1.01a Proof: structure of mathematical proof and logical steps

\includegraphics{figure_1} A smooth sphere \(P\) lies at rest on a smooth horizontal plane. A second identical sphere \(Q\), moving on the plane, collides with the sphere \(P\). Immediately before the collision the direction of motion of \(Q\) makes an angle \(\alpha\) with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(Q\) makes an angle \(\beta\) with the line joining the centres of spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\). Show that \((1-e) \tan \beta = 2 \tan \alpha\). [11]

$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).

1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]

Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),

1. find the value of \(k\). [2]
2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]

The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.

1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]

Triangle \(ABC\), with \(BC = a\), \(AC = b\) and \(AB = c\) is inscribed in a circle. Given that \(AB\) is a diameter of the circle and that \(a^2\), \(b^2\) and \(c^2\) are three consecutive terms of an arithmetic progression (arithmetic series),

1. express \(b\) and \(c\) in terms of \(a\), [4]
2. verify that \(\cot A\), \(\cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression. [3]

In an acute-angled triangle \(PQR\) the sides \(QR\), \(PR\) and \(PQ\) have lengths \(p\), \(q\) and \(r\) respectively.

1. Prove that $$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]

Given now that triangle \(PQR\) is such that \(p^2\), \(q^2\) and \(r^2\) are three consecutive terms of an arithmetic progression,

1. use the cosine rule to prove that $$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
2. Using the results given in parts \((c)\) and \((d)\), prove that \(\cot P\), \(\cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression. [3]

1. Determine the set of values of \(n\) for which \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) are positive integers. [3]

A 'Pythagorean triple' is a set of three positive integers \(a\), \(b\) and \(c\) such that \(a^2 + b^2 = c^2\).

1. Prove that, for the set of values of \(n\) found in part (a), the numbers \(n\), \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) form a Pythagorean triple. [2]

Prove that \(\sqrt{2} \cos(2\theta + 45°) = \cos^2 \theta - 2\sin \theta \cos \theta - \sin^2 \theta\), where \(\theta\) is measured in degrees. [3]

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_6} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Show that the area of triangle \(ABC\) is \(\frac{5}{2}\sqrt{9 - 2a}\). [7]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]

Given that \(y \in \mathbb{R}\), prove that $$(2 + 3y)^4 + (2 - 3y)^4 \geq 32$$ Fully justify your answer. [6 marks]

The diagram below shows a circle and four triangles. \(AB\) is a diameter of the circle. \(C\) is a point on the circumference of the circle. Triangles \(ABK\), \(BCL\) and \(CAM\) are equilateral. Prove that the area of triangle \(ABK\) is equal to the sum of the areas of triangle \(BCL\) and triangle \(CAM\). [5 marks]

Integers \(m\) and \(n\) are both odd. Prove that \(m^2 + n^2\) is a multiple of 2 but not a multiple of 4 [5 marks]

Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4. [5 marks]

Prove that the function \(f(x) = x^3 - 3x^2 + 15x - 1\) is an increasing function. [6 marks]

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

Prove that 23 is a prime number. [2 marks]

\(x^2 + bx + c\) and \(x^2 + dx + e\) have a common factor \((x + 2)\) Show that \(2(d - b) = e - c\) Fully justify your answer. [4 marks]

\(a\) and \(b\) are two positive irrational numbers. The sum of \(a\) and \(b\) is rational. The product of \(a\) and \(b\) is rational. Caroline is trying to prove \(\frac{1}{a} + \frac{1}{b}\) is rational. Here is her proof: Step 1 \quad \(\frac{1}{a} + \frac{1}{b} = \frac{2}{a + b}\) Step 2 \quad \(2\) is rational and \(a + b\) is non-zero and rational. Step 3 \quad Therefore \(\frac{2}{a + b}\) is rational. Step 4 \quad Hence \(\frac{1}{a} + \frac{1}{b}\) is rational.

1. 1. Identify Caroline's mistake. [1 mark]
   2. Write down a correct version of the proof. [2 marks]
2. Prove by contradiction that the difference of any rational number and any irrational number is irrational. [4 marks]

Two particles \(A\) and \(B\) are released from rest from different starting points above a horizontal surface. \(A\) is released from a height of \(h\) metres. \(B\) is released at a time \(t\) seconds after \(A\) from a height of \(kh\) metres, where \(0 < k < 1\) Both particles land on the surface \(5\) seconds after \(A\) was released. Assuming any resistance forces may be ignored, prove that $$t = 5(1 - \sqrt{k})$$ Fully justify your answer. [5 marks]

A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers. He claims that this will always be true. He attempts to prove his claim as follows: Step 1: Check first few cases \(3 + 5 = 8\) and \(8 = 3^2 - 1^2\) \(5 + 7 = 12\) and \(12 = 4^2 - 2^2\) \(7 + 9 = 16\) and \(16 = 5^2 - 3^2\) Step 2: Use pattern to predict and check a large example \(101 + 103 = 204\) subtract 1 and divide by 2 for the first number Add 1 and divide by two for the second number \(52^2 - 50^2 = 204\) it works! Step 3: Conclusion The first few cases work and there is a pattern, which can be used to predict larger numbers. Therefore, it must be true for all consecutive odd numbers.

1. Explain what is wrong with the student's "proof". [1 mark]
2. Prove that the student's claim is correct. [3 marks]

The three sides of a right-angled triangle have lengths \(a\), \(b\) and \(c\), where \(a, b, c \in \mathbb{Z}\) \includegraphics{figure_6}

1. State an example where \(a\), \(b\) and \(c\) are all even. [1 mark]
2. Prove that it is not possible for all of \(a\), \(b\) and \(c\) to be odd. [3 marks]

\(N\) is an integer that is not divisible by 3. Prove that \(N^2\) is of the form \(3p + 1\), where \(p\) is an integer. [5]

\(P\) and \(Q\) are consecutive odd positive integers such that \(P > Q\). Prove that \(P^2 - Q^2\) is a multiple of 8. [3]