Question 1

Let $f(x)=\frac{sin3x}{sinx}-\frac{cos3x}{cosx}$.
For what values of x does f(x) not exist?

Accepted Answer

$x=\frac{nπ}{2},n∈Z$

Answer

$x=2nπ,n∈Z$

Answer

$x=nπ,n∈Z$

Answer

$x=\frac{3nπ}{2},n∈Z$

Question 2

The triangle ABC is equilateral of side 3 cm. The point D lies on [BC] such that BD = 1 cm.
Find cosDAC.

Accepted Answer

$\frac{2}{\sqrt{7}}$

Answer

$\frac{2}{7}$

Answer

$\frac{\sqrt2}{7}$

Answer

$\sqrt{\frac{2}{7}}$

Question 3

From a vertex of an equilateral triangle of side 2x, a circular arc is drawn to divide the triangle into two regions, as shown in the diagram.
Given that the areas of the two regions are equal, find the radius of the arc in terms of x.

Accepted Answer

$r=x\sqrt{\frac{3\sqrt{3}}{π} }$

Answer

$r=x\sqrt{\frac{6\sqrt{3}}{π} }$

Answer

$r=x\sqrt{\frac{3}{π} }$

Answer

$r=x\sqrt{\frac{6}{π} }$

Question 4

The first three terms of a geometric sequence are $sinx, sin2x$ and  $4sinxcos^2 x, \frac{-π}{2}<x<\frac{π}{2}$.
Find the common ratio r.

Accepted Answer

$2cosx$

Answer

$\frac{1}{2cosx}$

Answer

$sin2$

Answer

$sinxcosx$

Question 5

The first three terms of a geometric sequence are $sinx, sin2x$ and  $4sinxcos^2 x, \frac{-π}{2}<x<\frac{π}{2}$. Consider $x=arccos(\frac{1}{4}), x>0$.
The sum to infinity of this series is:

Accepted Answer

$\frac{\sqrt{15}}{2}$

Answer

$\frac{15}{2}$

Answer

$\frac{\sqrt{2}}{15}$

Answer

$\frac{2}{15}$

Question 6

The logo, for a company that makes chocolate, is a sector of a circle of radius 2 cm, shown as shaded in the diagram. The area of the logo is $3π cm^2$.
Find, in radians, the value of the angle $θ$, as indicated on the diagram.

Accepted Answer

$θ=\frac{π}{2}$

Answer

$θ=\frac{2π}{3}$

Answer

$θ=π$

Answer

$θ=\frac{3π}{4}$

Question 7

The logo, for a company that makes chocolate, is a sector of a circle of radius 2 cm, shown as shaded in the diagram. The area of the logo is $3π cm^2$. Find the total length of the perimeter of the logo

Accepted Answer

$3π+4 $

Answer

$\frac{3π}{2}+4 $

Answer

$2π+4 $

Answer

$3.5π+4 $

Question 8

The diagram below shows a curve with equation $y=1+ksinx$ , defined for $0≤x≤3π$ .
The point $A(\frac{π}{6},-2)$ lies on the curve and $B(a, b)$ is the maximum point.  Find the value of $k$.

Accepted Answer

$-6$

Answer

$-4$

Answer

$-5$

Answer

$-7$

Question 9

The diagram below shows a curve with equation $y=1+ksinx$ , defined for $0≤x≤3π$ .
The point $A(\frac{π}{6},-2)$ lies on the curve and $B(a, b)$ is the maximum point.   Find the values of $a$ and $b$

Accepted Answer

$a=\frac{3π}{2}$ and $b=7$

Answer

$a=π$ and $b=7$

Answer

$a=\frac{π}{2}$ and $b=6$

Answer

$a=\frac{3π}{2}$ and $b=6$

Question 10

The diagram below shows the boundary of the cross-section of a water channel.
The equation that represents this boundary is $y=16sec(\frac{πx}{36})-32$ where x and y are both measured in cm. The top of the channel is level with the ground and has a width of 24 cm. The maximum depth of the channel is 16 cm. Find the width of the water surface in the channel when the water depth is 10 cm.
Give your answer in the form $aarccosb$ where $a, b∈R$

Accepted Answer

$x=\frac{72}{π} arccos (\frac{8}{13})$

Answer

$x=\frac{72}{π} arccos (\frac{13}{8})$

Answer

$x=\frac{72}{π} arccos (\frac{8}{21})$

Answer

$x=\frac{72}{π} arccos (\frac{21}{8})$

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