STEP (Sixth Term Examination Paper)

Flashcards for STEP Mathematics 2 and STEP Mathematics 3, split into Pure, Mechanics and Probability and Statistics and then by topic, following the STEP specification for examinations from June 2026 onwards (version 1.3). STEP 2 holds the content assumed for both papers together with STEP 2's own additions; the STEP 3 subdeck holds only what STEP 3 adds beyond STEP 2. There is no formulae booklet for STEP, so every formula here has to be recalled.

Adding it gives you your own copy, with every subdeck below. Each card then comes back just before you would forget it, and every one you get right or wrong counts towards your mastery of its topic. You can delete or suspend the parts you are not studying once it is yours.

Some of its cards

• To prove by induction that $\sum_{r=1}^n r=\frac12n(n+1)$, the base case checks [...], and the inductive step shows that adding $(k+1)$ to the $n=k$ sum gives $\frac12k(k+1)+(k+1)=\frac12(k+1)(k+2)$, which matches the formula at $n=k+1$.

  To prove by induction that $\sum_{r=1}^n r=\frac12n(n+1)$, the base case checks $n=1$: both sides equal $1$, and the inductive step shows that adding $(k+1)$ to the $n=k$ sum gives $\frac12k(k+1)+(k+1)=\frac12(k+1)(k+2)$, which matches the formula at $n=k+1$.
• What condition must two matrices satisfy before they can be multiplied together?

  The number of columns of the first must equal the number of rows of the second
• To solve $\dfrac{dy}{dx}+P(x)y=Q(x)$, find the integrating factor [...], multiply the equation through by $I$ so the left side becomes $\dfrac{d}{dx}(Iy)$, then integrate both sides with respect to $x$.

  To solve $\dfrac{dy}{dx}+P(x)y=Q(x)$, find the integrating factor $I=e^{\int P(x)\,dx}$, multiply the equation through by $I$ so the left side becomes $\dfrac{d}{dx}(Iy)$, then integrate both sides with respect to $x$.
• How is $\sinh x$ defined in terms of $e^x$?

  $\dfrac{e^x-e^{-x}}{2}$
• What does de Moivre's theorem state?

  $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$
• How is $f'(x)$ defined as a limit?

  $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$
• What is the general term in the binomial expansion of $(a+bx)^n$, for a positive integer $n$?

  $\binom{n}{r}a^{n-r}(bx)^r$
• What does it mean for random variables $X$ and $Y$ to be independent?

  The outcome of one has no effect on the probability distribution of the other
• What is $E(X)$ for a discrete random variable $X$?

  $\sum x_iP(X=x_i)$
• What does it mean for two events to be mutually exclusive?

  They cannot both happen at once: $P(A\cap B)=0$

And 499 more once you add the deck.