Section A: Real Numbers and Algebra
1. Real-number and algebraic structure
[15 marks]This question uses exact arithmetic, inequalities, binomial coefficients, and summation notation.
a) Prove that \(\sqrt{3}+\sqrt{7}\) is irrational.
[3 marks]Paper 1
a) Prove that \(\sqrt{3}+\sqrt{7}\) is irrational.
[3 marks]b) Solve the inequality \(\frac{2x+1}{x-3}\ge1\).
[4 marks]c) Find the coefficient of \(x^5\) in the expansion of \((2-x)^7\).
[4 marks]d) Evaluate \(\sum_{r=1}^{10}(2r^2-r)\).
[4 marks]a) Show that \(\frac{\sin(2x)}{1+\cos(2x)}=\tan x\) wherever both sides are defined.
[4 marks]b) Find the exact value of \(\sin\left(\arctan\left(\frac{5}{12}\right)\right)\).
[3 marks]c) Solve \(\arctan x+\arctan(2x)=\frac{\pi}{4}\).
[4 marks]d) Evaluate \(\lim_{h\to0}\frac{1-\cos(4h)}{h^2}\).
[4 marks]a) Determine when the volume is increasing and when it is decreasing.
[3 marks]b) Find an expression for the volume \(V(t)\).
[4 marks]c) Find the minimum and maximum volumes during \(0\le t\le5\).
[4 marks]d) Find the average volume over the interval \(0\le t\le5\).
[4 marks]a) Evaluate \(\frac{(2-i)(3+4i)}{1+i}\) in the form \(a+bi\).
[3 marks]b) Use De Moivre's theorem to evaluate \((1+i)^6\).
[4 marks]c) Find all complex solutions of \(z^3=27\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)\).
[4 marks]d) Describe the moduli and arguments of the roots from part (c), and state the geometric shape they form in the complex plane.
[4 marks]