • Differentiation
     • Differentiation from first principles
     • Tangents, normals and stationary points
     • Increasing and decreasing functions
   • Integration
     • Indefinite and definite integrals
     • Area under a curve
     • Reversal of differentiation as integration
2. 2

   Pure Mathematics 2

   6 topics

   • Advanced Algebra and Functions
     • Partial fractions
     • Binomial expansion for rational and negative indices
     • Logarithms and exponential functions
   • Further Trigonometry
     • Compound and double angle formulae
     • R sin/R cos form for a sin x + b cos x
     • Reciprocal and inverse trigonometric functions
   • Differentiation Techniques
     • Product, quotient and chain rules
     • Implicit and parametric differentiation
     • Differentiation of exponential, log and trig functions
   • Integration Techniques
     • Integration by substitution
     • Integration by parts
     • Integration using partial fractions and standard forms
   • Differential Equations
     • Formation of first-order equations
     • Solution by separation of variables
   • Numerical Methods
     • Location of roots by sign change
     • Iterative methods and convergence
     • Trapezium rule for numerical integration
3. 3

   Probability and Statistics

   5 topics

   • Data Representation and Summary
     • Histograms, cumulative frequency and box plots
     • Mean, median, mode and standard deviation
   • Probability
     • Addition and multiplication laws
     • Conditional probability and independence
     • Permutations and combinations
   • Discrete Random Variables
     • Probability distributions and expectation
     • Binomial distribution
     • Poisson distribution and its approximation to binomial
   • The Normal Distribution
     • Standardisation and use of tables
     • Normal approximation to binomial and Poisson
   • Sampling and Hypothesis Testing
     • Sampling methods and the distribution of the sample mean
     • Hypothesis tests for a population mean
     • Type I and Type II errors
4. 4

   Mechanics

   6 topics

   • Kinematics
     • Displacement, velocity and acceleration
     • Constant acceleration equations
     • Variable acceleration using calculus
   • Forces and Newton's Laws
     • Resolving forces and equilibrium
     • Newton's three laws of motion
     • Friction and the coefficient of friction
   • Connected Particles
     • Pulleys and tension in strings
     • Systems on inclined planes
   • Momentum and Impulse
     • Impulse-momentum principle
     • Conservation of momentum in collisions
   • Work, Energy and Power
     • Work done by a force
     • Kinetic and potential energy
     • Conservation of energy and power
   • Projectile Motion
     • Horizontal and vertical components
     • Range, time of flight and trajectory equation

Mathematics (Principal Subject) flashcards for Cambridge Pre-U

25 of 63 cards from the Mathematics (Principal Subject) deck — real questions with worked answers.

1. State the quadratic formula giving the roots of $ax^{2}+bx+c=0$ (where $a\neq 0$).

   $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$

2. In a quadratic $ax^{2}+bx+c=0$, what does the discriminant $b^{2}-4ac$ tell you about the roots?

   If $b^{2}-4ac>0$: two distinct real roots; if $=0$: one repeated real root; if $<0$: no real roots (two complex roots).

3. State the Remainder Theorem and the Factor Theorem for a polynomial $f(x)$.

   Remainder Theorem: the remainder when $f(x)$ is divided by $(x-a)$ is $f(a)$. Factor Theorem: $(x-a)$ is a factor of $f(x)$ if and only if $f(a)=0$.

4. What are the three laws of indices for $a^{m}$ combined with $a^{n}$ (multiply, divide, power of a power)?

   $$a^{m}\times a^{n}=a^{m+n},\quad \frac{a^{m}}{a^{n}}=a^{m-n},\quad (a^{m})^{n}=a^{mn}$$

5. Give the three logarithm laws (product, quotient, power).

   $$\log_b(xy)=\log_b x+\log_b y,\quad \log_b\!\left(\tfrac{x}{y}\right)=\log_b x-\log_b y,\quad \log_b(x^{k})=k\log_b x$$

6. How is the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ calculated?

   $$d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$$

7. What is the relationship between the gradients $m_1$ and $m_2$ of two perpendicular lines?

   $m_1 m_2=-1$ (equivalently $m_2=-\tfrac{1}{m_1}$). Parallel lines have equal gradients.

8. Give the equation of a circle with centre $(a,b)$ and radius $r$.

   $$(x-a)^{2}+(y-b)^{2}=r^{2}$$

9. Write the equation of a straight line with gradient $m$ passing through the point $(x_1,y_1)$.

   $$y-y_1=m(x-x_1)$$

10. State the formula for the $n$th term of an arithmetic sequence with first term $a$ and common difference $d$.

    $$u_n=a+(n-1)d$$

11. Give the sum of the first $n$ terms of an arithmetic series.

    $$S_n=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)$$ where $l$ is the last term.

12. State the $n$th term and the sum of $n$ terms of a geometric series with first term $a$ and common ratio $r$.

    $$u_n=ar^{n-1},\qquad S_n=\frac{a(1-r^{n})}{1-r}\;(r\neq 1)$$

13. When does an infinite geometric series converge, and what is its sum to infinity?

    It converges when $|r|<1$, with $$S_\infty=\frac{a}{1-r}$$

14. State the binomial expansion of $(a+b)^{n}$ for positive integer $n$.

    $$(a+b)^{n}=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r},\quad \binom{n}{r}=\frac{n!}{r!(n-r)!}$$

15. What are the exact values of $\sin$, $\cos$ and $\tan$ of $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$?

    $\sin30=\tfrac{1}{2},\cos30=\tfrac{\sqrt{3}}{2},\tan30=\tfrac{1}{\sqrt{3}}$; $\sin45=\cos45=\tfrac{1}{\sqrt{2}},\tan45=1$; $\sin60=\tfrac{\sqrt{3}}{2},\cos60=\tfrac{1}{2},\tan60=\sqrt{3}$.

16. State the three Pythagorean trigonometric identities.

    $$\sin^{2}\theta+\cos^{2}\theta=1,\quad 1+\tan^{2}\theta=\sec^{2}\theta,\quad 1+\cot^{2}\theta=\csc^{2}\theta$$

17. State the sine rule and the cosine rule for a triangle with sides $a,b,c$ and opposite angles $A,B,C$.

    Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$. Cosine rule: $a^{2}=b^{2}+c^{2}-2bc\cos A$.

18. Give the formula for the area of a triangle using two sides and the included angle.

    $$\text{Area}=\tfrac{1}{2}ab\sin C$$

19. State the double angle formulae for $\sin 2\theta$ and $\cos 2\theta$.

    $$\sin 2\theta=2\sin\theta\cos\theta,\qquad \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta$$

20. State the compound angle formula for $\sin(A\pm B)$ and $\cos(A\pm B)$.

    $$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\qquad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$

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

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

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

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

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

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

24. 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).

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

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

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