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### ANNACUH

#### Cambridge International AS and A Level

 Name of student ANNACUH Date Adm. number Year/grade Annacuh Stream Annacuh 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 10 9 9 9 10 56
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/15/8, Topic: Complex numbers The complex number$w$is defined by$\displaystyle w=\frac{22+4 \mathrm{i}}{(2-\mathrm{i})^{2}}$.$\text{(i)}$Without using a calculator, show that$w=2+4 \mathrm{i}$.$[3]\text{(ii)}$It is given that$p$is a real number such that$\frac{1}{4} \pi \leqslant \arg (w+p) \leqslant \frac{3}{4} \pi$. Find the set of possible values of$p$.$[3]\text{(iii)}$The complex conjugate of$w$is denoted by$w^{*}$. The complex numbers$w$and$w^{*}$are represented in an Argand diagram by the points$S$and$T$respectively. Find, in the form$|z-a|=k$, the equation of the circle passing through$S, T$and the origin.$[3]$Question 2 Code: 9709/32/M/J/15/8, Topic: Algebra Let$\displaystyle \mathrm{f}(x)=\frac{5 x^{2}+x+6}{(3-2 x)\left(x^{2}+4\right)}\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^{2}$.$[5]$Question 3 Code: 9709/33/M/J/15/8, Topic: Complex numbers Two planes have equations$x+3 y-2 z=4$and$2 x+y+3 z=5$. The planes intersect in the straight line$l$.$\text{(i)}$Calculate the acute angle between the two planes.$[4]\text{(ii)}$Find a vector equation for the line$l$.$[6]$Question 4 Code: 9709/31/O/N/15/8, Topic: Differential equations The variables$x$and$\theta$satisfy the differential equation $$\displaystyle\frac{\mathrm{d} x}{\mathrm{~d} \theta}=(x+2) \sin ^{2} 2 \theta$$ and it is given that$x=0$when$\theta=0$. Solve the differential equation and calculate the value of$x$when$\theta=\frac{1}{4} \pi$, giving your answer correct to 3 significant figures.$[9]$Question 5 Code: 9709/32/O/N/15/8, Topic: Differential equations Question 6 Code: 9709/33/O/N/15/8, Topic: Vectors A plane has equation$4 x-y+5 z=39$. A straight line is parallel to the vector$\mathbf{i}-3 \mathbf{j}+4 \mathbf{k}$and passes through the point$A(0,2,-8)$. The line meets the plane at the point$B$.$\text{(i)}$Find the coordinates of$B$.$[3]\text{(ii)}$Find the acute angle between the line and the plane.$[4]\text{(iii)}$The point$C$lies on the line and is such that the distance between$C$and$B$is twice the distance between$A$and$B$. Find the coordinates of each of the possible positions of the point$C$.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

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