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

24 sample cards from the Pure Mathematics: Algebra and Functions deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $\dfrac{1}{a}$.

2. 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.

3. 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 $A,B$.

4. 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.

5. What partial-fraction form corresponds to an irreducible quadratic factor $(x^{2}+px+q)$?

   A linear numerator over that factor: $\dfrac{Ax+B}{x^{2}+px+q}$.

6. 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.

7. 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 $r\neq 1$.

8. 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 $S_{\infty}=\dfrac{a}{1-r}$.

9. 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 $\sum_{r=1}^{n} r^{2}=\dfrac{n(n+1)(2n+1)}{6}$.

10. State the standard result for $\sum_{r=1}^{n} r^{3}$.

    $\sum_{r=1}^{n} r^{3}=\left(\dfrac{n(n+1)}{2}\right)^{2}=\left(\sum_{r=1}^{n} r\right)^{2}$.

11. 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.

12. 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.

13. Solve the linear recurrence $u_{n+1}=ku_{n}$ with $u_{1}=a$ in closed form.

    It is geometric: $u_{n}=a\,k^{\,n-1}$.

14. 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 $\binom{n}{r}=\dfrac{n!}{r!(n-r)!}$.

15. 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 $|x|<1$.

16. 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. $|x|<\dfrac{|a|}{|b|}$.

17. 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.

18. 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. $f(k+1)=Mf(k)+(\text{multiple of divisor})$.

19. 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 $x\to-\infty$.

20. 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 $\log(x^{k})=k\log x$.

21. State the change of base formula for logarithms.

    $\log_{a} x=\dfrac{\log_{b} x}{\log_{b} a}$, for any valid base $b$.

22. 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 $x=\dfrac{\ln b}{\ln a}=\log_{a} b$.

23. 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 $\log a$).

24. 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.

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Pure Mathematics: Algebra and Functions flashcards FAQ