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Some Applications of Trigonometry · Class 10 Mathematics: 115 MCQs with answers

All 115 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything Flashcards · 46 Quick Revision · 35

Warm-up · 24 questions

Q1. The straight line from the eye to the object is called the

Beginner
A horizontal B angle of elevation C vertical D line of sight
Show answer & explanation

line of sight

That is its definition.

Q2. The angle of elevation is measured from the

Beginner
A vertical B ground under the object C horizontal D line of sight
Show answer & explanation

horizontal

Both elevation and depression start at the horizontal.

Q3. The angle of depression is measured from the

Beginner
A vertical wall B ground C horizontal through the eye D object itself
Show answer & explanation

horizontal through the eye

It needs a horizontal drawn at the observer.

Q4. In the right triangle of a heights problem, the line of sight is the

Beginner
A opposite side B adjacent side C hypotenuse D vertical
Show answer & explanation

hypotenuse

It faces the right angle.

Q5. If the object is above the observer, the angle used is the angle of

Beginner
A elevation B depression C rotation D incidence
Show answer & explanation

elevation

Eyes lift.

Q6. Which ratio links the height of a tower to the distance of the observer from its foot?

Beginner
A $\sin$ B $\tan$ C $\cos$ D $\sec$
Show answer & explanation

$\tan$

Both are legs of the right triangle.

Q7. At an angle of elevation of 45°, the height of a tower compared with the observer's distance is

Beginner
A twice it B half it C $\sqrt3$ times it D equal to it
Show answer & explanation

equal to it

$\tan 45° = 1$.

Q8. A tower 20 m high has an elevation of 45° from a point on the ground. That point is

Beginner
A 10 m from the foot B $20\sqrt2$ m from the foot C 20 m from the foot D 40 m from the foot
Show answer & explanation

20 m from the foot

At 45° the legs are equal.

Q9. Which angles does this chapter allow in its problems?

Beginner
A any angle B 0° to 90° C 30° and 90° only D 30°, 45° and 60° only
Show answer & explanation

30°, 45° and 60° only

The syllabus restricts them to the standard three.

Q10. How many right triangles may a problem in this chapter involve?

Beginner
A exactly one B at most four C any number D at most two
Show answer & explanation

at most two

The syllabus caps it at two.

Q11. A tower casts a 20 m shadow when the sun's elevation is 45°. The tower is

Beginner
A 10 m high B $20\sqrt2$ m high C $20\sqrt3$ m high D 20 m high
Show answer & explanation

20 m high

Equal legs at 45°.

Q12. In a heights-and-distances figure, the right angle is always

Beginner
A at the observer's eye B at the top of the tower C where the vertical meets the horizontal ground D along the line of sight
Show answer & explanation

where the vertical meets the horizontal ground

The object stands vertically on level ground.

Q13. The very first step in answering any question in this chapter is to

Beginner
A write the final answer B draw and label the figure C choose a calculator D square both sides
Show answer & explanation

draw and label the figure

The figure carries marks of its own and prevents most mistakes.

Q14. Define the angle of elevation and the angle of depression, and state what both are measured from.

Beginner
Show answer & explanation

—

Q15. A boy 30 m from the foot of a tower sees its top at an angle of elevation of 30°. Find the height of the tower.

Beginner
Show answer & explanation

—

Q16. The angle of elevation of the top of a 20 m tower from a point on the ground is 45°. How far is the point from the foot?

Beginner
Show answer & explanation

—

Q17. Why must the answer $\frac{50}{\sqrt3}$ be rewritten, and what does it become?

Beginner
Show answer & explanation

—

Q18. A single angle of elevation is not enough to find a tower's height because

Beginner
A the angle may be wrong B tangent is hard to compute C there are two unknowns but only one equation D the ground may slope
Show answer & explanation

there are two unknowns but only one equation

Height and distance are both unknown.

Q19. The two right triangles in a two-observation problem always share the

Beginner
A vertical side B line of sight C angle at the observer D horizontal side
Show answer & explanation

vertical side

That shared height is what gets eliminated.

Q20. Of two points on the same side, the one with the LARGER angle of elevation is

Beginner
A further from the tower B the same distance C nearer the tower D on the other side
Show answer & explanation

nearer the tower

A smaller distance makes the ratio, and so the angle, larger.

Q21. Two observers stand on the same side, a gap $g$ apart. If the nearer is $d$ away, the further is

Beginner
A $g - d$ away B $d - g$ away C $dg$ away D $d + g$ away
Show answer & explanation

$d + g$ away

Same side means the distances differ by the gap.

Q22. After solving for the distance, what remains to be done?

Beginner
A substitute back to find the height B nothing C square the answer D convert to degrees
Show answer & explanation

substitute back to find the height

The question almost always wants the height too.

Q23. How many right triangles does the syllabus allow in one problem?

Beginner
A exactly one B at most three C any number D at most two
Show answer & explanation

at most two

Two is the cap.

Q24. The height found from the two equations should finally be

Beginner
A left as a fraction with a surd below B rationalised and given with its unit C rounded to the nearest 10 D given as an angle
Show answer & explanation

rationalised and given with its unit

That is what finishes the answer.

Building up · 43 questions

Q25. The angle of depression from the top of a tower and the angle of elevation of the top from the ground are

Intermediate
A equal B complementary C supplementary D unrelated
Show answer & explanation

equal

They are alternate angles between two parallel horizontals.

Q26. Which ratio would you use given the length of the line of sight and wanting the height?

Intermediate
A $\sin$ B $\cos$ C $\tan$ D $\cot$
Show answer & explanation

$\sin$

Opposite over hypotenuse.

Q27. Which ratio would you use given the line of sight and wanting the ground distance?

Intermediate
A $\sin$ B $\tan$ C $\operatorname{cosec}$ D $\cos$
Show answer & explanation

$\cos$

Adjacent over hypotenuse.

Q28. A tower is 30 m from an observer and its elevation is 30°. Its height is

Intermediate
A $30\sqrt3$ m B 15 m C $10\sqrt3$ m D $\frac{\sqrt3}{30}$ m
Show answer & explanation

$10\sqrt3$ m

$30\tan 30° = \frac{30}{\sqrt3} = 10\sqrt3$.

Q29. A pole is 15 m high and the elevation of its top from a point is 60°. The point is

Intermediate
A $5\sqrt3$ m away B $15\sqrt3$ m away C 30 m away D $\frac{15}{2}$ m away
Show answer & explanation

$5\sqrt3$ m away

$\frac{15}{\sqrt3} = 5\sqrt3$.

Q30. If the angle of elevation of the sun is 45°, a vertical pole's shadow is

Intermediate
A twice the pole B half the pole C as long as the pole D $\sqrt3$ times the pole
Show answer & explanation

as long as the pole

The shadow is the horizontal leg, equal to the height at 45°.

Q31. As the sun rises higher, the shadow of a fixed pole

Intermediate
A gets longer B gets shorter C stays the same D disappears at 45°
Show answer & explanation

gets shorter

A larger angle gives a larger tangent, so the same height needs less ground.

Q32. The most common mistake when handling an angle of depression is

Intermediate
A using degrees B forgetting to draw the horizontal through the eye C drawing the tower too tall D rounding too early
Show answer & explanation

forgetting to draw the horizontal through the eye

Without that line the angle is taken from the vertical.

Q33. Assertion (A): The angle of depression of a boat from a lighthouse equals the angle of elevation of the lighthouse top from the boat. Reason (R): The horizontal at the top and the horizontal at the sea are parallel, and the line of sight is a transversal.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

Alternate interior angles between parallels are equal, which is exactly R's argument.

Q34. Assertion (A): The angle of depression is measured from the vertical. Reason (R): A vertical line is easier to draw on a cliff figure than a horizontal one.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

A is false — depression is measured from the horizontal — and R is not a mathematical reason at all.

Q35. A triangle drawn from an eye-level angle gives the height

Intermediate
A above the ground B above the observer's eye C of the observer D of the line of sight
Show answer & explanation

above the observer's eye

Its base sits at eye level, not at ground level.

Q36. $\frac{40}{\sqrt3}$ rationalised is

Intermediate
A $\frac{40\sqrt3}{3}$ B $40\sqrt3$ C $\frac{\sqrt3}{40}$ D $\frac{40}{3}$
Show answer & explanation

$\frac{40\sqrt3}{3}$

Multiply above and below by $\sqrt3$.

Q37. From a window 10 m above the ground, the angle of depression of a point on the ground is 45°. The point is

Intermediate
A 10 m from the wall B 5 m from the wall C $10\sqrt2$ m from the wall D 20 m from the wall
Show answer & explanation

10 m from the wall

The elevation at the point is also 45°, and $\tan 45° = 1$.

Q38. If you know the height and the angle of elevation and want the ground distance, you should

Intermediate
A multiply the height by $\tan\theta$ B divide the height by $\tan\theta$ C divide the height by $\sin\theta$ D multiply by $\cos\theta$
Show answer & explanation

divide the height by $\tan\theta$

$\tan\theta = \frac{h}{d}$ rearranges to $d = \frac{h}{\tan\theta}$.

Q39. A tree 12 m tall is seen at an elevation of 30°. The observer's distance from it is

Intermediate
A $4\sqrt3$ m B 24 m C 6 m D $12\sqrt3$ m
Show answer & explanation

$12\sqrt3$ m

$\frac{12}{\tan 30°} = 12\sqrt3$.

Q40. The height of a tower whose top is seen at 60° from a point 8 m away is

Intermediate
A $\frac{8}{\sqrt3}$ m B 16 m C $8\sqrt3$ m D 4 m
Show answer & explanation

$8\sqrt3$ m

$8\tan 60° = 8\sqrt3$.

Q41. A statement that an answer is '$17.3$ m' rather than '$10\sqrt3$ m' is

Intermediate
A always wrong B always preferred C not a length at all D an approximation, acceptable if the question asks for it
Show answer & explanation

an approximation, acceptable if the question asks for it

Exact surd form is the default unless a decimal is requested.

Q42. Assertion (A): At an elevation of 45°, the height and the ground distance must be equal. Reason (R): $\tan 45^\circ = 0$, and the tangent is the height divided by the distance.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is true but R is false

The tangent of $45^\circ$ is 1, which is what forces the two lengths to agree. The assertion stands, the reason is false.

Q43. A boat is 100 m from the foot of a cliff and the angle of depression from the cliff top is 45°. The cliff is

Intermediate
A 100 m high B 50 m high C $100\sqrt2$ m high D 200 m high
Show answer & explanation

100 m high

Equal legs at 45°.

Q44. Why is a diagram worth drawing even when the arithmetic is obvious?

Intermediate
A it saves time B it fixes which side is opposite the angle C it replaces the working D it is decorative
Show answer & explanation

it fixes which side is opposite the angle

Most errors here are about which side is which, not about arithmetic.

Q45. A kite is flying at a height of 60 m. The string, assumed straight, makes an angle of 60° with the ground. Find the length of the string.

Intermediate
Show answer & explanation

—

Q46. Explain why the angle of depression of a boat from a lighthouse equals the angle of elevation of the lighthouse top from the boat.

Intermediate
Show answer & explanation

—

Q47. The sun's angle of elevation is 60°. Find the length of the shadow of a 9 m vertical pole.

Intermediate
Show answer & explanation

—

Q48. State the three ratios used in this chapter and say which quantities each one connects.

Intermediate
Show answer & explanation

—

Q49. A tower stands between two points a distance $g$ apart. If one is $x$ away, the other is

Intermediate
A $g + x$ away B $g - x$ away C $x - g$ away D $\frac{g}{x}$ away
Show answer & explanation

$g - x$ away

Opposite sides means the distances add to the gap.

Q50. From a point 20 m away, a tower top is at 45° and a flagstaff top on it at 60°. The tower's height is

Intermediate
A $20\sqrt3$ m B 20 m C $20(\sqrt3 - 1)$ m D 40 m
Show answer & explanation

20 m

$20\tan 45° = 20$.

Q51. The commonest error in a flagstaff problem is

Intermediate
A using degrees B giving the combined height as the flagstaff's height C forgetting the observer's height D drawing two figures
Show answer & explanation

giving the combined height as the flagstaff's height

The flagstaff alone is a difference.

Q52. The conjugate of $\sqrt3 - 1$ is

Intermediate
A $1 - \sqrt3$ B $-\sqrt3 - 1$ C $\sqrt3$ D $\sqrt3 + 1$
Show answer & explanation

$\sqrt3 + 1$

Change the sign between the two terms.

Q53. A moving-observer problem is the same as

Intermediate
A two points on opposite sides B a flagstaff problem C two points on the same side D a single-triangle problem
Show answer & explanation

two points on the same side

The two positions are the two points.

Q54. When should the standard values be substituted?

Intermediate
A in the very first line B never C only for 45° D after the height has been eliminated
Show answer & explanation

after the height has been eliminated

Arranging the algebra first keeps the surds manageable.

Q55. Assertion (A): Two observations from different points can determine both the height of a tower and its distance. Reason (R): Two independent equations are enough to solve for two unknowns.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

That is exactly why a second angle is worth taking.

Q56. Drawing both triangles in ONE figure matters because

Intermediate
A it shows how the two distances are related B it saves paper C it is required by the syllabus D two figures are impossible
Show answer & explanation

it shows how the two distances are related

The relationship between the distances is the whole problem.

Q57. A tower and a flagstaff on it are seen from one point. The two triangles share

Intermediate
A the same vertical height B nothing C the same horizontal distance D the same hypotenuse
Show answer & explanation

the same horizontal distance

One observer, one ground distance, two different tops.

Q58. Which word in a question tells you the two points are on opposite sides?

Intermediate
A beyond B along C towards D between
Show answer & explanation

between

A tower 'between' two points sits inside the gap.

Q59. Which check confirms the angles were assigned to the right points?

Intermediate
A the height is a whole number B both distances are equal C the larger angle came out with the smaller distance D the answer has no surd
Show answer & explanation

the larger angle came out with the smaller distance

Nearer means steeper, always.

Q60. Eliminating the height from the two equations leaves

Intermediate
A two equations in two distances B nothing useful C an identity D one equation in one distance
Show answer & explanation

one equation in one distance

That is the point of eliminating it.

Q61. If the answer to a same-side problem gives the 60° point further away than the 30° point, you should

Intermediate
A accept it B change the gap C use three triangles D recheck which angle was assigned to which point
Show answer & explanation

recheck which angle was assigned to which point

It contradicts the rule that the nearer point sees the larger angle.

Q62. Two coastguard stations locating a boat between them are using

Intermediate
A the flagstaff picture B the opposite-sides picture C a single triangle D no triangle at all
Show answer & explanation

the opposite-sides picture

The boat lies between the two known points.

Q63. Assertion (A): In a flagstaff problem the two right triangles have the same horizontal side. Reason (R): A flagstaff stands vertically on the top of the tower.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, but R is not the correct explanation of A

Both are true, but the flagstaff being vertical is not why the two triangles share a horizontal side; the common observation point is.

Q64. Explain why a problem asking for both the height of a tower and the distance of an observer must supply two angles.

Intermediate
Show answer & explanation

—

Q65. State the two ways in which the distances of two observers relate to the gap between them, and say when each applies.

Intermediate
Show answer & explanation

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Q66. Rationalise $\frac{30}{\sqrt3 - 1}$ and explain the step used.

Intermediate
Show answer & explanation

—

Q67. Why must the larger of two angles of elevation belong to the nearer point?

Intermediate
Show answer & explanation

—

Topper level · 48 questions

Q68. A kite flies at a height of 60 m with its straight string at 60° to the ground. The string is

Advanced
A $30\sqrt3$ m long B 120 m long C $40\sqrt3$ m long D $60\sqrt3$ m long
Show answer & explanation

$40\sqrt3$ m long

$\frac{60}{\sin 60°} = \frac{120}{\sqrt3} = 40\sqrt3$.

Q69. A ladder leans at 60° and reaches 6 m up a wall. Its length is

Advanced
A $6\sqrt3$ m B 12 m C $3\sqrt3$ m D $4\sqrt3$ m
Show answer & explanation

$4\sqrt3$ m

$\frac{6}{\sin 60°} = \frac{12}{\sqrt3} = 4\sqrt3$.

Q70. If the elevation of the sun is 60°, a 9 m pole casts a shadow of

Advanced
A $9\sqrt3$ m B 18 m C $3\sqrt3$ m D 4.5 m
Show answer & explanation

$3\sqrt3$ m

$\frac{9}{\sqrt3} = 3\sqrt3$.

Q71. At an elevation of 30°, the line of sight to the top of a tower compared with the tower's height is

Advanced
A twice it B half it C equal to it D $\sqrt3$ times it
Show answer & explanation

twice it

$\sin 30° = \frac12$, so the hypotenuse is twice the opposite side.

Q72. An observer 1.5 m tall measures the elevation of a tower top as 45° from a point 18.5 m away. The tower's height is

Advanced
A 18.5 m B 20 m C 17 m D $18.5\sqrt2$ m
Show answer & explanation

20 m

The triangle gives 18.5 m above eye level; add the 1.5 m eye height.

Q73. An aeroplane is 1200 m above the ground and its angle of depression from an observer's viewpoint is stated as 30°. The horizontal distance is

Advanced
A $400\sqrt3$ m B $1200\sqrt3$ m C 600 m D 2400 m
Show answer & explanation

$1200\sqrt3$ m

$\frac{1200}{\tan 30°} = 1200\sqrt3$.

Q74. A 6 m ladder rests against a wall at 30° to the ground. It reaches up the wall by

Advanced
A $3\sqrt3$ m B 3 m C $6\sqrt3$ m D 12 m
Show answer & explanation

3 m

$6\sin 30° = 3$.

Q75. That same 6 m ladder at 30° has its foot from the wall at

Advanced
A $3\sqrt3$ m B 3 m C 6 m D $6\sqrt3$ m
Show answer & explanation

$3\sqrt3$ m

$6\cos 30° = 3\sqrt3$.

Q76. Which of these questions does NOT need the line of sight?

Advanced
A height from string length and angle B ground distance from string length and angle C string length from height and angle D height from ground distance and angle
Show answer & explanation

height from ground distance and angle

Only the first involves the two legs alone.

Q77. Two observers stand at different distances from the same tower. The one standing nearer sees

Advanced
A a larger angle of elevation B a smaller angle of elevation C the same angle D an angle of depression
Show answer & explanation

a larger angle of elevation

A smaller denominator in $\frac{h}{d}$ means a larger tangent.

Q78. Assertion (A): If the sun's elevation increases, the shadow of a fixed pole gets longer. Reason (R): The shadow length is the height divided by the tangent of the elevation.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, but R is not the correct explanation of A

R is true, but it shows the opposite of A: a bigger tangent gives a SHORTER shadow.

Q79. A ladder makes an angle of 60° with the ground and reaches 6 m up a wall. Find the length of the ladder and the distance of its foot from the wall.

Advanced
Show answer & explanation

—

Q80. An observer 1.5 m tall stands 18.5 m from a tower and finds the angle of elevation of its top to be 45°. Find the height of the tower.

Advanced
Show answer & explanation

—

Q81. (a) Draw and describe the figure for an angle of depression, naming every line in it. (b) From the top of a 40 m lighthouse the angle of depression of a boat is 45°. Find the distance of the boat from the foot of the lighthouse. (c) Explain why the answer would be larger if the angle of depression were 30° instead.

Advanced
Show answer & explanation

—

Q82. (a) Explain how to decide which trigonometric ratio a heights-and-distances question needs. (b) A kite flies at a height of 45 m with its straight string at an angle of 30° to the ground. Find the length of the string, and the horizontal distance of the kite from the person holding it. (c) Check your two answers against Pythagoras.

Advanced
Show answer & explanation

—

Q83. (a) State what is special about elevations of 45° and of 30°. (b) A vertical pole casts a shadow 15 m long when the sun's elevation is 60°. Find the height of the pole. (c) Find the length of the shadow of the same pole when the sun's elevation drops to 30°, and comment on the ratio of the two shadows.

Advanced
Show answer & explanation

—

Q84. (a) An observer of height 1.6 m stands on level ground. Explain carefully what height the right triangle gives when the angle is measured at the observer's eye. (b) Such an observer stands $12\sqrt3$ m from a tower and measures the elevation of its top as 30°. Find the height of the tower. (c) State two other mistakes commonly made in this chapter and how to avoid each.

Advanced
Show answer & explanation

—

Q85. Two points 20 m apart on the same side give elevations of 60° and 30°. The nearer point is

Advanced
A 20 m from the foot B 30 m from the foot C $10\sqrt3$ m from the foot D 10 m from the foot
Show answer & explanation

10 m from the foot

$d\sqrt3 = \frac{d+20}{\sqrt3}$ gives $3d = d + 20$.

Q86. In that same problem, the height of the tower is

Advanced
A 10 m B $10\sqrt3$ m C 20 m D $20\sqrt3$ m
Show answer & explanation

$10\sqrt3$ m

$h = d\tan 60° = 10\sqrt3$.

Q87. A tower between two points 80 m apart is seen at 60° and 30°. Its height is

Advanced
A $20\sqrt3$ m B $40\sqrt3$ m C 40 m D 20 m
Show answer & explanation

$20\sqrt3$ m

$3x = 80 - x$ gives $x = 20$, so $h = 20\sqrt3$.

Q88. In that road problem, the two distances from the foot are

Advanced
A 20 m and 60 m B 40 m and 40 m C 30 m and 50 m D 10 m and 70 m
Show answer & explanation

20 m and 60 m

$x = 20$, so the other is $80 - 20 = 60$.

Q89. In that flagstaff problem, the height of the FLAGSTAFF alone is

Advanced
A $20(\sqrt3 - 1)$ m B $20\sqrt3$ m C 20 m D $20(\sqrt3 + 1)$ m
Show answer & explanation

$20(\sqrt3 - 1)$ m

$20\sqrt3 - 20$, the difference of the two verticals.

Q90. A man walks 40 m towards a tower and the elevation rises from 30° to 45°. The height is

Advanced
A $20(\sqrt3 - 1)$ m B $20(\sqrt3 + 1)$ m C 40 m D $40\sqrt3$ m
Show answer & explanation

$20(\sqrt3 + 1)$ m

$d\sqrt3 = d + 40$ gives $d = \frac{40}{\sqrt3 - 1} = 20(\sqrt3+1)$, and $h = d$ at 45°.

Q91. $\frac{40}{\sqrt3 - 1}$ rationalised is

Advanced
A $20(\sqrt3 - 1)$ B $40(\sqrt3 + 1)$ C $20(\sqrt3 + 1)$ D $\frac{40\sqrt3}{3}$
Show answer & explanation

$20(\sqrt3 + 1)$

Multiply by $\frac{\sqrt3+1}{\sqrt3+1}$; the denominator becomes $3 - 1 = 2$.

Q92. If both given angles were equal, the two equations would

Advanced
A contradict each other B say the same thing, leaving the problem unsolvable C give two heights D still be solvable
Show answer & explanation

say the same thing, leaving the problem unsolvable

Eliminating the height would leave $0 = 0$.

Q93. Two points on opposite sides of a 60 m road each see the tower between them at 45°. The tower stands

Advanced
A 30 m from each point B 20 m and 40 m from them C 15 m from one D at one end
Show answer & explanation

30 m from each point

Equal angles force equal distances, and they add to 60.

Q94. In that symmetric case, the height of the tower is

Advanced
A 60 m B 30 m C $30\sqrt2$ m D 15 m
Show answer & explanation

30 m

$\tan 45° = 1$, so the height equals the 30 m distance.

Q95. Assertion (A): In a same-side problem the two distances from the foot add up to the gap between the observers. Reason (R): The gap between two points is the difference of their distances from a point in line with them and beyond both.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

A is false — on the same side the distances DIFFER by the gap — while R states that correct fact.

Q96. Two points 30 m apart on the same side give elevations of 45° and 30°. The nearer distance is

Advanced
A $15(\sqrt3 + 1)$ m B $15(\sqrt3 - 1)$ m C 30 m D $30\sqrt3$ m
Show answer & explanation

$15(\sqrt3 + 1)$ m

$d\sqrt3 = d + 30$ gives $d = \frac{30}{\sqrt3-1} = 15(\sqrt3+1)$.

Q97. Two points 15 m apart on the same side see a tower at 60° and 30°. The height is

Advanced
A $\frac{15\sqrt3}{2}$ m B $15\sqrt3$ m C 7.5 m D $\frac{15}{2}$ m
Show answer & explanation

$\frac{15\sqrt3}{2}$ m

$3d = d + 15$ gives $d = 7.5$, so $h = 7.5\sqrt3 = \frac{15\sqrt3}{2}$.

Q98. A tower between two points 120 m apart is seen at 30° and 60°. The height is

Advanced
A $30\sqrt3$ m B $60\sqrt3$ m C 60 m D 30 m
Show answer & explanation

$30\sqrt3$ m

$3x = 120 - x$ gives $x = 30$, so $h = 30\sqrt3$.

Q99. From 30 m away a tower reads 30° and the top of a pole on it reads 45°. The pole's height is

Advanced
A $10\sqrt3$ m B 30 m C $30\sqrt3$ m D $30 - 10\sqrt3$ m
Show answer & explanation

$30 - 10\sqrt3$ m

Tower $= \frac{30}{\sqrt3} = 10\sqrt3$; total $= 30$; pole $= 30 - 10\sqrt3$.

Q100. Which pairing of angles leaves a surd in the denominator that needs a conjugate?

Advanced
A 30° and 60° B 45° and 45° C 30° and 45° D 60° and 60°
Show answer & explanation

30° and 45°

$\sqrt3$ against 1 gives $\sqrt3 - 1$; $\sqrt3$ against $\frac1{\sqrt3}$ does not.

Q101. With angles 30° and 60° and a gap of $g$, the nearer distance always works out to

Advanced
A $g$ B $\frac{g}{3}$ C $\frac{g}{2}$ D $g\sqrt3$
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$\frac{g}{2}$

$3d = d + g$ gives $d = \frac{g}{2}$.

Q102. In the 30°–60° pairing, the further distance is

Advanced
A twice the nearer one B three times the nearer one C equal to it D half of it
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three times the nearer one

$d$ and $d + 2d$, since the gap is $2d$.

Q103. A tower is seen at 45° from a point and at 30° from a point 50 m further back. The height is

Advanced
A $25(\sqrt3 - 1)$ m B 50 m C $25(\sqrt3 + 1)$ m D $50\sqrt3$ m
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$25(\sqrt3 + 1)$ m

$d\sqrt3 = d + 50$ gives $d = \frac{50}{\sqrt3-1} = 25(\sqrt3+1)$, and $h = d$.

Q104. Assertion (A): A problem giving only one angle of elevation and no distance can still be solved for the height. Reason (R): The tangent ratio relates the height to the horizontal distance.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
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A is true but R is false

R is true but A is false: with both the height and the distance unknown, one equation is not enough.

Q105. Two points 24 m apart on the same side see a tower at 60° and 30°. The height is

Advanced
A $24\sqrt3$ m B 12 m C 24 m D $12\sqrt3$ m
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$12\sqrt3$ m

$d = \frac{24}{2} = 12$, so $h = 12\sqrt3$.

Q106. Two points on level ground on the same side of a tower are 20 m apart. The angles of elevation of the top are 60° and 30°. Find the height of the tower.

Advanced
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Q107. A tower stands on a straight road between two points 80 m apart. The angles of elevation of its top from the two points are 60° and 30°. Find its height.

Advanced
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Q108. From a point 20 m from the foot of a tower, the angle of elevation of the top of the tower is 45° and of the top of a flagstaff on it is 60°. Find the height of the flagstaff.

Advanced
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Q109. A man walks 40 m directly towards a tower and finds that the angle of elevation of its top has increased from 30° to 45°. Find the height of the tower.

Advanced
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Q110. A tower is seen at 60° from a point and at 30° from a point 24 m further back. Find its height.

Advanced
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Q111. What would go wrong if a question gave the same angle of elevation from both points?

Advanced
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Q112. (a) Explain the method for solving a problem with two observation points, listing the steps. (b) Two points on the same side of a tower and in line with its foot are 20 m apart. The angles of elevation of the top from them are 60° and 30°. Find the height of the tower and the distance of each point from the foot. (c) Verify your answer by checking both original angles.

Advanced
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Q113. (a) Describe the figure for a flagstaff standing on a tower, observed from one point on the ground. (b) From a point 20 m from the foot of a tower, the angle of elevation of the top of the tower is 45° and that of the top of a flagstaff on it is 60°. Find the height of the tower and of the flagstaff. (c) Explain the mistake a student makes who answers '$20\sqrt3$ m' for the flagstaff.

Advanced
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Q114. (a) Explain why a tower standing between two observers gives a different pair of equations from one standing beyond both. (b) A tower stands on a straight road between two points 120 m apart. The angles of elevation of its top from the two points are 60° and 30°. Find the height of the tower and the distance of each point from its foot. (c) If instead both points were on the same side and still 120 m apart with the same two angles, find the height and compare.

Advanced
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Q115. (a) A problem gives angles of 30° and 45°. Explain why its algebra is messier than one giving 30° and 60°, and how the mess is cleared. (b) A tower is seen at 45° from a point and at 30° from a point 50 m further back on the same side. Find the height. (c) State two checks you would apply to your answer before stopping.

Advanced
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