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Name of student | JARODNOM | Date | |||

Adm. number | Year/grade | 1981 | Stream | Jarodnom | |

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 | 8 | 8 | 8 | 9 | 9 | 8 | 50 |

Score |

Question 1 Code: 9709/31/M/J/10/6, Topic: Numerical solutions of equations

The diagram shows a semicircle $A C B$ with centre $O$ and radius $r$. The angle $B O C$ is $x$ radians. The area of the shaded segment is a quarter of the area of the semicircle.

$\text{(i)}$ Show that $x$ satisfies the equation $[3]$

$$ x=\frac{3}{4} \pi-\sin x $$$\text{(ii)}$ This equation has one root. Verify by calculation that the root lies between $1.3$ and $1.5$. $[2]$

$\text{(iii)}$ Use the iterative formula

$$ \displaystyle x_{n+1}=\frac{3}{4} \pi-\sin x_{n} $$to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

Question 2 Code: 9709/32/M/J/10/6, Topic: Differentiation

The equation of a curve is

$$ x \ln y=2 x+1 $$$\text{(i)}$ Show that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{y}{x^{2}}$. $[4]$

$\text{(ii)}$ Find the equation of the tangent to the curve at the point where $y=1$, giving your answer in the form $a x+b y+c=0$. $[4]$

Question 3 Code: 9709/33/M/J/10/6, Topic: Numerical solutions of equations

The curve $\displaystyle y=\displaystyle\frac{\ln x}{x+1}$ has one stationary point.

$\text{(i)}$ Show that the $x$-coordinate of this point satisfies the equation

$$ x=\frac{x+1}{\ln x} $$and that this $x$-coordinate lies between 3 and 4. $[5]$

$\text{(ii)}$ Use the iterative formula

$$ \displaystyle x_{n+1}=\frac{x_{n}+1}{\ln x_{n}} $$to determine the $x$-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $[3]$

Question 4 Code: 9709/31/O/N/10/6, Topic: Complex numbers

The complex number $z$ is given by

$$ z=(\sqrt{3})+\mathrm{i} $$$\text{(i)}$ Find the modulus and argument of $z$. $[2]$

$\text{(ii)}$ The complex conjugate of $z$ is denoted by $z^{*}$. Showing your working, express in the form $x+\mathrm{i} y$, where $x$ and $y$ are real,

$\text{(a)}$ $2 z+z^{*}$,

$\text{(b)}$ $\displaystyle\frac{\mathrm{i} z^{*}}{z}$

$4$

$\text{(iii)}$ On a sketch of an Argand diagram with origin $O$, show the points $A$ and $B$ representing the complex numbers $z$ and $\mathrm{i} z^{*}$ respectively. Prove that angle $A O B=\frac{1}{6} \pi$. $[3]$

Question 5 Code: 9709/32/O/N/10/6, Topic: Complex numbers

Question 6 Code: 9709/33/O/N/10/6, Topic: Vectors

The straight line $l$ passes through the points with coordinates $(-5,3,6)$ and $(5,8,1)$. The plane $p$ has equation $2 x-y+4 z=9$.

$\text{(i)}$ Find the coordinates of the point of intersection of $l$ and $p$. $[4]$

$\text{(ii)}$ Find the acute angle between $l$ and $p$. $[4]$