P1: TrigonometryEdexcel International A Level Maths: Topic test

1. 1
   A sector of a circle has radius $12$ cm and its angle at the centre is $\frac{5\pi}{6}$ radians.

   (a)

   Find the length of the arc of the sector.

   [1 mark]

   • A$60\pi$ cm
   • B$\frac{5\pi}{72}$ cm
   • C$10\pi$ cm
   • D$120\pi$ cm

   (b)

   Find the area of the sector.

   [1 mark]

   • A$60\pi$ cm$^2$
   • B$120\pi$ cm$^2$
   • C$10\pi$ cm$^2$
   • D$5\pi$ cm$^2$

   (c)

   Find the perimeter of the sector, giving your answer in the form $a+b\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=3\cos2x$ for $0\le x\le2\pi$, where $x$ is in radians.

   (a)

   What is the period of the curve $C$?

   [1 mark]

   • A$2\pi$
   • B$\pi$
   • C$4\pi$
   • D$\frac{\pi}{2}$

   (b)

   How many times does $C$ meet the line $y=2$?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Find the smallest positive value of $x$ for which $3\cos2x=2$, giving your answer in radians to $3$ significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle $KLM$, $KL=7.2$ cm, $LM=9.5$ cm and angle $KLM=64^\circ$.

   (a)

   Find the length of $KM$, giving your answer to $3$ significant figures.

   [3 marks]

   (b)

   Find the area of the triangle and the size of angle $KML$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden is in the shape of a sector $OAB$ of a circle with centre $O$ and radius $20$ m. The arc $AB$ has length $26$ m and angle $AOB=\theta$ radians.

   (a)

   (i) Find the value of $\theta$.
   (ii) Find the area of the garden.
   (iii) Find the area of the segment bounded by the chord $AB$ and the arc $AB$, giving your answer to $3$ significant figures.

   [6 marks]

   (b)

   A straight path runs along the chord $AB$.
   (i) Find the length of the path, giving your answer to $3$ significant figures.
   (ii) Find the perimeter of the segment bounded by the chord $AB$ and the arc $AB$.
   The radius stays $20$ m but $\theta$ can vary between $0$ and $\pi$.
   (iii) State the value of $\theta$ for which the area of triangle $OAB$ is greatest, and find this greatest area.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In triangle $XYZ$, angle $YXZ=48^\circ$, angle $XYZ=75^\circ$ and $XZ=14$ cm.

   (a)

   Find the size of angle $XZY$.

   [1 mark]

   • A$123^\circ$
   • B$27^\circ$
   • C$42^\circ$
   • D$57^\circ$

   (b)

   Find the length of $YZ$.

   [1 mark]

   • A$18.2$ cm
   • B$10.8$ cm
   • C$12.4$ cm
   • D$10.4$ cm

   (c)

   Hence find the area of triangle $XYZ$, giving your answer to $3$ significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The curve $C$ has equation $y=\cos\left(x+\frac{\pi}{4}\right)$ for $0\le x\le2\pi$, where $x$ is in radians.

   (a)

   Find the value of $y$ where $C$ meets the $y$-axis.

   [1 mark]

   • A$\frac{\sqrt2}{2}$
   • B$\frac12$
   • C$1$
   • D$0$

   (b)

   Which of the following transforms the curve $y=\cos x$ onto $C$?

   [1 mark]

   • Aa translation of $\frac{\pi}{4}$ in the positive $x$-direction
   • Ba stretch in the $x$-direction with scale factor $\frac14$
   • Ca translation of $\frac{\pi}{4}$ in the negative $x$-direction
   • Da translation of $\frac{\pi}{4}$ in the positive $y$-direction

   (c)

   Find the values of $x$ at which $C$ meets the $x$-axis.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   In triangle $DEF$, $DE=9$ cm, $DF=6$ cm and angle $DEF=38^\circ$.

   (a)

   Show that there are two possible sizes of angle $DFE$, and find them.

   [3 marks]

   (b)

   Find the two possible areas of triangle $DEF$, giving each answer to $3$ significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The depth $d$ metres of water at a harbour entrance, $t$ hours after midnight, is modelled by $d=6+2\sin\left(\frac{\pi t}{6}\right)$ for $0\le t\le24$, where the angle is in radians.

   (a)

   (i) Find the maximum depth predicted by the model and the first time at which it occurs.
   (ii) Find the period of the model.
   (iii) Find the depth at $t=1$.
   (iv) Find the times in the first $12$ hours at which the depth is $7$ m.

   [6 marks]

   (b)

   A boat can enter the harbour only when the depth is at least $7$ m.
   (i) Find the minimum depth predicted by the model and the first time at which it occurs.
   (ii) Find the total length of time in the first $12$ hours for which the boat can enter.
   (iii) Find the times between $t=12$ and $t=24$ at which the boat can enter.

   [6 marks]

   Total for question 8: 12 marks

End of questions