Ch15-Trigonometry...Page: ABBASI MOHAMMED ASIM [email protected] Ch15-Trigonometry 1. NOT TO...
Transcript of Ch15-Trigonometry...Page: ABBASI MOHAMMED ASIM [email protected] Ch15-Trigonometry 1. NOT TO...
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Ch15-Trigonometry
1.
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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.
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(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°.
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(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.
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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.
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Answer ……………………………… cm2 [2]
6.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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