1.01a Proof: structure of mathematical proof and logical steps

194 questions

Sort by: Default | Easiest first | Hardest first
OCR MEI C1 Q4
3 marks Standard +0.3
Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]
OCR MEI C1 Q8
3 marks Moderate -0.5
Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]
OCR MEI C1 Q9
2 marks Moderate -0.8
\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]
OCR MEI C1 Q4
5 marks Moderate -0.3
  1. Prove that 12 is a factor of \(3n^2 + 6n\) for all even positive integers \(n\). [3]
  2. Determine whether 12 is a factor of \(3n^2 + 6n\) for all positive integers \(n\). [2]
Edexcel C2 Q6
12 marks Moderate -0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]
The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find
  1. the value of \(r\) and the value of \(a\), [5]
  2. the sum to infinity of the series. [3]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,
  1. prove that \(\theta = 1 + \frac{1}{4}\pi\). [5]
Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
  1. the perimeter of shape \(X\), [2]
  2. the perimeter of shape \(Y\). [3]
  3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]
Edexcel C2 Q8
12 marks Standard +0.3
A geometric series is \(a + ar + ar^2 + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]
The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
  1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
  2. Find the sum to infinity of \(G\). [2]
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
  1. Find the exact value of the common ratio of this series. [3]
Edexcel C2 Q7
9 marks Standard +0.3
  1. Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac{a(1-r^n)}{1-r}.$$ [4]
  2. Evaluate \(\sum_{r=1}^{12} (5 \times 2^r)\). [5]
Edexcel C3 Q4
6 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta$$ [6]
Edexcel C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
  2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
OCR C3 Q9
12 marks Standard +0.8
  1. Prove the identity $$\tan(\theta + 60°) \tan(\theta - 60°) \equiv \frac{\tan^2 \theta - 3}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Solve, for \(0° < \theta < 180°\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = 4 \sec^2 \theta - 3,$$ giving your answers correct to the nearest \(0.1°\). [5]
  3. Show that, for all values of the constant \(k\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = k^2$$ has two roots in the interval \(0° < \theta < 180°\). [3]
OCR C3 2013 January Q9
10 marks Standard +0.8
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
  3. It is given that there are two values of \(\theta\), where \(-90° < \theta < 90°\), satisfying the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \frac{2}{3}\theta - \sin \frac{2}{3}\theta) = k,$$ where \(k\) is a constant. Find the set of possible values of \(k\). [3]
OCR MEI C3 2011 January Q4
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
OCR MEI C3 2011 June Q7
4 marks Standard +0.3
  1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
  2. Hence prove that if \(n\) is a positive integer then \(3^{2n} - 1\) is divisible by 8. [3]
OCR MEI C3 2016 June Q7
4 marks Standard +0.8
You are given that \(n\) is a positive integer. By expressing \(x^{2n} - 1\) as a product of two factors, prove that \(2^{2n} - 1\) is divisible by 3. [4]
OCR MEI C3 Q1
4 marks Standard +0.3
Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]
OCR MEI C3 Q2
4 marks Moderate -0.3
  1. Disprove the following statement: $$3^n + 2 \text{ is prime for all integers } n \geqslant 0.$$ [2]
  2. Prove that no number of the form \(3^n\) (where \(n\) is a positive integer) has 5 as its final digit. [2]
OCR MEI C3 Q3
4 marks Moderate -0.3
  1. Factorise fully \(n^3 - n\). [2]
  2. Hence prove that, if \(n\) is an integer, \(n^3 - n\) is divisible by 6. [2]
OCR MEI C3 Q5
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
OCR MEI C3 Q6
4 marks Standard +0.3
  1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
  2. Hence prove that if \(n\) is a positive integer then \(3^{2n} - 1\) is divisible by 8. [3]
OCR MEI C3 Q8
3 marks Moderate -0.8
  1. Disprove the following statement. $$\text{'If } p > q, \text{ then } \frac{1}{p} < \frac{1}{q}.$$ [2]
  2. State a condition on \(p\) and \(q\) so that the statement is true. [1]
OCR MEI C3 Q9
7 marks Standard +0.3
  1. Show that
    1. \((x - y)(x^2 + xy + y^2) = x^3 - y^3\),
    2. \((x + \frac{1}{2}y)^2 + \frac{3}{4}y^2 = x^2 + xy + y^2\). [4]
  2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x^3 > y^3\). [3]
OCR MEI C3 Q10
4 marks Easy -1.2
  1. Verify the following statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{ less than 11'.} [2]
  2. Calculate \(23 \times 89\), and hence disprove this statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{'.} [2]
OCR MEI C3 Q11
3 marks Moderate -0.5
Use the method of exhaustion to prove the following result. No 1- or 2-digit perfect square ends in 2, 3, 7 or 8 State a generalisation of this result. [3]
OCR MEI C3 Q13
6 marks Standard +0.3
Positive integers \(a\), \(b\) and \(c\) are said to form a Pythagorean triple if \(a^2 + b^2 = c^2\).
  1. Given that \(t\) is an integer greater than 1, show that \(2t\), \(t^2 - 1\) and \(t^2 + 1\) form a Pythagorean triple. [3]
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2t\), \(t^2 - 1\) and \(t^2 + 1\). [3]