Sixth Term Examination Paper (STEP) Pure Mathematics: Algebra and Functions Syllabus

Pure Mathematics: Algebra and Functions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Pure Mathematics: Algebra and Functions in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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

Pure Mathematics: Algebra and Functions flashcards for Sixth Term Examination Paper (STEP)

23 of 50 cards from the Pure Mathematics: Algebra and Functions deck — real questions with worked answers.

1. 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 $\left(a^{m}\right)^{n}=a^{mn}$.

2. 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 $a\neq 0$.

3. How do you rationalise the denominator of $\dfrac{1}{a+\sqrt{b}}$?

   Multiply numerator and denominator by the conjugate $a-\sqrt{b}$, giving $\dfrac{a-\sqrt{b}}{a^{2}-b}$.

4. 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 $p(a)=0$.

5. 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 $\deg r < \deg d$.

6. For $ax^{2}+bx+c=0$, give the quadratic formula and the discriminant.

   $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with discriminant $\Delta=b^{2}-4ac$.

7. 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).

8. 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 $\left(-\dfrac{b}{2a},\,c-\dfrac{b^{2}}{4a}\right)$.

9. For a quadratic with roots $\alpha,\beta$, state Vieta's formulae.

   $\alpha+\beta=-\dfrac{b}{a}$ and $\alpha\beta=\dfrac{c}{a}$.

10. For a cubic $ax^{3}+bx^{2}+cx+d=0$ with roots $\alpha,\beta,\gamma$, give the relations between roots and coefficients.

    $\alpha+\beta+\gamma=-\dfrac{b}{a}$, $\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a}$, $\alpha\beta\gamma=-\dfrac{d}{a}$.

11. Express $\alpha^{2}+\beta^{2}$ for a quadratic in terms of the symmetric functions $s=\alpha+\beta$ and $p=\alpha\beta$.

    $\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta=s^{2}-2p$.

12. 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 $p_{k}=\sum \alpha_{i}^{k}$.

13. Give Newton's identity for $p_{3}$ (three-variable case) in terms of $e_{1},e_{2},e_{3}$.

    $p_{3}=e_{1}p_{2}-e_{2}p_{1}+3e_{3}=e_{1}^{3}-3e_{1}e_{2}+3e_{3}$.

14. 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 $p^{(m)}(a)\neq 0$.

15. 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 $p'$.

16. 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.

17. 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 $f$.

18. 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.

19. 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 $f^{-1}(x)$.

20. 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 $x$.

21. For $k>0$, solve $|x|<k$ and $|x|>k$ as inequalities in $x$.

    $|x|<k \iff -k<x<k$; and $|x|>k \iff x<-k \text{ or } x>k$.

22. 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).

23. 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).

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