ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   In the given diagram, the angle of elevation of the top of a tower (labelled AB) is 30° from a point C on the ground, 120 m away from the foot of the tower (point B). Which labelled part represents the height of the tower?

   1. A. BC
   2. B. AB
   3. C. ∠ACB
   4. D. AC
   [1]
2. 2.
   A circle artist is climbing a $20\,\text{m}$ long rope, which is tightly stretched and tied from the top of a vertical pole to the ground, then the height of pole, if the angle made by the rope with the ground level is $30^{\circ}$, is

   1. a. $5\,\text{m}$
   2. b. $10\,\text{m}$
   3. c. $15\,\text{m}$
   4. d. $20\,\text{m}$
   [1]
3. 3.
   If the angle of depression of an object from a $75\,\mathrm{m}$ high tower is $30^{\circ}$, then the distance of the object from the tower is

   1. a. $25\sqrt{3}\,\mathrm{m}$
   2. b. $50\sqrt{3}\,\mathrm{m}$
   3. c. $75\sqrt{3}\,\mathrm{m}$
   4. d. $150\,\mathrm{m}$
   [1]
4. 4.
   A $6\mathrm{m}$ high tree cast a $4\mathrm{m}$ long shadow. At the same time, a flag pole cast a shadow $50\mathrm{m}$ long. How long is the flag pole?

   1. A. $75\mathrm{m}$
   2. B. $100\mathrm{m}$
   3. C. $150\mathrm{m}$
   4. D. $50\mathrm{m}$
   [1]
5. 5.
   An electric pole is 10 m high. A steel wire tied to the top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 45° with the horizontal through the foot of the pole, what is the length of the wire? (Use √2 = 1.414)

   1. A. 10 m
   2. B. 12.14 m
   3. C. 14.14 m
   4. D. 20 m
   [1]
6. 6.
   The length of a string between a kite and a point on the ground is 85 m. The string makes an angle θ with the level ground, as shown in the diagram , such that tan θ = 15/8. Which labelled part in the diagram represents the height of the kite above the ground?

   1. A. The length of the string (85 m)
   2. B. The vertical distance from the ground to the kite
   3. C. The horizontal distance from the point on the ground to the point below the kite
   4. D. The angle θ between the string and the ground
   [1]
7. 7.
   The horizontal distance between two towers is 60 m. The angle of elevation of the top of the taller tower as seen from the top of the shorter one is 30°. If the height of the taller tower is 150 m, what is the height of the shorter tower?

   1. A. 150 m
   2. B. 150 - 20√3 m
   3. C. 130 m
   4. D. 120 m
   [1]
8. 8.
   In the given diagram, identify the length of BC. Which of the following correctly represents the length of BC?

   1. A. 10 m
   2. B. 20 m
   3. C. 10√3 m
   4. D. 20√3 m
   [1]
9. 9.
   Two men on either side of a 75 m high building and in line with the base of the building observe the angles of elevation of the top of the building as 30° and 60°. What is the distance between the two men? (Use √3 = 1.73)

   1. A. 100 m
   2. B. 150 m
   3. C. 173 m
   4. D. 200 m
   [1]
10. 10.
    From two points A and B on the same side of a building, the angles of elevation of the top of the building are 30° and 60° respectively. If the height of the building is 10 m, find the distance between A and B correct to two decimal places.

    1. A. 5.77 m
    2. B. 10.00 m
    3. C. 11.54 m
    4. D. 17.32 m
    [1]
11. 11.
    As observed from the top of a 80 m tall lighthouse, the angles of depression of two ships, on the same side of the lighthouse in horizontal line with its base, are 30° and 40° respectively. What is the distance between the two ships, correct to the nearest metre?

    1. A. 32 m
    2. B. 43 m
    3. C. 55 m
    4. D. 139 m
    [1]
12. 12.
    From the top of a hill, the angles of depression of two consecutive kilometre stones due east are 30° and 45°. What are the distances of the two stones from the foot of the hill?

    1. A. 1000 m and 2000 m
    2. B. (1000√3) m and (1000 + 1000√3) m
    3. C. 500 m and 1500 m
    4. D. 1000 m and 1000√3 m
    [1]
13. 13.
    Find the height of a tree when it is found that on walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60° to 30°.

    1. A. 10 m
    2. B. 10√3 m
    3. C. 20 m
    4. D. 20√3 m
    [1]
14. 14.
    A man observes the angle of elevation of the top of a building to be 30°. He walks towards it in a horizontal line through its base. On covering 60 m, the angle of elevation changes to 60°. Find the height of the building correct to the nearest metre.

    1. A. 30 m
    2. B. 52 m
    3. C. 60 m
    4. D. 104 m
    [1]
15. 15.
    An aeroplane, at an altitude of 250 m, observes the angles of depression of two boats on the opposite banks of a river to be 45° and 60° respectively. Find the width of the river correct to the nearest whole number.

    1. A. 394 m
    2. B. 106 m
    3. C. 500 m
    4. D. 644 m
    [1]
16. 16.
    An observer, $1.5\,\text{m}$ tall is $20.5\,\text{m}$ away from a tower $22\,\text{m}$ high, then the angle of elevation of the top of the tower from the eye of observer is

    1. a. $30^{\circ}$
    2. b. $45^{\circ}$
    3. c. $60^{\circ}$
    4. d. $90^{\circ}$
    [1]
17. 17.
    From the figure given below, identify the segment whose length equals 10(√3 + 1) m.

    1. A. AB
    2. B. CD
    3. C. BD
    4. D. AC
    [1]
18. 18.
    The shadow of a vertical tower on a level ground increases by 10 m when the altitude of the sun changes from 45° to 30°. What is the height of the tower correct to two decimal places?

    1. A. 10 m
    2. B. 13.66 m
    3. C. 20 m
    4. D. 27.32 m
    [1]
19. 19.
    A boy standing on a vertical cliff in a jungle observes two rest houses in line with him on opposite sides deep in the jungle below. The angles of depression to the rest houses are $19^{\circ}$ and $26^{\circ}$, and the distance between them is $222 \text{ m}$. Which labeled segment in the diagram represents the height of the cliff?

    1. A. The horizontal distance between the two rest houses
    2. C. The slant line from the boy to one of the rest houses
    3. D. The horizontal line from the base of the cliff to one rest house
    [1]
20. 20.
    A man in a boat rowing away from a 100 m high lighthouse takes 2 minutes to change the angle of elevation of the top from $60^{\circ}$ to $30^{\circ}$. What is the speed of the boat in metres per minute? (Use $\sqrt{3} = 1.732$)

    1. A. $28.87 \mathrm{m/min}$
    2. B. $57.74 \mathrm{m/min}$
    3. C. $115.47 \mathrm{m/min}$
    4. D. $200 \mathrm{m/min}$
    [1]