Pure Mathematics Year 1 (AS) Unit Test 3: Further Algebra
1
Use the factor theorem and division to factorise f(x) completely. (6 marks)
2 a Expand in ascending powers of x, up to and including the term in ,
simplifying each coefficient in the expansion. (4 marks)
b Showing your working clearly, use your expansion to find, to 5 significant figures,
an approximation for (3 marks)
3 a Find the first four terms, in ascending powers of x, of the binomial expansion
of (4 marks)
Given that the coefficient of the x3 term in the expansion is −84
b i Find the value of p. (2 marks)
ii Find the numerical values for the coefficients of the x and x2 terms. (2 marks)
4 a Fully expand (2 marks)
A fair four-sided die, numbered 1, 2, 3 and 4, is rolled 5 times. Let p represent
the probability that the number 4 is rolled on a given roll and let q represent the
probability that the number 4 is not rolled on a given roll.
b Using the first three terms of the binomial expansion from part a, or otherwise,
find the probability that the number 4 is rolled at least 3 times. (5 marks)
5 where p and q are constants
Given that f(5) = 0 and f(−3) = 8
a find the values of p and q. (7 marks)
b factorise f(x) completely. (5 marks)
6 Prove that, for all values of x, (4 marks)
7 a Prove that if then x > 0 (4 marks)
b Show, by means of a counter example, that the inequality
is not true for all vaues of x. (2 marks)
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1
Use the factor theorem and division to factorise f(x) completely. (6 marks)
2 a Expand in ascending powers of x, up to and including the term in ,
simplifying each coefficient in the expansion. (4 marks)
b Showing your working clearly, use your expansion to find, to 5 significant figures,
an approximation for (3 marks)
3 a Find the first four terms, in ascending powers of x, of the binomial expansion
of (4 marks)
Given that the coefficient of the x3 term in the expansion is −84
b i Find the value of p. (2 marks)
ii Find the numerical values for the coefficients of the x and x2 terms. (2 marks)
4 a Fully expand (2 marks)
A fair four-sided die, numbered 1, 2, 3 and 4, is rolled 5 times. Let p represent
the probability that the number 4 is rolled on a given roll and let q represent the
probability that the number 4 is not rolled on a given roll.
b Using the first three terms of the binomial expansion from part a, or otherwise,
find the probability that the number 4 is rolled at least 3 times. (5 marks)
5 where p and q are constants
Given that f(5) = 0 and f(−3) = 8
a find the values of p and q. (7 marks)
b factorise f(x) completely. (5 marks)
6 Prove that, for all values of x, (4 marks)
7 a Prove that if then x > 0 (4 marks)
b Show, by means of a counter example, that the inequality
is not true for all vaues of x. (2 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
