Question : A person from the top of a hill observes a vehicle moving towards him at a uniform speed. It takes 10 minutes for the angle of depression to change from $45^{\circ}$ to $60^{\circ}$. After this, the time required by the vehicle to reach the bottom of the hill is:
Option 1: 12 minutes 20 seconds
Option 2: 13 minutes
Option 3: 13 minutes 40 seconds
Option 4: 14 minutes 24 seconds
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Correct Answer: 13 minutes 40 seconds
Solution : Let, $CD$ = 10 units In $\Delta ABC$, $\tan 45^{\circ} = \frac{AB}{BC}$ $\Rightarrow BC = AB \quad................(1)$ In $\Delta ABD$, $\tan 60^{\circ} = \frac{AB}{BD}$ $\Rightarrow \sqrt{3} = \frac{AB}{BD}$ $\Rightarrow AB = \sqrt{3} BD$ $\Rightarrow BC = \sqrt{3} BD \quad [\text{using (1)}]$ $\Rightarrow BC = BD + CD$ $\Rightarrow \sqrt{3} BD - BD = CD$ $\Rightarrow BD(\sqrt{3} - 1) = 10$ $\Rightarrow BD = \frac{10}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$ $\Rightarrow BD = \frac{10(\sqrt{3} + 1)}{2}$ $\Rightarrow BD = 5(1.732 + 1)$ $\Rightarrow BD = 5 \times 2.732$ $\Rightarrow BD = 13.66$ units $\therefore $ Time required to travel 10 units = 10 minutes $\Rightarrow$ Time required to travel 13.66 units = 13.66 minutes = 13 minutes 40 seconds approx. Hence, the correct answer is 13 minutes 40 seconds.
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Question : A man on the top of a tower, standing on the seashore, finds that a boat coming towards him takes 10 minutes for the angle of depression to change from 30° to 60°. How soon does the boat reach the seashore?
Option 1: 5 minutes
Option 2: 7 minutes
Option 3: 10 minutes
Option 4: 15 minutes
Question : In 6 minutes, $\frac{4}{13}$ of a bucket is filled. How much time will it take to fill the remaining bucket?
Option 1: 13 minutes 30 seconds
Option 2: 14 minutes 30 seconds
Option 3: 11 minutes 30 seconds
Option 4: 12 minutes 30 seconds
Question : $\angle A, \angle B$ and $\angle C$ are three angles of a triangle. If $\angle A- \angle B=15^{\circ}, \angle B - \angle C=30^{\circ}$, then $\angle A, \angle B$ and $\angle C$ are:
Option 1: $80^{\circ},60^{\circ},$ and $40^{\circ}$
Option 2: $70^{\circ},50^{\circ},$ and $60^{\circ}$
Option 3: $80^{\circ},65^{\circ},$ and $35^{\circ}$
Option 4: $80^{\circ},55^{\circ},$ and $45^{\circ}$
Question : In $\triangle A B C, \angle A=66^{\circ}$ and $\angle B=50^{\circ}$. If the bisectors of $\angle B$ and $\angle C$ meet at P, then, $\angle B P C-\angle P C A=$?
Option 1: $93^\circ$
Option 2: $91^\circ$
Option 3: $83^\circ$
Option 4: $81^\circ$
Question : A boat is moving away from an observation tower. It makes an angle of depression of 60° with an observer's eye when at a distance of 50 metres from the tower. After 8 seconds, the angle of depression becomes 30º. By assuming that it is running in still water, the approximate speed of the boat is:
Option 1: 33 km/h
Option 2: 42 km/h
Option 3: 45 km/h
Option 4: 50 km/h
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