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Question 1

HLPaper 2

A graph has the degree sequence:

\{5,5,4,3,3,2,2,2\}

1.

Is this a valid simple graph? Justify using the Havel-Hakimi algorithm.

[4]

2.

How many edges does the graph contain?

[1]

Question 2

HLPaper 1

Prove the identity:

\sin ^{2} \theta-\cos ^{2} \theta=-\cos 2 \theta

1.

\sin ^{2} \theta-\cos ^{2} \theta=-\cos 2 \theta

[4]

Question 3

HLPaper 1

A line L has the vector equation

\vec{r}=\left[\begin{array}{c} 1 \\ 2 \\ -3 \end{array}\right]+\lambda\left[\begin{array}{c} 2 \\ -1 \\ 4 \end{array}\right]

1.

Find the coordinates of the point on line L that is closest to the point $\mathrm{A}=(3$ , $-2,1)$ .

[6]

Question 4

SLPaper 2

A circular garden has a radius of 7 m . A path runs along a quarter of the perimeter.

1.

Find the length of the path.

[3]

Question 5

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

1.

Find the equation of line $A B$ and calculate the length of the segment $P Q$ .

[5]

Question 6

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2 x+y=5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

1.

Calculate the ratio $P A: A Q$ .

[7]

Question 7

HLPaper 1

Given vectors $\vec{a}=\left[\begin{array}{c}4 \\ -2\end{array}\right]$ and $\vec{b}=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ , find the vector $\vec{r}$ such that $\vec{r}$ is perpendicular to both $\vec{a}$ and $\vec{b}-\vec{a}$ .

1.

Given vectors $\vec{a}=\left[\begin{array}{c}4 \\ -2\end{array}\right]$ and $\vec{b}=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ , find the vector $\vec{r}$ such that $\vec{r}$ is perpendicular to both $\vec{a}$ and $\vec{b}-\vec{a}$ .

[5]

Question 8

HLPaper 1

A line L is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

1.

Find the value of $\lambda$ forwhichthepointonthelineliesontheplanex $+y+z=5$ .

[4]

Question 9

SLPaper 1

Given that the points $A(2,1), B(t, 2 t)$ , and $C\left(t^{2}, 5 t-3\right)$ are collinear, find all possible values of $t$ .

1.

Given that the points $A(2,1), B(t, 2 t)$ , and $C\left(t^{2}, 5 t-3\right)$ are collinear, find all possible values of $t$ .

[7]

Question 10

SLPaper 2

The arc length of a sector is 20 cm and the radius is r .

1.

If the central angle is $100^{\circ}$ , find r.

[5]

Geometry and Trigonometry Questionbank