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Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions Syllabus
The Pure Mathematics: Algebra and Functions syllabus for Sixth Term Examination Paper (STEP) in full: 4 chapters, 24 topics and 16 sub-topics, every one listed below, plus 50 Pure Mathematics: Algebra and Functions flashcards with answers.
Pure Mathematics: Algebra and Functions syllabus: all 4 chapters
Expand any chapter to see its topics and sub-topics.
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Polynomials, Surds and Indices
6 topics- Laws of indices and surd manipulation
- Rationalising denominators with conjugates
- Fractional and negative exponents
- Polynomial division and the factor/remainder theorems
- Synthetic division techniques
- Factorising cubics and quartics
- Quadratics: discriminant, completing the square, vertex form
- Symmetric functions of roots and Newton's identities
- Sums of powers of roots
- Forming new equations with transformed roots
- Relations between roots and coefficients (Vieta's formulae)
- Repeated roots and the factor structure of derivatives
- Laws of indices and surd manipulation
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Functions, Mappings and Inequalities
6 topics- Domain, range, composition and inverse functions
- Restricting domains to ensure invertibility
- Self-inverse functions
- The modulus function and equations/inequalities with |x|
- Algebraic and graphical solution of inequalities
- Rational inequalities via critical values
- Sign tables and interval analysis
- Curve sketching from structure (asymptotes, intercepts, symmetry)
- Transformations of graphs: translations, stretches, reflections
- Combining multiple transformations in order
- Effect on f(|x|) and |f(x)|
- Partial fractions including repeated and quadratic factors
- Domain, range, composition and inverse functions
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Sequences, Series and the Binomial Theorem
7 topics- Arithmetic and geometric progressions and their sums
- Convergence of geometric series and sum to infinity
- Sigma notation and standard summation results
- Sums of r, r-squared, r-cubed
- Method of differences (telescoping)
- Recurrence relations and closed forms
- Binomial expansion for positive integer index
- Binomial series for rational/negative index and validity range
- Approximations and error of truncation
- Proof by induction for summation and divisibility results
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Exponentials, Logarithms and Modelling
5 topics- Properties of the exponential function e^x
- Laws of logarithms and change of base
- Solving exponential and logarithmic equations
- Linearising power and exponential models with logs
- Log-log and log-linear plots
- Exponential growth and decay models
Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions flashcards — 50 questions with answers
State the three basic laws of indices for $a^{m}$ combined with $a^{n}$ (product, quotient, power).
$a^{m}\cdot a^{n}=a^{m+n}$, $\dfrac{a^{m}}{a^{n}}=a^{m-n}$, and $\left(a^{m}\right)^{n}=a^{mn}$.
How do you express $a^{-n}$ and $a^{m/n}$ in alternative forms?
$a^{-n}=\dfrac{1}{a^{n}}$ and $a^{m/n}=\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}$, with $a^{0}=1$ for $a\neq 0$.
How do you rationalise the denominator of $\dfrac{1}{a+\sqrt{b}}$?
Multiply numerator and denominator by the conjugate $a-\sqrt{b}$, giving $\dfrac{a-\sqrt{b}}{a^{2}-b}$.
State the factor theorem and the remainder theorem for a polynomial $p(x)$.
Remainder theorem: dividing $p(x)$ by $(x-a)$ leaves remainder $p(a)$. Factor theorem: $(x-a)$ is a factor of $p(x)$ if and only if $p(a)=0$.
When $p(x)$ is divided by a divisor $d(x)$, what identity relates quotient $q(x)$ and remainder $r(x)$, and what constrains $r(x)$?
$p(x)=d(x)\,q(x)+r(x)$, where $\deg r < \deg d$.
For $ax^{2}+bx+c=0$, give the quadratic formula and the discriminant.
$x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with discriminant $\Delta=b^{2}-4ac$.
How does the discriminant $\Delta=b^{2}-4ac$ classify the roots of a real quadratic?
$\Delta>0$: two distinct real roots; $\Delta=0$: one repeated real root; $\Delta<0$: no real roots (a complex conjugate pair).
Write $ax^{2}+bx+c$ in completed-square (vertex) form and state the vertex.
$a\left(x+\dfrac{b}{2a}\right)^{2}+c-\dfrac{b^{2}}{4a}$; the vertex is at $\left(-\dfrac{b}{2a},\,c-\dfrac{b^{2}}{4a}\right)$.
For a quadratic with roots $\alpha,\beta$, state Vieta's formulae.
$\alpha+\beta=-\dfrac{b}{a}$ and $\alpha\beta=\dfrac{c}{a}$.
For a cubic $ax^{3}+bx^{2}+cx+d=0$ with roots $\alpha,\beta,\gamma$, give the relations between roots and coefficients.
$\alpha+\beta+\gamma=-\dfrac{b}{a}$, $\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a}$, $\alpha\beta\gamma=-\dfrac{d}{a}$.
Express $\alpha^{2}+\beta^{2}$ for a quadratic in terms of the symmetric functions $s=\alpha+\beta$ and $p=\alpha\beta$.
$\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta=s^{2}-2p$.
State Newton's identity giving the power sum $p_{1}$ and $p_{2}$ in terms of elementary symmetric polynomials $e_{1},e_{2}$.
$p_{1}=e_{1}$ and $p_{2}=e_{1}p_{1}-2e_{2}=e_{1}^{2}-2e_{2}$, where $p_{k}=\sum \alpha_{i}^{k}$.
Give Newton's identity for $p_{3}$ (three-variable case) in terms of $e_{1},e_{2},e_{3}$.
$p_{3}=e_{1}p_{2}-e_{2}p_{1}+3e_{3}=e_{1}^{3}-3e_{1}e_{2}+3e_{3}$.
How can you detect a repeated root of a polynomial $p(x)$ using its derivative?
$x=a$ is a repeated root if and only if $p(a)=0$ and $p'(a)=0$; more generally a root of multiplicity $m$ satisfies $p(a)=p'(a)=\dots=p^{(m-1)}(a)=0$ but $p^{(m)}(a)\neq 0$.
If $p(x)=(x-a)^{m}q(x)$ with $q(a)\neq 0$, what factor does $p'(x)$ contain?
$p'(x)$ contains the factor $(x-a)^{m-1}$, so a root of multiplicity $m$ in $p$ is a root of multiplicity $m-1$ in $p'$.
Define the domain and range of a function $f$.
The domain is the set of input values $x$ for which $f(x)$ is defined; the range is the set of all output values $f(x)$ attained as $x$ ranges over the domain.
What is the composite function $(f\circ g)(x)$, and what condition makes it valid?
$(f\circ g)(x)=f\big(g(x)\big)$: apply $g$ first, then $f$. It requires the range of $g$ to lie within the domain of $f$.
What condition must $f$ satisfy to have an inverse $f^{-1}$, and how do their graphs relate?
$f$ must be one-to-one (injective) on its domain. The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y=x$, with domain and range swapped.
How do you find the inverse function $f^{-1}(x)$ algebraically?
Set $y=f(x)$, swap $x$ and $y$ (or solve for $x$), then rearrange to make $y$ the subject; the result is $f^{-1}(x)$.
Define the modulus function $|x|$ as a piecewise expression.
$|x|=\begin{cases} x & x\geq 0\\ -x & x<0\end{cases}$, giving the non-negative magnitude of $x$.
Show the other 30 Pure Mathematics: Algebra and Functions flashcards
For $k>0$, solve $|x|<k$ and $|x|>k$ as inequalities in $x$.
$|x|<k \iff -k<x<k$; and $|x|>k \iff x<-k \text{ or } x>k$.
How can the equation $|x-a|=|x-b|$ be interpreted and solved?
It states $x$ is equidistant from $a$ and $b$, so $x=\dfrac{a+b}{2}$ (the midpoint).
What is a general strategy for solving a rational inequality such as $\dfrac{f(x)}{g(x)}>0$?
Bring all terms to one side, find the critical values where numerator or denominator are zero, then test the sign of the expression on each resulting interval (never multiply across by $g(x)$ unless its sign is known).
When sketching $y=\dfrac{p(x)}{q(x)}$, how do you locate vertical and horizontal asymptotes?
Vertical asymptotes occur where $q(x)=0$ (and $p\neq 0$). The horizontal asymptote is found from $\lim_{x\to\pm\infty}\dfrac{p(x)}{q(x)}$, e.g. the ratio of leading coefficients when degrees are equal, or $y=0$ when $\deg p<\deg q$.
How do you test a curve $y=f(x)$ for symmetry about the $y$-axis and about the origin?
Even (symmetric in $y$-axis) if $f(-x)=f(x)$; odd (symmetric about the origin) if $f(-x)=-f(x)$.
Describe the graph transformations $y=f(x)+a$ and $y=f(x+a)$.
$y=f(x)+a$ translates the graph by $a$ vertically (up if $a>0$); $y=f(x+a)$ translates it by $a$ horizontally to the left (right if $a<0$).
Describe the transformations $y=af(x)$ and $y=f(ax)$ for $a>0$.
$y=af(x)$ is a vertical stretch by factor $a$; $y=f(ax)$ is a horizontal stretch by factor $\dfrac{1}{a}$.
What transformations do $y=-f(x)$ and $y=f(-x)$ represent?
$y=-f(x)$ reflects the graph in the $x$-axis; $y=f(-x)$ reflects it in the $y$-axis.
Into what partial-fraction form does $\dfrac{1}{(x-a)(x-b)}$ decompose?
$\dfrac{1}{(x-a)(x-b)}=\dfrac{A}{x-a}+\dfrac{B}{x-b}$ for suitable constants $A,B$.
What partial-fraction form is used for a repeated linear factor $(x-a)^{2}$?
$\dfrac{A}{x-a}+\dfrac{B}{(x-a)^{2}}$: each power up to the multiplicity appears.
What partial-fraction form corresponds to an irreducible quadratic factor $(x^{2}+px+q)$?
A linear numerator over that factor: $\dfrac{Ax+B}{x^{2}+px+q}$.
Give the $n$-th term and sum of the first $n$ terms of an arithmetic progression with first term $a$ and common difference $d$.
$u_{n}=a+(n-1)d$ and $S_{n}=\dfrac{n}{2}\big(2a+(n-1)d\big)=\dfrac{n}{2}(a+l)$ where $l$ is the last term.
Give the $n$-th term and sum of the first $n$ terms of a geometric progression with first term $a$ and common ratio $r$.
$u_{n}=ar^{n-1}$ and $S_{n}=\dfrac{a(1-r^{n})}{1-r}$ for $r\neq 1$.
When does a geometric series converge, and what is its sum to infinity?
It converges if and only if $|r|<1$, with sum to infinity $S_{\infty}=\dfrac{a}{1-r}$.
State the standard summation results for $\sum_{r=1}^{n} r$ and $\sum_{r=1}^{n} r^{2}$.
$\sum_{r=1}^{n} r=\dfrac{n(n+1)}{2}$ and $\sum_{r=1}^{n} r^{2}=\dfrac{n(n+1)(2n+1)}{6}$.
State the standard result for $\sum_{r=1}^{n} r^{3}$.
$\sum_{r=1}^{n} r^{3}=\left(\dfrac{n(n+1)}{2}\right)^{2}=\left(\sum_{r=1}^{n} r\right)^{2}$.
How can sigma notation be used to evaluate $\sum_{r=1}^{n}(ar^{2}+br+c)$ via linearity?
Split it: $a\sum r^{2}+b\sum r+c\,n$, then substitute the standard results, since $\sum$ is linear.
What is a recurrence relation, and what is meant by its closed form?
A recurrence defines each term using previous terms, e.g. $u_{n+1}=f(u_{n})$ with initial value(s). A closed form expresses $u_{n}$ directly as a function of $n$ without reference to earlier terms.
Solve the linear recurrence $u_{n+1}=ku_{n}$ with $u_{1}=a$ in closed form.
It is geometric: $u_{n}=a\,k^{\,n-1}$.
State the binomial theorem for a positive integer index $n$.
$(a+b)^{n}=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r}$, where $\binom{n}{r}=\dfrac{n!}{r!(n-r)!}$.
Give the binomial series for $(1+x)^{n}$ with rational/negative $n$, and its validity range.
$(1+x)^{n}=1+nx+\dfrac{n(n-1)}{2!}x^{2}+\dfrac{n(n-1)(n-2)}{3!}x^{3}+\cdots$, valid for $|x|<1$.
How do you expand $(a+bx)^{n}$ for non-integer $n$ using the standard binomial series, and what is its validity range?
Factor out $a^{n}$: $a^{n}\left(1+\dfrac{b}{a}x\right)^{n}$, then apply the series; valid for $\left|\dfrac{b}{a}x\right|<1$, i.e. $|x|<\dfrac{|a|}{|b|}$.
Outline the steps of proof by induction for a summation result $P(n)$.
Base case: verify $P(1)$ (or smallest $n$). Inductive step: assume $P(k)$ true, then prove $P(k+1)$. Conclude $P(n)$ holds for all $n\geq 1$ by induction.
In a divisibility proof by induction, what algebraic step typically completes the inductive step?
Write the $(k+1)$ expression in terms of the $k$ expression (assumed divisible) plus a remaining term that is also a multiple of the divisor, e.g. $f(k+1)=Mf(k)+(\text{multiple of divisor})$.
State key properties of the exponential function $e^{x}$ (derivative, value at 0, and series).
$\dfrac{d}{dx}e^{x}=e^{x}$, $e^{0}=1$, and $e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}$; it is always positive with $e^{x}\to 0$ as $x\to-\infty$.
State the three laws of logarithms (product, quotient, power).
$\log(xy)=\log x+\log y$, $\log\!\left(\dfrac{x}{y}\right)=\log x-\log y$, and $\log(x^{k})=k\log x$.
State the change of base formula for logarithms.
$\log_{a} x=\dfrac{\log_{b} x}{\log_{b} a}$, for any valid base $b$.
How do you solve an exponential equation such as $a^{x}=b$ (with $a,b>0$)?
Take logs of both sides: $x\ln a=\ln b$, so $x=\dfrac{\ln b}{\ln a}=\log_{a} b$.
How do you linearise the power model $y=ax^{n}$ and the exponential model $y=ab^{x}$ using logarithms?
Power: $\log y=\log a+n\log x$ — plot $\log y$ against $\log x$ (gradient $n$, intercept $\log a$). Exponential: $\log y=\log a+x\log b$ — plot $\log y$ against $x$ (gradient $\log b$, intercept $\log a$).
Write the exponential growth/decay model and interpret its constants.
$N=N_{0}e^{kt}$, where $N_{0}$ is the initial amount and $k$ is the rate constant: $k>0$ gives growth, $k<0$ gives decay; the quantity changes by a constant factor over equal time intervals.
Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions FAQ
What is in the Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions syllabus?
Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions is split into 4 chapters: Polynomials, Surds and Indices, Functions, Mappings and Inequalities, Sequences, Series and the Binomial Theorem and Exponentials, Logarithms and Modelling. Together they contain 24 topics and 16 sub-topics.
How many chapters are in Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions?
4 chapters, containing 24 topics and 16 sub-topics. The largest chapter is Sequences, Series and the Binomial Theorem, with 7 topics.
Are there flashcards for Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions?
Yes — a 50-card Pure Mathematics: Algebra and Functions deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.