Further algebra and functions questions

Questions to attempt

6 questions on Further algebra and functions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-4.1 · Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   The non-zero roots of $2x^4-3x^3-5x^2+7x-4=0$ are $\alpha,\beta,\gamma,\delta$. Without solving the equation, find $\alpha+\beta+\gamma+\delta$ and $\frac1\alpha+\frac1\beta+\frac1\gamma+\frac1\delta$.

   (4)

   (Total for Question 1 is 4 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

2. CP-4.2 · Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   The roots of $2t^3-t^2+4t-3=0$ are $\alpha,\beta,\gamma$. Form a polynomial equation whose roots are $3\alpha-1$, $3\beta-1$ and $3\gamma-1$.

   (4)

   (Total for Question 2 is 4 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

3. CP-4.3 · Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   Show that $\sum_{r=1}^{n}(3r^2-2r+4)=\frac{n(2n^2+n+7)}2$.

   (4)

   (Total for Question 3 is 4 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

4. CP-4.4 · Understand and use the method of differences for summation of series including use of partial fractions.

   Q4

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   4.

   Find a closed form for $\sum_{r=1}^{n}\frac1{r(r+2)}$. Hence evaluate the sum for $n=10$.

   (6)

   (Total for Question 4 is 6 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

5. CP-4.5 · Find the Maclaurin series of a function including the general term.

   Q5

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   5.

   Find the Maclaurin series of $e^{2t}$ up to and including the term in $t^4$, and state the general term.

   (4)

   (Total for Question 5 is 4 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

6. CP-4.6 · Recognise and use the Maclaurin series for e^x, ln(1+x), sin x, cos x and (1+x)^n, and be aware of the range of values of x for which they are valid (proof not required).

   Q6

   ORIGINAL

   Year 2

   Year 2: this is content the exam board adds beyond the AS subject content, for the full A-level.

   6.

   Write the first four terms of the expansion of $(1-2x)^{-1/2}$, and state the range of values of $x$ for which it is valid.

   (4)

   (Total for Question 6 is 4 marks)

   Reveal answer and method

   Report this question

   Ask Oscar about this question

Progress is saved on this device for guests and accounts right now; cross-device account sync is not live yet.

CP-4 Further algebra and functions revision notes