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### MIKE STEPHEN

#### Cambridge International AS and A Level

 Name of student MIKE STEPHEN Date Adm. number Year/grade 1981 Stream Mike Stephen Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 10 10 10 10 10 10 60
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/19/9, Topic: Vectors The diagram shows a set of rectangular axes$O x, O y$and$O z$, and four points$A, B, C$and$D$with position vectors$\overrightarrow{O A}=3 \mathbf{i}, \overrightarrow{O B}=3 \mathbf{i}+4 \mathbf{j}, \overrightarrow{O C}=\mathbf{i}+3 \mathbf{j}$and$\overrightarrow{O D}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}$.$\text{(i)}$Find the equation of the plane$B C D$, giving your answer in the form$a x+b y+c z=d$.$[6]\text{(ii)}$Calculate the acute angle between the planes$B C D$and$O A B C$.$[4]$Question 2 Code: 9709/32/M/J/19/9, Topic: Vectors The points$A$and$B$have position vectors$\mathbf{i}+2 \mathbf{j}-\mathbf{k}$and$3 \mathbf{i}+\mathbf{j}+\mathbf{k}$respectively. The line$l$has equation$\mathbf{r}=2 \mathbf{i}+\mathbf{j}+\mathbf{k}+\mu(\mathbf{i}+\mathbf{j}+2 \mathbf{k})\text{(i)}$Show that$l$does not intersect the line passing through$A$and$B$.$[5]\text{(ii)}$The plane$m$is perpendicular to$A B$and passes through the mid-point of$A B$. The plane$m$intersects the line$l$at the point$P$. Find the equation of$m$and the position vector of$P$.$[5]$Question 3 Code: 9709/33/M/J/19/9, Topic: Algebra Let$\displaystyle\mathrm{f}(x)=\frac{2 x(5-x)}{(3+x)(1-x)^{2}}\text{(i)}$Express$\mathrm{f}(x)$in partial fractions.$[5]\text{(ii)}$Hence obtain the expansion of$\mathrm{f}(x)$in ascending powers of$x$up to and including the term in$x^{3}$.$[5]$Question 4 Code: 9709/31/O/N/19/9, Topic: Trigonometry, Integration$\text{(i)}$By first expanding$\cos (2 x+x)$, show that$\cos 3 x \equiv 4 \cos ^{3} x-3 \cos x$.$[4]\text{(ii)}$Hence solve the equation$\cos 3 x+3 \cos x+1=0$, for$0 \leqslant x \leqslant \pi$.$[2]\text{(iii)}$Find the exact value of$\displaystyle\int_{\frac{1}{6} \pi}^{\frac{1}{3} \pi} \cos ^{3} x \mathrm{~d} x$.$[4]$Question 5 Code: 9709/32/O/N/19/9, Topic: Integration, Numerical solutions of equations It is given that$\displaystyle\int_{0}^{a} x \cos \frac{1}{3} x \mathrm{~d} x=3$, where the constant$a$is such that$0 < a < \frac{3}{2} \pi$.$\text{(i)}$Show that$a$satisfies the equation $$a=\frac{4-3 \cos \frac{1}{3} a}{\sin \frac{1}{3} a}$$$\text{(ii)}$Verify by calculation that$a$lies between$2.5$and 3.$\text{(iii)}$Use an iterative formula based on the equation in part$\text{(i)}$to calculate$a$correct to 3 decimal places. Give the result of each iteration to 5 decimal places.$[3]$Question 6 Code: 9709/33/O/N/19/9, Topic: Differential equations The variables$x$and$t$satisfy the differential equation$\displaystyle 5 \displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=(20-x)(40-x).$It is given that$x=10$when$t=0$.$\text{(i)}$Using partial fractions, solve the differential equation, obtaining an expression for$x$in terms of$t$.$[9]\text{(ii)}$State what happens to the value of$x$when$t$becomes large.$[1]\$

Worked solutions: P1, P3 & P6 (S1)

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