SECTION C — Short Answer Type Questions [3 Marks Each]
- 3 Marks Q26. Two men on opposite sides of a $75\text{ m}$ high tower observe the angles of elevation of the top of the tower to be $30^\circ$ and $60^\circ$ respectively. Find the distance between the two men.
- 3 Marks Q27. A vertical flagstaff of length $h$ stands on the top of a building. From a point on the ground, the angle of elevation of the bottom of the flagstaff is $45^\circ$ and the angle of elevation of the top of the flagstaff is $60^\circ$. Find the height of the building in terms of $h$.
- 3 Marks Q28. From a window ($h$ meters high above the ground) of a house in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are $\theta$ and $\varphi$ respectively. Show that the height of the opposite house is $h(1 + \cot \theta \tan \varphi)$.
- 3 Marks Q29. A vertical tower stands on a horizontal ground and is surmounted by a vertical flag-post of height $h$. At a point on the ground, the angles of elevation of the bottom and the top of the flag-post are $\alpha$ and $\beta$ respectively. Prove that the height of the tower is $\frac{h \tan \alpha}{\tan \beta - \tan \alpha}$.
- 3 Marks Q30. An airplane, when flying at a height of $4000\text{ m}$ from the ground, passes vertically above another airplane at an instant when the angles of elevation of the two planes from the same point on the ground are $60^\circ$ and $45^\circ$ respectively. Find the vertical distance between the two planes at that instant.
- 3 Marks Q31. The angle of elevation of the top of a vertical tower from a point on the ground is $60^\circ$. From another point $10\text{ m}$ vertically above the first, the angle of elevation is $30^\circ$. Find the height of the tower and its horizontal distance from the observation point.
- 3 Marks Q32. A flagstaff $6\text{ m}$ high is placed on the top of a tower. The angle of elevation of the top of the flagstaff is $60^\circ$ and the angle of depression of the same point from the top of the flagstaff is $30^\circ$ (measured appropriately or via alternate points). Find the height of the tower.
- 3 Marks Q33. Two pillars of equal heights stand on either side of a road, which is $100\text{ m}$ wide. From a point on the road between the pillars, the angles of elevation of the top of the pillars are $60^\circ$ and $30^\circ$. Find the height of the pillars and the position of the point from the road sides.
- 3 Marks Q34. A statue $1.6\text{ m}$ tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point, the angle of elevation of the top of the pedestal is $45^\circ$. Find the heig ht of the pedestal.
- 3 Marks Q35. The angle of elevation of the top of a hill at the foot of a tower is $60^\circ$ and the angle of elevation of the top of the tower from the foot of the hill is $30^\circ$. If the tower is $50\text{ m}$ high, find the height of the hill.
- 3 Marks Q36. The angle of elevation of a jet plane from a point $A$ on the ground is $60^\circ$. After a flight of $30\text{ seconds}$, the angle of elevation changes to $30^\circ$. If the jet plane is flying at a constant height of $3000\sqrt{3}\text{ m}$, find the speed of the jet plane in $\text{km/h}$.
- 3 Marks Q37. A boy standing on a horizontal plane finds a bird flying at a distance of $100\text{ m}$ from him at an elevation of $30^\circ$. A girl standing on the roof of a $20\text{ m}$ high building finds the elevation of the same bird to be $45^\circ$. Both are on the opposite sides of the bird. Find the horizontal distance between the boy and the girl
- 3 Marks Q38. A vertical tower is $20\text{ m}$ high. A man looks at the top of the tower from a point on the ground at a distance of $20\text{ m}$ from the base. What is the angle of elevation, and if he moves $20(\sqrt{3} - 1)\text{ m}$ closer, what is the new angle of elevation?
- 3 Marks Q39. An observer on the top of a cliff $200\text{ m}$ high observes the angles of depression of the top and bottom of a tower standing on the same horizontal plane to be $30^\circ$ and $60^\circ$ respectively. Find the height of the tower.
- 3 Marks Q40. A round balloon of radius $r$ subtends an angle $\alpha$ at the eye of the observer while the angle of elevation of its center is $\beta$. Prove that the height of the center of the balloon is $r \csc(\alpha/2) \sin \beta$.
- 3 Marks Q41. A vertical pole and a vertical tower are on the same level ground. The angle of elevation of the top of the tower from the foot of the pole is $60^\circ$ and the angle of elevation of the top of the pole from the foot of the tower is $30^\circ$. If the height of the pole is $20\text{ m}$, find the height of the tower.
- 3 Marks Q42. A cloud is $h$ meters above the lake. The angle of elevation of the cloud from a point $p$ meters above the lake is $\theta$ and the angle of depression of its reflection in the lake is $\phi$. Prove that the distance of the cloud from the point of observation is $\frac{2h \csc \theta}{\cot \theta - \cot \phi}$ (formulation variant).
- 3 Marks Q43. A person on a tour boat spots a lighthouse at an angle of elevation of $30^\circ$. After sailing $100\text{ m}$ towards the lighthouse, the angle of elevation becomes $60^\circ$. Find the height of the lighthouse and the initial distance of the boat from it.
- 3 Marks Q44. From a point $P$ on the ground, the angle of elevation of the top of a $10\text{ m}$ tall building is $30^\circ$. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from $P$ is $45^\circ$. Find the length of the flagstaff and the distance of point $P$ from the building.
- 3 Marks Q45. A vertical electric pole is $15\text{ m}$ high. A steel wire tied to the top of the pole is fixed at a point on the ground and makes an angle of $45^\circ$ with the horizontal ground. Due to slack adjustment, if the wire is lengthened by $2\text{ m}$, find the new angle the wire makes with the ground.
- 3 Marks Q46. The angle of elevation of the top of a tower from a point $A$ on the ground is $30^\circ$. If the tower height is increased by $20\text{ m}$, the angle of elevation from $A$ becomes $60^\circ$. Find the original height of the tower and the distance of $A$ from the base.
- 3 Marks Q47. An aero-plane at an altitude of $1200\text{ m}$ observes two ships sailing towards it in the same direction on the sea. The angles of depression of the ships as viewed from the aero-plane are $60^\circ$ and $30^\circ$ respectively. Find the distance between the two ships.
- 3 Marks Q48. A tower of $x$ meters height casts a shadow of length $x/\sqrt{3}$ meters. Find the angle of elevation of the sun, and determine the new shadow length if the height of the tower is doubled while the sun angle remains the same
- 3 Marks Q49. A window of a house is $h$ meters above the ground. From the window, the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are $45^\circ$ and $30^\circ$ respectively. Find the height of the opposite house in terms of $h$.
- 3 Marks Q50. Two stations due south of a leaning tower which tilts towards the north are at distances $a$ and $b$ from its base. If the angles of elevation of the top of the tower from these stations are $\alpha$ and $\beta$ respectively, prove that the inclination $\theta$ of the tower to the vertical is given by $\cot \theta = \frac{b \cot \alpha - a \cot \beta}{b - a}$.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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