Geometry and trigonometry: exam review Exam review: second paper

A road climbs a hillside from A to C. ABC is a right-angled triangle, with the right angle at B. The horizontal distance AB is 5.6 km and the height BC is 1.9 km. [7]

1. (a) Find the size of angle BAC. [2]
2. (b) A second road has the same horizontal distance AB and rises at an angle BAC = 12°. Find its height BC. [2]
3. (c) Two straight roads on a map have equations \(y = 2x\) and \(y = -1.5x\). Find the acute angle between the roads. [3]

A triangular sail PQR has PQ = 5.3 m, PR = 6.8 m and angle QPR = 58°. [7]

1. (a) Find the length of QR. [3]
2. (b) Find the size of angle PQR. [2]
3. (c) Find the area of triangle PQR. [2]

A shipping crate ABCDEFGH is a cuboid with AB = 1.2 m, BC = 0.9 m and CG = 0.7 m. [5]

1. (a) Find the length of AG. [2]
2. (b) Find the angle between AG and the base ABCD. [3]

A slice of cake is a sector of a circle with radius 14 cm and angle 0.9 radians at the centre. [7]

1. (a) Write 0.9 radians in degrees, and write 135° in radians as a multiple of \(\pi\). [3]
2. (b) Find the arc length of the sector. [2]
3. (c) Find the area of the sector. [2]

A garden bed is a sector OAB of a circle with centre O, radius 6.8 m and angle AOB = 110°. [5]

1. (a) Find the length of the arc AB. [2]
2. (b) Find the area of the segment between the chord AB and the arc AB. [3]

In triangle PQR, angle P = 35°, QR = 8.4 cm and PR = 12.5 cm. [6]

1. (a) Calculate 12.5 sin 35° and hence explain why there are two possible triangles. [2]
2. (b) Find the two possible values of angle PQR. [4]

A surveyor stands at P and measures the angle of elevation of the top T of a wind turbine as 18°. He walks 60 m directly towards the turbine to Q and measures the angle of elevation as 29°. His eyes are 1.7 m above the ground. N is the point at the height of his eyes directly below T. [5]

1. (a) Find the size of angle PTQ. [1]
2. (b) Use the sine rule in triangle PQT to find QT. [2]
3. (c) Find the height of the turbine above the ground. [2]

A triangular sail PQR has PQ = 11 m, PR = 8 m and area 35 m². Angle QPR is acute. The point S lies on QR with QS = 6 m. [7]

1. (a) Find the size of angle QPR. [2]
2. (b) Find the length of QR. [2]
3. (c) Find the length of PS. [3]

The angle \(\theta\) is obtuse and \(\sin\theta = \frac{8}{17}\). [4]

1. (a) Find the exact value of \(\cos\theta\). [2]
2. (b) Find the exact value of \(\tan\theta\). [1]
3. (c) Find the size of \(\theta\), in degrees, to the nearest degree. [1]

A goat is tied to the corner P of a rectangular barn, 14 m by 9 m, standing on flat grass. The diagram shows the barn from above. The goat cannot go into the barn. [12]

1. (a) The goat is tied at P by a rope of length 8 m. Find the area of grass the goat can reach. [3]
2. (b) The rope is now 20 m long. After the rope has passed the two corners of the shed, find the radius of each of the two quarter circles the goat can reach beyond them. [2]
3. (c) Find the total area of grass the goat can reach with the 20 m rope. [3]
4. (d) The rope is moved to M, the midpoint of the 14 m wall, and shortened to 11 m. Find the area of grass the goat can now reach. [4]

A rectangular paddock ABCD has AB = 22 m and AD = 12 m. A goat is tied at A by a rope of length 17 m. The paddock has a fence on all four sides, and the rope is long enough to reach the side DC. [12]

1. (a) The rope just reaches the side DC at E, and AE = 17 m. Find the length DE and the size of angle EAB. [3]
2. (b) The goat can reach the region ADE and the sector EAF, where F is on AB and AF = AE. Find the total area of the region the goat can reach. [4]
3. (c) Find the percentage of the field that the goat can reach. [2]
4. (d) The rope length r m is changed, where 12 < r < 22. The goat can then reach half of the field. Write down an equation for r and solve it using technology. [3]

A hiker leaves camp P and walks 12.5 km on a bearing of 112° to Q. The hiker then walks 9.6 km on a bearing of 215° to R. [12]

1. (a) Find the size of angle PQR. [2]
2. (b) Find the distance PR. [3]
3. (c) Find the size of angle QPR. [3]
4. (d) Find the bearing of P from R. Give your answer to the nearest degree. [3]
5. (e) Write down the bearing of Q from R. [1]

A tent VABCD is in the shape of a pyramid. The base ABCD is a horizontal square of side 10 m and each sloping edge is 13 m. M is the centre of the base, and V is directly above M. [12]

1. (a) Find the length of AC. [2]
2. (b) Find the height VM of the pyramid. [2]
3. (c) Find the angle between the edge VA and the base ABCD. [2]
4. (d) N is the midpoint of AB. Find the angle between the face VAB and the base. [2]
5. (e) Find the volume of the pyramid. [2]
6. (f) Find the total area of the four triangular faces. [2]

The height of a pod on a wheel, \(h\) metres, \(t\) minutes after the wheel starts is modelled by \(h = 5.5 + 3\sin(0.45t)\), for \(0 \le t \le 12\), where the angle is in radians. [8]

1. (a) Write down the greatest height and the least height. [2]
2. (b) Find the period of the model. [1]
3. (c) Find the times, in the first 12 minutes, when the height is 6.5 m. [3]
4. (d) Find the length of time, in the first 12 minutes, for which the height is at least 6.5 m. [2]

A paper cup is a cone with its point at the bottom. The radius of the top is 6 cm and the perpendicular height is 15 cm. The cup is held with its axis vertical and filled with water to a depth of 10 cm. [11]

1. (a) Find the volume of the cup when it is full, in cm³ and in litres. [2]
2. (b) Find the radius of the surface of the liquid. [2]
3. (c) Find the volume of the liquid, and write it as a percentage of the volume of the full cup. [3]
4. (d) The liquid is poured into a cylinder with radius 5 cm. Find the depth of the liquid in the cylinder. [2]
5. (e) Instead, the liquid is frozen into a solid hemisphere. Find the radius of the hemisphere. [2]