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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 4 5 6 Total
Marks 9 8 8 9 9 8 51
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/10/7, Topic: Complex numbers The complex number$2+2 \mathrm{i}$is denoted by$u$.$\text{(i)}$Find the modulus and argument of$u$.$[3]\text{(ii)}$Sketch an Argand diagram showing the points representing the complex numbers 1 , i and$u$. Shade the region whose points represent the complex numbers$z$which satisfy both the inequalities$|z-1| \leqslant|z-\mathrm{i}|$and$|z-u| \leqslant 1[4]\text{(iii)}$Using your diagram, calculate the value of$|z|$for the point in this region for which$\arg z$is least.$[3]$Question 2 Code: 9709/32/M/J/10/7, Topic: Differentiation The variables$x$and$t$are related by the differential equation $$\mathrm{e}^{2 t} \displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=\cos ^{2} x$$ where$t \geqslant 0$. When$t=0, x=0$.$\text{(i)}$Solve the differential equation, obtaining an expression for$x$in terms of$t$.$[6]\text{(ii)}$State what happens to the value of$x$when$t$becomes very large.$[1]\text{(iii)}$Explain why$x$increases as$t$increases.$[1]$Question 3 Code: 9709/33/M/J/10/7, Topic: Trigonometry, Integration$\text{(i)}$Prove the identity$\cos 3 \theta \equiv 4 \cos ^{3} \theta-3 \cos \theta$.$[4]\text{(ii)}$Using this result, find the exact value of$[4]$$$\displaystyle\int_{\frac{1}{3} \pi}^{\frac{1}{2} \pi} \cos ^{3} \theta \mathrm{d} \theta$$ Question 4 Code: 9709/31/O/N/10/7, Topic: Vectors With respect to the origin$O$, the points$A$and$B$have position vectors given by$\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}$and$\overrightarrow{O B}=3 \mathbf{i}+4 \mathbf{j}.$The point$P$lies on the line$A B$and$O P$is perpendicular to$A B$.$\text{(i)}$Find a vector equation for the line$A B$.$[1]\text{(ii)}$Find the position vector of$P$.$[4]\text{(iii)}$Find the equation of the plane which contains$A B$and which is perpendicular to the plane$O A B$, giving your answer in the form$a x+b y+c z=d$.$[4]$Question 5 Code: 9709/32/O/N/10/7, Topic: Vectors Question 6 Code: 9709/33/O/N/10/7, Topic: Integration, Numerical solutions of equations$\text{(i)}$Given that$\displaystyle\int_{1}^{a} \frac{\ln x}{x^{2}} \mathrm{~d} x=\frac{2}{5}$, show that$a=\frac{5}{3}(1+\ln a)$.$[5]\text{(ii)}$Use an iteration formula based on the equation$a=\frac{5}{3}(1+\ln a)$to find the value of$a$correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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