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I was looking for pre-university books that would help me prepare for the Sixth Term Examination Paper (STEP). STEP is admission test for mathematics designed by the University of Cambridge.

Books designed for maths contests and the Olympiads does not help because of the contents. Olympiad focuses on inequalities, functional equations, modular arithmetic, Diophantine equations and Advanced Geometry, none of which help with the STEP. Not to mention, Olympiad problems are significantly harder. I bought this book called Art of Problem Solving: Volume 2 and then returned it after 2 days. The only book in this category, I found helpful was Complex Numbers A to Z.

Apparently the only way to prepare is by doing past papers. Unfortunately I am not ready for the past papers yet. There is this book called Advanced Problem in Mathematics by Silkos which is basically selected problems from the actual examination. This is not what I am looking for.

I am looking for Algebra, Calculus and Coordinate Geometry books with a lot of problems of similar difficulty to the problems I have posted below.

1.

In this question, r and θ are polar coordinates with $$r \geq 0$$ and $$ −π < θ \leq π$$ and a and b are positive constants. Let L be a fixed line and let A be a fixed point not lying on L. Then the locus of points that are a fixed distance (call it d) from L measured along lines through A is called a conchoid of Nicomedes.

(i) If |r − a sec θ| = b , (∗)

where a > b, then sec θ > 0. Show that all points with coordinates satisfying (∗) lie on a certain conchoid of Nicomedes (you should identify L, d and A).

Sketch the locus of these points.

(ii) In the case a < b, sketch the curve (including the loop for which sec θ < 0) given by

|r − a sec θ| = b .

Find the area of the loop in the case a = 1 and b = 2.

2.

Use the factor theorem to show that a+b-c is a factor of

$$(a+b+c)^3 -6(a+b+c)(a^2+b^2+c^2) +8(a^3+b^3+c^3)$$

Hence factorise ($*$) completely.

Use the result above to solve the equation $$ (x+1)^3 -3 (x+1)(2x^2 +5) +2(4x^3+13)=0\,. $$

By setting $d+e=c$, or otherwise, show that $(a+b-d-e)$ is a factor of $$ (a+b+d+e)^3 -6(a+b+d+e)(a^2+b^2+d^2+e^2) +8(a^3+b^3+d^3+e^3) $$ and factorise this expression completely.

Hence solve the equation $$ (x+6)^3 - 6(x+6)(x^2+14) +8(x^3+36)=0\,. $$

3.

In this question, $a$ is a positive constant. Express $\cosh a$ in terms of exponentials.

By using partial fractions, prove that $$ \int_0^1 \frac 1{ x^2 +2x\cosh a +1} \, d x = \frac a {2\sinh a}\,. $$

Find, expressing your answers in terms of hyperbolic functions, $$ \int_1^\infty \frac 1 {x^2 +2x \sinh a -1} \,d x \, $$

and

$$ \int_0^\infty \frac 1 {x^4 +2x^2\cosh a +1} \,d x. $$

Thank you!!

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Perhaps you could try these books

$\rightarrow$Putnam and Beyond by Razvan Gelca

$\rightarrow$Problem-Solving Through Problems by Loren C. Larson

$\rightarrow$Linear Algebra Problem Book by Halmos.

$\rightarrow$Halmos' book on Hilbert spaces is pretty tough.

$\rightarrow$The Putnam archive is pretty good to find tricky questions

This post might be of some help to you.

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You could try Further Mathematics by R I Porter. It has lots of old 'S' Level questions in it and it should be useful for at least some things. It used to be very good for Cambridge Entrance Exams in the 70s and 80s. https://www.amazon.co.uk/Further-Mathematics-RI-Porter/dp/B003HAQDJC

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