step questions by a level chapter - … step questions by a level chapter the following pure step...

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STEP Questions by A Level Chapter

The following pure STEP questions are aligned to the chapters of Pearson’s A Level

textbooks for the new 2017 A Level Maths. I’ve only included a question in a chapter if

students would have already covered all the skills involved, hence why chapters towards the

end of the textbook have more associated questions.

Year 1

A. Algebraic Expressions

Question 1 (STEP I 2006 Q1)

Solutions: (i) 157×211 (ii) 𝑚 = 16382

Question 3 (STEP I 2005 Q7i,ii)

Solutions: (i) 𝑛 + 1 (ii) 𝑛+1

B. Quadratics (solving, completing square, graphs, discriminant)

Solution: (i) 1 real root if 𝑎 < −1

4. 2 distinct real roots if 𝑎 = 2

(ii) If 𝑏 = −1

4, one solution. No solutions if 𝑏 < −

4. Two solutions if −

4< 𝑏 ≤ 0

Solutions: (i) Example: 𝑚 = 441, 𝑛 = 225 (ii) Example: 𝑚 = 14, 𝑦 = 7

C. Equations & Inequalities (simultaneous eqns, quadratic inequalities,

graphs)

Question 1 (STEP 2010 Q1)

D. Graphs & Transformations (cubics, quartics, reciprocal, transforming)

E. Straight Line Graphs

F. Equations of Circles

Solution: (ii) (𝑥 − 2)2 + (𝑦 − 1)2 = 1

G. Algebraic Methods (alg fractions, dividing polys, factor theorem, proof)

Solutions: (vi) 6

Solution: (ii) 15

Solutions: (ii) 210 arrangements

Solutions: (i) (𝑥 − 3)(𝑥 + 𝑦 − 2)(𝑥 + 𝑦) (ii) (𝑦 + 2)(𝑥 − 2𝑦 + 1)(𝑥 − 3𝑦 + 5)

H. Binomial Expansion (including factorial notation)

Solutions: (iii) 𝐵𝑛 = 𝐹𝑛+1 for all 𝑛.

Solutions: (ii) (𝑝, 𝑞) = (−1,0), (−1, −2), (0, −1)

I. Trigonometric Ratios (sine/cosine rule, areas, graphs)

Question 1 (STEP I 2009 Q4i)

Solution: (ii) 9: 2

J. Trigonometric Identities & Equations

K. Vectors (only magnitude/direction, position vectors)

L. Differentiation (tangents/normal, stationary points, sketching grad

funcs)

Solution: (i) (𝑎𝑏, (1 − 𝑎)(1 − 𝑏))

Solution: (2𝑎+4

𝑎−1, [

2𝑎+4

𝑎−1]

3) (i) Touch at (2,8), intersect at (8,512), no turning points (ii) touch at (2,8),

do not intersect elsewhere, and has no turning points, (iii) Touch at (2,8), intersect at (0,0), turns at

𝑥 = 2 ± √2

Solution:

M. Integration (integrating polynomials, areas under/between curves)

N. Exponentials & Logarithms (exponential modelling, solving log

equations)

Hints:

Year 2

A. Algebraic Methods (proof by contradiction, partial fractions)

B. Functions & Graphs (modulus, mapping, composite, inverse,

solving modulus equations)

C. Sequences & Series (arithmetic, geometric, sigma notation,

recurrences)

Solutions: (iii) (2[𝑎] − 1) + (𝑎 − [𝑎])×2[𝑎]

Solutions: (ii) 𝑏𝑛 = (−2)𝑛 + 4(3)𝑛

(Note: Knowledge of method of Proof By Induction required for this question)

D. Binomial Expansion (using partial fractions, n negative or

fractional)

Solutions: 8 and 12

E. Radians (including small angle approximations)

F. Trig Functions (reciprocal funcs + identities)

G. Trigonometry & Modelling (addition formulae)

Solutions: (a) −117

125 (b) 𝑡𝑎𝑛 𝜃 = 8 + √75

Solutions: (i) 𝑛 + 1 (ii) 𝑛+1

2𝑛 (iii) 1

H. Parametric Equations (parametric Cartesian, sketching,

points of intersection)

Solution: (i) ℎ2 = |𝑦| ℎ3 =1

2|𝑦 + √3𝑥 − √3|

I. Differentiation (trig, exp/log, chain/quotient/product rule,

parametric, implicit, rates of change)

Question 4 (STEP I 2011)

Question 7 (STEP 1 2011 Q7)

Solution: (ii) 𝑝 = 20, 𝑞 = 17, 𝑟 = 7, 𝑠 = 14

Solution: (ii) ℎ = √2𝑟

Solution: (i) ln(𝑥 + 𝑏 + √𝑥2 + 2𝑏𝑥 + 𝑐) + 𝑘

Solutions: (i) 0 < 𝑥 <𝜋

6< 𝑥 < 2𝜋

(ii) 𝜋

2− 𝛼 < 𝑥 <

J. Numerical methods (iteration, Newton-Raphson)

K. Integration (everything, including trapezium rule)

Solutions: (ii) 𝜋

2𝑙𝑛 2

Solutions: (ii) 𝑦 = 𝑥√6𝑥2 − 2𝑥 for 𝑥 >1

Solutions: First part - Use by parts, polynomial division or a suitable substitution.

Second part: 2 − 𝑒 + 𝐸

(i) 𝐸 (ii) 𝑒𝐸

Solution: (iii)(a) 3

2+ 4𝑙𝑛 2 (b)

8+ 𝑙𝑛 3

Solution: (ii) 44

325(𝑒2𝜋 − 1)

Solution: (ii) 𝑥2 + 𝑦2 = 1

Solutions: (i) Both 3𝜋

16 (ii) Both

Solutions: (i) When 𝑘 ≠ 0, 𝐼 =2𝑘−1

𝑘 When 𝑘 = 0, 𝐼 = 𝑙𝑛 2

(ii) If 𝑚 ≠ −1, −2, 𝐼 =𝑚2𝑚+1+1

(𝑚+2)(𝑚+1) If 𝑚 = −1, 𝐼 = 1 −

𝑥+1

If 𝑚 = −2, 𝐼 = 𝑙𝑛 2 −1

Solutions: (i) 𝑦 = ±(𝑥 + 1) (ii) 𝑦2 =(ln 𝑥)2+1

Solutions: (ii) 𝑎𝑟𝑠𝑒𝑐(𝑥 + 2) + 𝑐 (iii) 1

3arcsec (

𝑥+2

3) + 𝑐

L. Vectors (3D cords)

M. Vectors FM (vector equations of lines – no longer in

standard A Level)

Solutions: (i) 𝐷 = √6, (3,2, −1), (7,4,3) (ii) 𝐷 = √50 if lines are parallel, 𝐷 = 5 otherwise

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