Core Pure: Further algebra and functionsEdexcel A-Level Further Maths: Topic test

1. 1
   The cubic equation $x^3-6x^2+11x-7=0$ has roots $\alpha$, $\beta$ and $\gamma$.

   (a)

   What is the value of $\alpha^2+\beta^2+\gamma^2$?

   [1 mark]

   • A$58$
   • B$14$
   • C$25$
   • D$36$

   (b)

   What is the value of $\frac1\alpha+\frac1\beta+\frac1\gamma$?

   [1 mark]

   • A$\frac67$
   • B$\frac{7}{11}$
   • C$-\frac{11}{7}$
   • D$\frac{11}{7}$

   (c)

   Find the value of $(\alpha+2)(\beta+2)(\gamma+2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{f}(n)=\sum_{r=1}^{n}\left(r^3-2r\right)$, where $n$ is a positive integer.

   (a)

   What is the value of $\mathrm{f}(3)$?

   [1 mark]

   • A$36$
   • B$12$
   • C$24$
   • D$48$

   (b)

   Which expression is equal to $\mathrm{f}(n)$?

   [1 mark]

   • A$\frac14n(n+1)\left(n^2+n-4\right)$
   • B$\frac14n(n+1)\left(n^2+n+4\right)$
   • C$\frac14n^2(n+1)^2-n$
   • D$\frac14n^2(n+1)^2-2$

   (c)

   Hence find $\sum_{r=11}^{20}\left(r^3-2r\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $u_r=\frac{1}{(2r-1)(2r+1)}$ for positive integers $r$.

   (a)

   Express $u_r$ in partial fractions.

   [3 marks]

   (b)

   Use the method of differences to show that $\sum_{r=1}^{n}u_r=\frac{n}{2n+1}$, and hence find the sum to infinity of the series.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=\sqrt{1+x}$.

   (a)

   Find the Maclaurin series for $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^3$, by differentiation.

   [6 marks]

   (b)

   Use the series from part (a) to find a series for $\sqrt{1-2x}$ up to and including the term in $x^3$, and state the range of values of $x$ for which it is valid. Use $x=0.02$ to estimate $\sqrt{0.96}$ to 6 decimal places.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $\mathrm{h}(x)=\sin2x$ is expanded as a Maclaurin series in ascending powers of $x$.

   (a)

   What is the coefficient of $x^3$ in the series?

   [1 mark]

   • A$-\frac13$
   • B$\frac43$
   • C$-\frac{8}{3}$
   • D$-\frac43$

   (b)

   For which values of $x$ is the series for $\sin2x$ valid?

   [1 mark]

   • A$|x|<1$
   • B$-1<x\leqslant1$
   • CAll real values of $x$
   • D$|x|<\frac12$

   (c)

   Use the series to show that $\lim_{x\to0}\frac{\sin2x-2x}{x^3}=-\frac43$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The cubic equation $2x^3+x^2-8x+5=0$ has roots $\alpha$, $\beta$ and $\gamma$.

   (a)

   What is the value of $\alpha^2+\beta^2+\gamma^2$?

   [1 mark]

   • A$\frac{33}{4}$
   • B$-\frac{31}{4}$
   • C$\frac14$
   • D$\frac{17}{4}$

   (b)

   Which equation has roots $2\alpha$, $2\beta$ and $2\gamma$?

   [1 mark]

   • A$16y^3+4y^2-16y+5=0$
   • B$y^3+y^2-16y+20=0$
   • C$y^3+y^2-4y+5=0$
   • D$y^3+y^2-16y-20=0$

   (c)

   Find the value of $(\alpha+1)(\beta+1)(\gamma+1)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $S_n=\sum_{r=1}^{n}(r+1)(r+4)$.

   (a)

   Show that $S_n=\frac13n(n+4)(n+5)$.

   [3 marks]

   (b)

   Hence find $\sum_{r=n+1}^{2n}(r+1)(r+4)$, giving your answer in the form $\frac13n(n+a)(bn+c)$ where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   For positive integers $r$, let $u_r=\ln\left(1+\frac1r\right)$.

   (a)

   Use the method of differences to show that $\sum_{r=1}^{n}u_r=\ln(n+1)$.

   [6 marks]

   (b)

   Use the first three non-zero terms of the Maclaurin series for $\ln(1+x)$ with $x=\frac1r$ to estimate $u_5$ to 4 decimal places. State the range of $x$ for which the series is valid and explain why it can be used for every positive integer $r$. Explain why your estimate is greater than the true value of $u_5$.

   [6 marks]

   Total for question 8: 12 marks

End of questions