Ch15-Trigonometry...Page: ABBASI MOHAMMED ASIM [email protected] Ch15-Trigonometry 1. NOT TO...

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ABBASI MOHAMMED ASIM Page: 1 [email protected]

Ch15-Trigonometry

1.

NOT TO SCALE

A shop has a wheelchair ramp to its entrance from the pavement. The ramp is 3.17 metres long and is inclined at 5° to the horizontal. Calculate the height, h metres, of the entrance above the pavement. Show all your working.

Answer ……….………………….…… m [2]

2. A square ABCD, of side 8 cm, has another square, PQRS, drawn inside it. P, Q, R and S are at the midpoints of each side of the square ABCD, as shown in the

diagram.

NOT TO SCALE

(a) Calculate the length of PQ.

Answer (a) ……….………………….…… cm [2]

(b) Calculate the area of the square PQRS.

Answer (b) ……….………………….…… cm2 [1]

pavem ent

entrance

5°

3.17 m

h m

A B

D C

P

S

R

Q

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3. A plane flies from Auckland (A) to Gisborne (G) on a bearing of 115°. The plane then flies on to Wellington (W). Angle AGW = 63°.

NOT TO SCALE

(a) Calculate the bearing of Wellington from Gisborne.

Answer (a) ………..………………….…… [2]

(b) The distance from Wellington to Gisborne is 400 kilometres. The distance from Auckland to Wellington is 410 kilometres. Calculate the bearing of Wellington from Auckland.

Answer (b) ………..………………….…… [4]

4.

NOT TO SCALE

A, B, C and D lie on a circle, centre O, radius 8 cm. AB and CD are tangents to a circle, centre O, radius 4 cm. ABCD is a rectangle.

(a) Calculate the distance AE. Answer (a) AE = …………….………… cm [2]

(b) Calculate the shaded area.

Answer (b) ………………….………… cm2 [3]

115°

63°

A

G

N orth

N orth

W

400 km

410 km

A B

D C

O

E

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5. In triangle ABC, AB = 6 cm, AC = 8 cm and BC = 12 cm. Angle ACB = 26.4°. Calculate the area of the triangle ABC.

NOT TO SCALE

Answer ……………………………… cm2 [2]

6.

NOT TO SCALE

The right-angled triangle in the diagram has sides of length 7x cm, 24x cm and 150 cm.

(a) Show that x2 = 36 [2]

(b) Calculate the perimeter of the triangle. Answer (b) ….………………… cm [1]

7.

NOT TO SCALE

ABCD is a trapezium.

(a) Find the area of the trapezium in terms of x and simplify your answer.

Answer (a) …………………………. cm2 [2]

(b) Angle BCD = y°. Calculate the value of y. Answer (b) y = ………..………… [2]

A

B

C

6 cm

8 cm

12 cm26.4°

150 cm

24x cm

7x cm

5x cm

13x cm

y°

A B

CD

17x cm 12x cm

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8.

NOT TO SCALE

The diagram shows three touching circles. A is the centre of a circle of radius x centimetres. B and C are the centres of circles of radius 3.8 centimetres. Angle ABC = 70°. Find the value of x.

Answer x = ………….………… [3]

9.

NOT TO SCALE

The diagram shows a trapezium ABCD. AB = 12 cm, DC = 9 cm and the perpendicular distance between these parallel sides

is 7 cm. AD = BC.

(a) Approximately halfway down your page, draw a line AB of length 12 cm. [1]

(b) Using a straight edge and compasses only, construct the perpendicular bisector of AB.

[2]

(c) Complete an accurate drawing of the trapezium ABCD. [2]

(d) Measure angle ABC, giving your answer correct to the nearest degree. [1]

70°

A

B C

D

A B

C9 cm

12 cm

7 cm

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(e) Use trigonometry to calculate angle ABC. Show all your working and give your answer correct to 1 decimal place.

[2]

(f) On your diagram,

(i) draw the locus of points inside the trapezium which are 5 cm from D, [1]

(ii) using a straight edge and compasses only, construct the locus of points equidistant from DA and from DC,

[2] (iii) shade the region inside the trapezium containing points which are less

than 5 cm from D and nearer to DA than to DC. [1]

10.

NOT TO SCALE

OABCDE is a regular hexagon. With O as origin the position vector of C is c and the position vector of D is d.

(a) Find, in terms of c and d,

(i) [1]

(ii) [2]

(iii) the position vector of B. [2]

(b) The sides of the hexagon are each of length 8 cm.

Calculate

(i) the size of angle ABC, [1]

(ii) the area of triangle ABC, [2]

(iii) the length of the straight line AC, [3]

(iv) the area of the hexagon. [3]

C

D

E

O

A

B

cd

,DC

,OE

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11. NOT TO SCALE

The diagram shows a pencil of length 18 cm. It is made from a cylinder and a cone. The cylinder has diameter 0.7 cm and length 16.5 cm. The cone has diameter 0.7 cm and length 1.5 cm.

(a) Calculate the volume of the pencil.

[The volume, V, of a cone of radius r and height h is given by V =

[3]

(b)

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Twelve of these pencils just fit into a rectangular box of length 18 cm, width w cm and height x cm.

The pencils are in 2 rows of 6 as shown in the diagram.

(i) Write down the values of w and x. [2]

(ii) Calculate the volume of the box. [2]

(iii) Calculate the percentage of the volume of the box occupied by the pencils.

(c) Showing all your working, calculate

(i) the slant height, l, of the cone, [2]

(ii) the total surface area of one pencil, giving your answer correct to 3 significant figures.

[The curved surface area, A, of a cone of radius r and slant height l is given by A = πrl.] [6]

0.7 cm

16.5 cm 1.5 cm

h

l

.3

1 2hr

18 cm

w cm

x cm

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12.

NOT TO SCALE

The quadrilateral PQRS shows the boundary of a forest. A straight 15 kilometre road goes due East from P to R.

(a) The bearing of S from P is 030° and PS = 7 km.

(i) Write down the size of angle SPR. [1]

(ii) Calculate the length of RS. [4]

(b) Angle RPQ = 55° and QR = 14 km.

(i) Write down the bearing of Q from P. [1]

(ii) Calculate the acute angle PQR. [3]

(iii) Calculate the length of PQ. [3]

(c) Calculate the area of the forest, correct to the nearest square kilometre. [4]

30°

55°

N orth

P R

Q

S

7 km

15 km

14 km

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13.

NOT TO SCALE

The diagram shows a pyramid on a rectangular base ABCD, with AB = 6 cm and AD = 5 cm.

The diagonals AC and BD intersect at F. The vertical height FP = 3 cm.

(a) How many planes of symmetry does the pyramid have? [1]

(b) Calculate the volume of the pyramid.

[The volume of a pyramid is × area of base × height.] [2]

(c) The mid-point of BC is M. Calculate the angle between PM and the base. [2]

(d) Calculate the angle between PB and the base. [4]

(e) Calculate the length of PB. [2]

14.

NOT TO SCALE

The diagram shows a pyramid on a horizontal rectangular base ABCD. The diagonals of ABCD meet at E. P is vertically above E. AB = 8 cm, BC = 6 cm and PC = 13 cm.

F

P

CD

BA

5 cm

6 cm

3 cm

M

3

1

8 cm

6 cm

13 cm

A B

CD

E

P

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(a) Calculate PE, the height of the pyramid. [3]

(b) Calculate the volume of the pyramid.

[The volume of a pyramid is given by × area of base × height.]

[2]

(c) Calculate angle PCA.

(d) M is the mid-point of AD and N is the mid-point of BC. Calculate angle MPN.

[3]

(e) (i) Calculate angle PBC. [2]

(ii) K lies on PB so that BK = 4 cm. Calculate the length of KC.

[3]

15.

NOT TO SCALE

The diagram shows a right-angled triangle. The lengths of the sides are given in terms of y.

(i) Show that 2y2 – 8y – 3 = 0. [3]

(ii) Solve the equation 2y2 – 8y – 3 = 0, giving your answers to 2 decimal places. [4]

(iii) Calculate the area of the triangle. [2]

3

1

y

y + 2

2 – 1y

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16.

NOT TO SCALE

The diagram shows the positions of four cities in Africa, Windhoek (W), Johannesburg (J), Harari (H) and Lusaka (L).

WL = 1400 km and WH = 1600 km. Angle LWH = 13°, angle HWJ = 36° and angle WJH = 95°.

(a) Calculate the distance LH. [4]

(b) Calculate the distance WJ. [4]

(c) Calculate the area of quadrilateral WJHL. [3]

(d) The bearing of Lusaka from Windhoek is 060°. Calculate the bearing of

(i) Harari from Windhoek, [1]

(ii) Windhoek from Johannesburg. [1]

(e) On a map the distance between Windhoek and Harari is 8 cm. Calculate the scale of the map in the form 1 : n. [2]

1400 km

1600 km

13°

36°

95°

L

H

J

W

N orth

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17.

NOT TO SCALE

A, B, C and D lie on a circle. AC and BD intersect at X. Angle ABX = 55° and angle AXB = 92°. BX = 26.8 cm, AX = 40.3 cm and XC = 20.1 cm.

(i) Calculate the area of triangle AXB You must show your working.

[2]

(ii) Calculate the length of AB. You must show your working.

[3]

(iii) Write down the size of angle ACD. Give a reason for your answer. [2]

(iv) Find the size of angle BDC. [1]

(v) Write down the geometrical word which completes the statement

“Triangle AXB is ___________ to triangle DXC.” [1]

(vi) Calculate the length of XD. You must show your working.

[2]

A

X

B

D

C

55

92

40.3 cm

20.1 cm

26.8 cm

20.1 cm

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18.

NOT TO SCALE

OBCD is a rhombus with sides of 25 cm. The length of the diagonal OC is 14 cm.

(a) Show, by calculation, that the length of the diagonal BD is 48 cm. [3]

(b) Calculate, correct to the nearest degree,

(i) angle BCD, [2]

(ii) angle OBC. [1]

(c) = 2p and = 2q. Find, in terms of p and q,

(i) , [1]

(ii) . [1]

(d) BE is parallel to OC and DCE is a straight line.

Find, in its simplest form, in terms of p and q. [2]

(e) M is the mid-point of CE.

Find, in its simplest form, in terms of p and q. [2]

EB

O C

D

14 cm

25 cmM

DB OC

OB

OD

OE

OM

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(f) O is the origin of a co-ordinate grid. OC lies along the x-axis and q =

( is vertical and | | = 48.) Write down as column vectors

(i) p, [1]

(ii) . [2]

(g) Write down the value of | |. [1]

19.

NOT TO SCALE

The diagram above shows the net of a pyramid.

The base ABCD is a rectangle 8 cm by 6 cm.

All the sloping edges of the pyramid are of length 7 cm.

M is the mid-point of AB and N is the mid-point of BC.

(a) Calculate the length of

(i) QM, [2]

(ii) RN. [1]

(b) Calculate the surface area of the pyramid. [2]

.0

7

DB DB

BC

DE

A B

D C

S

Q

RP

M

N

8 cm

6 cm

7 cm

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(c)

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The net is made into a pyramid, with P, Q, R and S meeting at P.

The mid-point of CD is G and the mid-point of DA is H.

The mid-point of CD is G and the mid-point of DA is H.

The diagonals of the rectangle ABCD meet at x.

(i) Show that the height, PX, of the pyramid is 4.90 cm, correct to 2 decimal places.

[2]

(ii) Calculate angle PNX. [2]

(iii) Calculate angle HPN. [2]

(iv) Calculate the angle between the edge PA and the base ABCD. [3]

(v) Write down the vertices of a triangle which is a plane of symmetry of the pyramid.

[1]

A B

D C

P

X

G

N

M

H

8 cm

6 cm

7 cm

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20.

NOT TO SCALE

The diagram shows the plan of a garden.

The garden is a trapezium with AB = 26 metres, DC = 18 metres and angle DAB = 80°.

A straight path from B to D has a length of 30 metres.

(a) (i) Using a scale of 1: 200, draw an accurate plan of the garden. [3]

(ii) Measure and write down the size of angle ADB and the size of angle DCB. [2]

(iii) A second path is such that all points on it are equidistant from AB and from AD.

Using a straight edge and compasses only, construct this path on your plan.

[2]

(iv) A third path is such that all points on it are equidistant from A and from D.

Using a straight edge and compasses only, construct this path on your plan.

[2]

(v) In the garden, vegetables are grown in the region which is nearer to AB than to AD and nearer to A than to D.

Shade this region on your plan. [1]

b) Use trigonometry, showing all your working, to calculate

(i) angle ADB, [3]

(ii) the length of BC, [4]

(iii) the area of the garden. [3]

80º

D C

BA26 m

30 m

18 m

Ch15-Trigonometry...Page: ABBASI MOHAMMED ASIM [email protected] Ch15-Trigonometry 1. NOT TO...

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