Questions — OCR MEI C3 (386 questions)

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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 Q7
3 marks Moderate -0.8
State whether the following statements are true or false; if false, provide a counter-example.
  1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
  2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
  3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational. [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 Q12
2 marks Moderate -0.8
Prove that the following statement is false. For all integers \(n\) greater than or equal to 1, \(n^2 + 3n + 1\) is a prime number. [2]
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]