Further algebra: roots, series and MaclaurinAQA A-Level Further Maths: Topic test

1. 1
   The roots of the equation $2x^3+3x^2-8x+5=0$ are $\alpha$, $\beta$ and $\gamma$.

   (a)

   Find the value of $\alpha+\beta+\gamma$.

   [1 mark]

   • A$-\frac32$
   • B$\frac32$
   • C$-3$
   • D$3$

   (b)

   Find the value of $\alpha\beta\gamma$.

   [1 mark]

   • A$\frac52$
   • B$-5$
   • C$-\frac52$
   • D$-4$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sequence has $r$th term $u_r=3r^2-2r$.

   (a)

   Find $\sum_{r=1}^{4}u_r$.

   [1 mark]

   • A$80$
   • B$70$
   • C$110$
   • D$280$

   (b)

   Which expression equals $\sum_{r=1}^{n}u_r$?

   [1 mark]

   • A$\frac{n(n+1)(2n+1)}{2}$
   • B$\frac{n(n+1)(2n-1)}{6}$
   • C$\frac{n(n+1)(2n-3)}{2}$
   • D$\frac{n(n+1)(2n-1)}{2}$

   (c)

   Find $\sum_{r=11}^{20}u_r$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $u_r=\dfrac{1}{r(r+3)}$ and $S_n=\sum_{r=1}^{n}u_r$.

   (a)

   Express $u_r$ in partial fractions.

   [3 marks]

   (b)

   Use the method of differences to find $S_n$ in terms of $n$, and find the value of $\lim_{n\to\infty}S_n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\mathrm{e}^{2x}\ln(1+x)$ for $x>-1$.

   (a)

   Use standard Maclaurin series to find the series for $f(x)$ up to and including the term in $x^3$, and state the range of values of $x$ for which the expansion of $f(x)$ is valid.

   [6 marks]

   (b)

   Use l'Hôpital's rule to evaluate $\lim_{x\to0}\dfrac{f(x)-x}{x^2}$, and explain how your answer relates to your series in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The roots of the equation $x^3-5x^2+2x+8=0$ are $\alpha$, $\beta$ and $\gamma$.

   (a)

   Find the value of $\alpha+\beta+\gamma$.

   [1 mark]

   • A$-5$
   • B$2$
   • C$-8$
   • D$5$

   (b)

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

   [1 mark]

   • A$x^3-10x^2+8x+64=0$
   • B$2x^3-5x^2+x+2=0$
   • C$x^3-10x^2+4x+16=0$
   • D$x^3-10x^2+8x-64=0$

   (c)

   Find the value of $\dfrac1\alpha+\dfrac1\beta+\dfrac1\gamma$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(x)=(1-3x)^{2/3}$.

   (a)

   Find the coefficient of $x^2$ in the Maclaurin series of $f(x)$.

   [1 mark]

   • A$-\frac19$
   • B$1$
   • C$-1$
   • D$-2$

   (b)

   For which values of $x$ is the Maclaurin series of $f(x)$ valid?

   [1 mark]

   • A$-1<x<1$
   • B$-\frac13<x<\frac13$
   • C$-3<x<3$
   • D$x<\frac13$

   (c)

   Use the first three terms of the series to estimate $0.97^{2/3}$. Give your answer to $4$ decimal places.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $f(x)=\ln(3+x)$ for $x>-3$.

   (a)

   Find the Maclaurin series of $f(x)$ up to and including the term in $x^2$.

   [3 marks]

   (b)

   Find the general term, in $x^r$ for $r\geq1$, of the Maclaurin series of $f(x)$, and state the range of values of $x$ for which the series is valid.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ are the roots of the equation $x^4-2x^3+3x^2-4x+1=0$. For positive integers $r$, $u_r=\dfrac{2r+1}{r^2(r+1)^2}$.

   (a)

   Find $\alpha^2+\beta^2+\gamma^2+\delta^2$ and hence show that the equation $x^4-2x^3+3x^2-4x+1=0$ does not have four real roots.

   [6 marks]

   (b)

   (i) Show that $u_r=\dfrac1{r^2}-\dfrac1{(r+1)^2}$. (ii) Hence find $\sum_{r=1}^{n}u_r$ in terms of $n$. (iii) Write down the value of $\sum_{r=1}^{\infty}u_r$.

   [6 marks]

   Total for question 8: 12 marks

End of questions