Height and distance problems Practice Questions - NDA Mathematics Question Bank

Question 1 of 10 Authentic UPSC NDA Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are q and 2q respectively. What is the height of the tower?

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Question 2 of 10 Authentic UPSC NDA Question

The shadow of a tower is found to be x metre longer, when the angle of elevation of the sun changes from 60° to 45°. If the height of the tower is 5(3 + √3) m, then what is x equal to?

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Question 3 of 10 Authentic UPSC NDA Question

A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75°. What is the height of the flagstaff?

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Question 4 of 10 Authentic UPSC NDA Question

What is the length of the chord of a unit circle which subtends at the centre of the circle an angle of 45°?

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Solution

Option (d) is correct.

Explanation:

The length of the chord \( L \) in a circle can be calculated using the formula:

\[ L = 2r \sin\left(\frac{\theta}{2}\right) \]

For a unit circle, the radius \( r = 1 \) and the subtended angle \( \theta = 45^\circ \): Substituting values:

\[ L = 2 \cdot 1 \cdot \sin\left(\frac{45^\circ}{2}\right) \]

Calculating \( \sin\left(22.5^\circ\right) \): Using the identity \( \sin{A} = \frac{\sqrt{1 - \cos{2A}}}{2} \):

\[ = 2 \cdot \sqrt{2 - \sqrt{2}} \, \text{units} \]

Thus, the length of the chord is:

\[ L = \sqrt{2} + \sqrt{2} \, \text{units} \]

This simplifies into:

\[ L = 2\sqrt{2} \, \text{units} \]

And finally, confirming the available choices: - A: \( \sqrt{2} + \sqrt{2} \) units - B: \( 2\sqrt{2} + \sqrt{2} \) units - C: \( \sqrt{2} - \sqrt{2} \) units - D: \( 2\sqrt{2} \) units The length of the chord subtended by a \( 45^\circ \) angle at the center of a unit circle is \( 2\sqrt{2} \, \text{units} \).

📌 Hints / Properties Used:

Circle chord length formula: \( L = 2r \sin\left(\frac{\theta}{2}\right) \)
For unit circle, \( r = 1 \)
Trigonometric identities for \( \sin \)

📊 Visual Diagram Suggestion:

A diagram illustrating a unit circle with a central angle of \( 45^\circ \), showing the chord connecting endpoints on the circle, highlighting the radius and right triangle formed with the angle.

📖 Factual Verification & Reference:

Verified against NCERT Class XII Mathematics, Chapter on Circles.

Question 5 of 10 Authentic UPSC NDA Question

A plane is observed to be approaching the airport. It is at a distance of 10 km from the point of observation and makes an angle of elevation of 67.5°.

What is the height of the plane above the ground?

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💡 Solution

Option (c) is correct.

Explanation:

Let 'h' be the height of the plane above the ground and 'd' be the distance of the plane from the point of observation. Given:

\[ d = 10 \text{ km} \] \[ \text{Angle of elevation} (\theta) = 67.5^\circ \]

In a right-angled triangle formed by the plane, the observation point, and the point on the ground directly below the plane, we have:

\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{h}{d} \]

Therefore, the height 'h' is:

\[ h = d \sin(\theta) \] \[ h = 10 \sin(67.5^\circ) \]

We know the half-angle formula for sine:

\[ \sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos x}{2}} \]

Here, \( x = 135^\circ \), so \( \frac{x}{2} = 67.5^\circ \). Since \( 67.5^\circ \) is in the first quadrant, \( \sin(67.5^\circ) \) is positive.

\[ \sin(67.5^\circ) = \sqrt{\frac{1 - \cos(135^\circ)}{2}} \]

We know that \( \cos(135^\circ) = \cos(180^\circ - 45^\circ) = -\cos(45^\circ) = -\frac{1}{\sqrt{2}} \). Substituting this value:

\[ \sin(67.5^\circ) = \sqrt{\frac{1 - \left(-\frac{1}{\sqrt{2}}\right)}{2}} = \sqrt{\frac{1 + \frac{1}{\sqrt{2}}}{2}} \] \[ = \sqrt{\frac{\frac{\sqrt{2} + 1}{\sqrt{2}}}{2}} = \sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}} \] \[ = \sqrt{\frac{(\sqrt{2} + 1)\sqrt{2}}{2\sqrt{2}\sqrt{2}}} = \sqrt{\frac{2 + \sqrt{2}}{4}} \] \[ = \frac{\sqrt{2 + \sqrt{2}}}{2} \]

Now, substitute this back into the equation for 'h':

\[ h = 10 \times \frac{\sqrt{2 + \sqrt{2}}}{2} \] \[ h = 5\sqrt{2 + \sqrt{2}} \text{ km} \]

💡 Hints & Visual Guide

📌 Hints / Properties Used:

Trigonometric relationship: \( \sin(\theta) = \text{Opposite / Hypotenuse} \).
Half-angle identity for sine: \( \sin(x/2) = \sqrt{(1 - \cos x) / 2} \) for angles in the first quadrant.
Value of \( \cos(135^\circ) = -1/\sqrt{2} \).

📊 Visual Diagram Suggestion:

Draw a right-angled triangle. Label the hypotenuse as the distance from the observation point to the plane (10 km). Label the vertical side opposite to the angle of elevation as the height 'h' of the plane. Label the angle of elevation as 67.5°. This visual representation clearly shows how the sine function relates the height, distance, and angle.

Question 6 of 10 Authentic UPSC NDA Question

Direction for Questions (49-50): Consider the following for the two (02) items that follow: The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly a long the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30°, 45° and 60°, respectively. Let PQ = a and QR = b.

What is MN equal to?

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Question 7 of 10 Authentic UPSC NDA Question

Direction for Questions (49-50): Consider the following for the two (02) items that follow: The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly a long the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30°, 45° and 60°, respectively. Let PQ = a and QR = b.

What is PN equal to?

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Question 8 of 10 Authentic UPSC NDA Question

A man at M standing 100 m away from the base (P) of a chimney of height 50 m. He observes the angle of elevation of the highest point (Q) of the smoke to be 45°. The highest point of the chimney is at R. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?

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Question 9 of 10 Authentic UPSC NDA Question

Consider the following for the next two (02) items that follow :A flagstaff 20 m long standing on a pillar 10 mhigh subtends an angle tan−1(0.5) at a point Pon the ground. Let q be the angle subtended by the pillar at this point P .

What is a possible value of tanθ ?

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Question 10 of 10 Authentic UPSC NDA Question

Consider the following for the next two (02) items that follow :A flagstaff 20 m long standing on a pillar 10 mhigh subtends an angle tan−1(0.5) at a point Pon the ground. Let q be the angle subtended bythe pillar at this point P .

If x is the distance of P from bottom of the pillar,then consider the following statements:

1. x can take two values which are in the ratio 1 : 3.

2. x can be equal to height of the flagstaff.

Which of the statements given above is/are correct ?

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Height and distance problems Questions

Solution

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