2019 Nov P2 Q6

Question 5

The graphs of $f(x)=-\frac{\displaystyle 1}{\displaystyle 2}\cos x$ and $g(x)=\sin (x+30^{\circ})$ , for the interval $x\in[0^{\circ}\,;\,180^{\circ}]$ , are drawn below. $A(130,9^{\circ}\,;\,0,33)$ is the approximate point of intersection of the two graphs.

$5.1$	Write down the period of $g$ .	$(1)$
$5.2$	Write down the amplitude of $f$ .	$(1)$
$5.3$	Determine the value of $f(180^{\circ})-g(180^{\circ})$	$(1)$
$5.4$	Use the graphs to determine the values of $x$ , in the interval $x\in[0^{\circ}\,;\,180^{\circ}]$ , for which:
$\quad 5.4.1$	$f(x-10^{\circ})=g(x-10^{\circ})$	$(1)$
$\quad 5.4.2$	$\sqrt{3}\sin x+\cos x \geq 1$	$(4)$
		$\textbf{[8]}$

Question 6

In the diagram, the graphs of

f(x)=\sin x-1

and

g(x)=\cos 2x

are drawn for the interval

x\in[-90^{\circ}\,;\,360^{\circ}]

. Graphs

f

and

g

intersect at

A

B(360^{\circ}\,;\,-1)

is a point on

f

6.1

Write down the range of

f

(2)

6.2

Write down the values of

x

in the interval

x\in[-90^{\circ}\,;\,360^{\circ}]

for which graph

f

is decreasing.

(2)

6.3

P

and

Q

are points on the graphs

g

and

f

respectively such that

PQ

is parallel to the

y

-axis. If

PQ

lies between

A

and

B

, determine the value(s) of

x

for which

PQ

will be a maximum.

(6)

\textbf{[10]}

In the diagram, the graphs of $f(x)=\sin x-1$ and $g(x)=\cos 2x$ are drawn for the interval $x\in[-90^{\circ}\,;\,360^{\circ}]$ . Graphs $f$ and $g$ intersect at $A$ . $B(360^{\circ}\,;\,-1)$ is a point on $f$ .

$6.1$	Write down the range of $f$ .	$(2)$
$6.2$	Write down the values of $x$ in the interval $x\in[-90^{\circ}\,;\,360^{\circ}]$ for which graph $f$ is decreasing.	$(2)$
$6.3$	$P$ and $Q$ are points on the graphs $g$ and $f$ respectively such that $PQ$ is parallel to the $y$ -axis. If $PQ$ lies between $A$ and $B$ , determine the value(s) of $x$ for which $PQ$ will be a maximum.	$(6)$
		$\textbf{[10]}$