Proof worksheet
10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.
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4 marks
(a)
The smallest of eleven consecutive integers is \(n\).
Write an expression, in terms of \(n\), for the sum of the eleven integers. Give your answer in its simplest form.
(b)Prove that the sum of any eleven consecutive integers is always a multiple of 11.
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4 marks
(a)
Two whole numbers differ by 9. The smaller of them is \(k\).
Expand and simplify \((k + 9)^2 - k^2\).
(b)Prove that the difference between the squares of any two whole numbers that differ by 9 is always a multiple of 9.
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4 marks
(a)
Expand and simplify \(m(m + 14) + 49\).
(b)Hence prove that \(m(m + 14) + 49\) is a square number for every whole number \(m\).
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4 marks
(a)
Factorise fully \(n^2 + 9n\).
(b)Prove that \(n^2 + 9n\) is always even when \(n\) is a whole number.
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4 marks
(a)
\(2k - 5\) is an odd number for every whole number \(k\).
Multiply out the brackets and simplify \((2k - 5)^2\).
(b)Prove that the square of any odd number is odd.
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4 marks
(a)
Write \(x^2 + 8x + 35\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are integers.
(b)Hence prove that \(x^2 + 8x + 35\) is always positive.
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4 marks
(a)
The smallest of nine consecutive integers is \(k\).
Write an expression, in terms of \(k\), for the sum of the nine integers. Give your answer in its simplest form.
(b)Prove that the sum of any nine consecutive integers is always a multiple of 9.
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4 marks
(a)
Two whole numbers differ by 11. The smaller of them is \(m\).
Expand and simplify \((m + 11)^2 - m^2\).
(b)Prove that the difference between the squares of any two whole numbers that differ by 11 is always a multiple of 11.
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4 marks
(a)
Expand and simplify \(t(t + 8) + 16\).
(b)Hence prove that \(t(t + 8) + 16\) is a square number for every whole number \(t\).
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4 marks
(a)
Factorise fully \(r^2 + 5r\).
(b)Prove that \(r^2 + 5r\) is always even when \(r\) is a whole number.
Answers
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(a) \(11n + 55\)
(b) \(11n + 55 = 11(n + 5)\). \(n + 5\) is a whole number, so the sum is 11 times a whole number, which is what being a multiple of 11 means. -
(a) \(18k + 81\)
(b) \(18k + 81 = 9(2k + 9)\). \(2k + 9\) is a whole number, so the difference is 9 times a whole number. -
(a) \(m^2 + 14m + 49\)
(b) \(m^2 + 14m + 49 = (m + 7)^2\), and \(m + 7\) is a whole number, so the expression is the square of a whole number. -
(a) \(n(n + 9)\)
(b) \(n^2 + 9n = n(n + 9)\). The two factors differ by 9, which is odd, so one of them is even and the other is odd. A product with an even factor is even. -
(a) \(4k^2 - 20k + 25\)
(b) \(4k^2 - 20k + 25 = 2(2k^2 - 10k + 12) + 1\), which is two times a whole number plus one, so it is odd. -
(a) \((x + 4)^2 + 19\)
(b) \((x + 4)^2\) is a square, so it is never negative. The smallest the whole expression can be is \(19\), which is greater than zero, so \(x^2 + 8x + 35\) is always positive. -
(a) \(9k + 36\)
(b) \(9k + 36 = 9(k + 4)\). \(k + 4\) is a whole number, so the sum is 9 times a whole number, which is what being a multiple of 9 means. -
(a) \(22m + 121\)
(b) \(22m + 121 = 11(2m + 11)\). \(2m + 11\) is a whole number, so the difference is 11 times a whole number. -
(a) \(t^2 + 8t + 16\)
(b) \(t^2 + 8t + 16 = (t + 4)^2\), and \(t + 4\) is a whole number, so the expression is the square of a whole number. -
(a) \(r(r + 5)\)
(b) \(r^2 + 5r = r(r + 5)\). The two factors differ by 5, which is odd, so one of them is even and the other is odd. A product with an even factor is even.