Further Pure 1: Further calculusEdexcel A-Level Further Maths: Topic test

1. 1
   The function $\mathrm{f}(x)=\sqrt{x}$ is expanded as a Taylor series in ascending powers of $(x-4)$.

   (a)

   Find $\mathrm{f}''(4)$.

   [1 mark]

   • A$-\frac{1}{16}$
   • B$-\frac{1}{32}$
   • C$\frac{1}{32}$
   • D$-\frac14$

   (b)

   What is the coefficient of $(x-4)^3$ in the series?

   [1 mark]

   • A$\frac{3}{256}$
   • B$\frac{1}{256}$
   • C$-\frac{1}{512}$
   • D$\frac{1}{512}$

   (c)

   Use the series up to the term in $(x-4)^3$ to estimate $\sqrt{4.4}$ to 4 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $y=x^2\sin x$.

   (a)

   Using Leibnitz's theorem with $u=x^2$ and $v=\sin x$, how many non-zero terms are there in the expression for $\frac{d^4y}{dx^4}$?

   [1 mark]

   • A$3$
   • B$2$
   • C$4$
   • D$5$

   (b)

   What is the value of $\frac{d^4y}{dx^4}$ when $x=\frac{\pi}{2}$?

   [1 mark]

   • A$\frac{\pi^2}{4}+12$
   • B$\frac{\pi^2}{4}$
   • C$\frac{\pi^2}{4}-12$
   • D$\frac{\pi^2}{4}-4\pi-12$

   (c)

   Use Leibnitz's theorem to find the value of $\frac{d^3y}{dx^3}$ when $x=\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The integral $I=\int_0^{\pi/2}\frac{1}{2+\cos x}\,\mathrm{d}x$ is evaluated using the substitution $t=\tan\frac{x}{2}$.

   (a)

   Show that $I=\int_0^1\frac{2}{3+t^2}\,\mathrm{d}t$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $L=\lim_{x\to0}\dfrac{\sin x-x\cos x}{x^3}$.

   (a)

   Use L'Hospital's rule to show that $L=\frac13$.

   [6 marks]

   (b)

   Use series expansions of $\sin x$ and $\cos x$ up to the term in $x^5$ to verify that $L=\frac13$, and hence find $\lim_{x\to0}\dfrac{\sin x-x\cos x-\frac13x^3}{x^5}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider $J=\int_0^{\pi/2}\frac{1}{1+\cos x}\,\mathrm{d}x$, evaluated using $t=\tan\frac{x}{2}$.

   (a)

   Which integral is equal to $J$ after the substitution?

   [1 mark]

   • A$\int_0^{\pi/2}1\,\mathrm{d}t$
   • B$\int_0^1\frac{2}{1+t^2}\,\mathrm{d}t$
   • C$\int_0^1\frac{1+t^2}{2}\,\mathrm{d}t$
   • D$\int_0^1 1\,\mathrm{d}t$

   (b)

   What is the value of $J$?

   [1 mark]

   • A$\frac{\pi}{2}$
   • B$1$
   • C$\frac12$
   • D$\ln2$

   (c)

   Hence find the exact value of $\int_{\pi/3}^{2\pi/3}\frac{1}{1+\cos x}\,\mathrm{d}x$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $\mathrm{f}(x)=\tan x$ is expanded as a Taylor series in ascending powers of $\left(x-\frac{\pi}{4}\right)$.

   (a)

   Find $\mathrm{f}''\left(\frac{\pi}{4}\right)$.

   [1 mark]

   • A$2$
   • B$2\sqrt2$
   • C$4$
   • D$8$

   (b)

   What is the coefficient of $\left(x-\frac{\pi}{4}\right)^3$ in the series?

   [1 mark]

   • A$\frac83$
   • B$16$
   • C$\frac{16}{3}$
   • D$\frac23$

   (c)

   Use the series up to the term in $\left(x-\frac{\pi}{4}\right)^3$ to estimate $\tan\left(\frac{\pi}{4}+0.1\right)$ to 3 decimal places.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the limit $\lim_{x\to0^+}x\ln x$.

   (a)

   Use L'Hospital's rule to show that the limit is $0$.

   [3 marks]

   (b)

   Hence find $\lim_{x\to0^+}x^x$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   For $0<a\le\frac{\pi}{2}$, let $F(a)=\int_0^a\frac{1}{1+\sin x+\cos x}\,\mathrm{d}x$.

   (a)

   Use the substitution $t=\tan\frac{x}{2}$ to show that $F(a)=\ln\left(1+\tan\frac{a}{2}\right)$, and deduce the exact value of $F\left(\frac{\pi}{2}\right)$.

   [6 marks]

   (b)

   (i) Use L'Hospital's rule to find $\lim_{a\to0^+}\dfrac{F(a)}{a}$. [3]
   (ii) Use series expansions to find the first two non-zero terms of the expansion of $F(a)$ in ascending powers of $a$. [3]

   [6 marks]

   Total for question 8: 12 marks

End of questions