Heights and distances — practice questions

Question 1 · difficulty L2 · understanding

The angle of elevation of the top $$\mathrm{P}$$ of a tower from the feet of one person standing due South of the tower is $$45^{\circ}$$ and from the feet of another person standing due west of the tower is $$30^{\circ}$$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

• A. 10
• B. $$\frac{5}{2} \sqrt{5}$$
• C. $$5 \sqrt{5}$$
• D. 5

Question 2 · difficulty L2 · understanding

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60$$^\circ$$. The pole subtends an angle 30$$^\circ$$ at the top of the tower. Then the height of the tower is :

• A. $$15\sqrt 3$$
• B. $$20\sqrt 3$$
• C. 20 + $$10\sqrt 3$$
• D. 30

Question 3 · difficulty L2 · understanding

The angle of elevation of a jet plane from a point A on the ground is 60$$^\circ$$. After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30$$^\circ$$. If the jet plane is flying at a constant height, then its height is :

• A. $$3600\sqrt 3$$ m
• B. $$1200\sqrt 3$$ m
• C. $$1800\sqrt 3$$ m
• D. $$2400\sqrt 3$$ m

Question 4 · difficulty L2 · understanding

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30 o . After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60 o . Then the time taken (in minutes) by him, from B to reach the pillar, is :

• A. 6
• B. 10
• C. 20
• D. 5

Question 5 · difficulty L3 · understanding

From the top $$\mathrm{A}$$ of a vertical wall $$\mathrm{AB}$$ of height $$30 \mathrm{~m}$$, the angles of depression of the top $$\mathrm{P}$$ and bottom $$\mathrm{Q}$$ of a vertical tower $$\mathrm{PQ}$$ are $$15^{\circ}$$ and $$60^{\circ}$$ respectively, $$\mathrm{B}$$ and $$\mathrm{Q}$$ are on the same horizontal level. If $$\mathrm{C}$$ is a point on $$\mathrm{AB}$$ such that $$\mathrm{CB}=\mathrm{PQ}$$, then the area (in $$\mathrm{m}^{2}$$ ) of the quadrilateral $$\mathrm{BCPQ}$$ is equal to :

• A. $$200(3-\sqrt{3})$$
• B. $$300(\sqrt{3}-1)$$
• C. $$300(\sqrt{3}+1)$$
• D. $$600(\sqrt{3}-1)$$

Question 6 · difficulty L3 · understanding

The angle of elevation of the top of a tower from a point A due north of it is $$\alpha$$ and from a point B at a distance of 9 units due west of A is $$\cos ^{-1}\left(\frac{3}{\sqrt{13}}\right)$$. If the distance of the point B from the tower is 15 units, then $$\cot \alpha$$ is equal to :

• A. $$\frac{6}{5}$$
• B. $$\frac{9}{5}$$
• C. $$\frac{4}{3}$$
• D. $$\frac{7}{3}$$

Question 7 · difficulty L3 · understanding

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $$45^{\circ}$$. Let R be a point on AQ and from a point B, vertically above $$\mathrm{R}$$, the angle of elevation of $$\mathrm{P}$$ is $$60^{\circ}$$. If $$\angle \mathrm{BAQ}=30^{\circ}, \mathrm{AB}=\mathrm{d}$$ and the area of the trapezium $$\mathrm{PQRB}$$ is $$\alpha$$, then the ordered pair $$(\mathrm{d}, \alpha)$$ is :

• A. $$(10(\sqrt{3}-1), 25)$$
• B. $$\left(10(\sqrt{3}-1), \frac{25}{2}\right)$$
• C. $$(10(\sqrt{3}+1), 25)$$
• D. $$\left(10(\sqrt{3}+1), \frac{25}{2}\right)$$

Question 8 · difficulty L3 · understanding

Let a vertical tower $$A B$$ of height $$2 h$$ stands on a horizontal ground. Let from a point $$P$$ on the ground a man can see upto height $$h$$ of the tower with an angle of elevation $$2 \alpha$$. When from $$P$$, he moves a distance $$d$$ in the direction of $$\overrightarrow{A P}$$, he can see the top $$B$$ of the tower with an angle of elevation $$\alpha$$. If $$d=\sqrt{7} h$$, then $$\tan \alpha$$ is equal to

• A. $$\sqrt{5}-2$$
• B. $$\sqrt{3}-1$$
• C. $$\sqrt{7}-2$$
• D. $$\sqrt{7}-\sqrt{3}$$

Question 9 · difficulty L3 · understanding

A tower PQ stands on a horizontal ground with base $$Q$$ on the ground. The point $$R$$ divides the tower in two parts such that $$Q R=15 \mathrm{~m}$$. If from a point $$A$$ on the ground the angle of elevation of $$R$$ is $$60^{\circ}$$ and the part $$P R$$ of the tower subtends an angle of $$15^{\circ}$$ at $$A$$, then the height of the tower is :

• A. $$5(2 \sqrt{3}+3) \,\mathrm{m}$$
• B. $$5(\sqrt{3}+3) \,\mathrm{m}$$
• C. $$10(\sqrt{3}+1) \,\mathrm{m}$$
• D. $$10(2 \sqrt{3}+1) \,\mathrm{m}$$

Question 10 · difficulty L3 · understanding

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let $${\pi \over 8}$$ and $$\theta$$ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan 2 $$\theta$$ is equal to

• A. $${{3 - 2\sqrt 2 } \over 2}$$
• B. $${{3 + \sqrt 2 } \over 2}$$
• C. $${{3 - 2\sqrt 2 } \over 4}$$
• D. $${{3 - \sqrt 2 } \over 4}$$