STEP questions by topic

Original problems in the papers' style, sorted into 14 topics, each with hints in stages, a worked solution and a mark scheme.

Questions	12 a paper: 8 Pure, 2 Mechanics, 2 Probability and Statistics
Marking	20 marks a question; your six best answers count, out of 120

The topics

Integration is free to try. Every other topic comes with the course.

Topic	Section
Algebra, inequalities and functions	Pure Mathematics
Sequences, series and induction	Pure Mathematics
Differentiation and curve sketching	Pure Mathematics
Integration	Pure Mathematics
Trigonometry and coordinate geometry	Pure Mathematics
Complex numbers and matrices	Pure Mathematics
Vectors	Pure Mathematics
Proof and number	Pure Mathematics
Hyperbolic functions, polar curves and Maclaurin series	Pure Mathematics
Differential equations	Pure Mathematics
Forces, motion and centre of mass	Mechanics
Energy, collisions and oscillations	Mechanics
Probability and discrete distributions	Probability and Statistics
Continuous distributions	Probability and Statistics

Paper 2 · Integration · Accessible · 20 marks

(i)
Let $a$ $>0$ , let $g$ be a continuous even function and let $h$ be a continuous odd function, both defined on $[-a,a]$ . By using the substitution $x$ $=-u$ , show that $\int_{-a}^{a}\frac{g(x)}{1+e^{h(x)}}\,\mathrm{d}x=\int_0^a g(x)\,\mathrm{d}x.$
7 marks
(ii)
Find $\displaystyle\int_{-1}^{1}\frac{1}{(1+x^2)(1+e^{x^3})}\,\mathrm{d}x$ .
3 marks
(iii)
Find $\displaystyle\int_{-\pi/3}^{\pi/3}\frac{\sec x}{1+e^{\tan x}}\,\mathrm{d}x$ .
4 marks
(iv)
By adapting the method of part (i) to the substitution $x$ $=\pi-u$ , find $\displaystyle\int_{0}^{\pi}\frac{\sin x}{1+e^{\cos x}}\,\mathrm{d}x$ .
6 marks

The STEP specification on OCR's site, on the OCR website

Are these real STEP questions?

No. Every problem here is original, written in the style of the papers and checked by computation and by an independent solve.