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Name of student | Date | ||||

Adm. number | Year/grade | Stream | |||

Subject | Pure Mathematics 3 (P3) | Variant(s) | P33 | ||

Start time | Duration | Stop time |

Qtn No. | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|

Marks | 4 | 7 | 6 | 8 | 9 | 9 | 43 |

Score |

Question 1 Code: 9709/33/O/N/20/2, Topic: Complex numbers

On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $|z| \geqslant 2$ and $|z-1+\mathrm{i}| \leqslant 1$. $[4]$

Question 2 Code: 9709/33/O/N/11/3, Topic: Trigonometry

$\text{(i)}$ Express $8 \cos \theta+15 \sin \theta$ in the form $R \cos (\theta-\alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$. Give the value of $\alpha$ correct to 2 decimal places. $[3]$

$\text{(ii)}$ Hence solve the equation $8 \cos \theta+15 \sin \theta=12$, giving all solutions in the interval $0^{\circ}< \theta <360^{\circ}$. $[4]$

Question 3 Code: 9709/33/O/N/12/4, Topic: Differential equations

The variables $x$ and $y$ are related by the differential equation

$$ \left(x^{2}+4\right) \displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=6 x y. $$It is given that $y=32$ when $x=0$. Find an expression for $y$ in terms of $x$. $[6]$

Question 4 Code: 9709/33/O/N/12/6, Topic: Numerical solutions of equations

The diagram shows the curve $y=x^{4}+2 x^{3}+2 x^{2}-4 x-16$, which crosses the $x$-axis at the points $(\alpha, 0)$ and $(\beta, 0)$ where $\alpha<\beta$. It is given that $\alpha$ is an integer.

$\text{(i)}$ Find the value of $\alpha$. $[2]$

$\text{(ii)}$ Show that $\beta$ satisfies the equation $x=\sqrt[3]{(8-2 x)}$. $[3]$

$\text{(iii)}$ Use an iteration process based on the equation in part $\text{(ii)}$ to find the value of $\beta$ correct to 2 decimal places. Show the result of each iteration to 4 decimal places. $[3]$

Question 5 Code: 9709/33/O/N/18/7, Topic: Differentiation, Integration

The diagram shows the curve $y=5 \sin ^{2} x \cos ^{3} x$ for $0 \leqslant x \leqslant \frac{1}{2} \pi$, and its maximum point $M$. The shaded region $R$ is bounded by the curve and the $x$-axis.

$\text{(i)}$ Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places. $[5]$

$\text{(ii)}$ Using the substitution $u=\sin x$ and showing all necessary working, find the exact area of $R$. $[4]$

Question 6 Code: 9709/33/O/N/19/7, Topic: Vectors

The plane $m$ has equation $x+4 y-8 z=2$. The plane $n$ is parallel to $m$ and passes through the point $P$ with coordinates $(5,2,-2)$.

$\text{(i)}$ Find the equation of $n$, giving your answer in the form $a x+b y+c z=d$. $[2]$

$\text{(ii)}$ Calculate the perpendicular distance between $m$ and $n$. $[3]$

$\text{(iii)}$ The line $l$ lies in the plane $n$, passes through the point $P$ and is perpendicular to $O P$, where $O$ is the origin. Find a vector equation for $l$. $[4]$