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### HENRYTAIGO

#### Cambridge International AS and A Level

 Name of student HENRYTAIGO Date Adm. number Year/grade HenryTaigo Stream HenryTaigo Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 Total
Marks 10 9 9 28
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 3 questions Question 1 Code: 9709/31/M/J/18/8, Topic: Integration, Numerical solutions of equations The positive constant$a$is such that$\displaystyle\int_{0}^{a} x \mathrm{e}^{-\frac{1}{2} x} \mathrm{~d} x=2$.$\text{(i)}$Show that$a$satisfies the equation$a=2 \ln (a+2)$.$[5]\text{(ii)}$Verify by calculation that$a$lies between 3 and$3.5$.$[2]\text{(iii)}$Use an iteration based on the equation in part$\text{(i)}$to determine$a$correct to 2 decimal places. Give the result of each iteration to 4 decimal places.$[3]$Question 2 Code: 9709/32/M/J/18/8, Topic: Differentiation, Integration The diagram shows the curve$y=(x+1) \mathrm{e}^{-\frac{1}{3} x}$and its maximum point$M$.$\text{(i)}$Find the$x$-coordinate of$M$.$[4]\text{(ii)}$Find the area of the shaded region enclosed by the curve and the axes, giving your answer in terms of e.$[5]$Question 3 Code: 9709/33/O/N/18/8, Topic: Complex numbers$\text{(a)}$Showing all necessary working, express the complex number$\displaystyle\frac{2+3 \mathrm{i}}{1-2 \mathrm{i}}$in the form$r \mathrm{e}^{\mathrm{i} \theta}$, where$r>0$and$-\pi<\theta \leqslant \pi$. Give the values of$r$and$\theta$correct to 3 significant figures.$[5]\text{(b)}$On an Argand diagram sketch the locus of points representing complex numbers$z$satisfying the equation$|z-3+2 i|=1$. Find the least value of$|z|$for points on this locus, giving your answer in an exact form.$[4]\$

Worked solutions: P1, P3 & P6 (S1)

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