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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 |

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 $C$. $[3]$