✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 9: Some Applications of Trigonometry
Assertion-Reason Question Bank
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Assertion (A)
If a \(1.5\text{ m}\) tall person stands \(28.5\text{ m}\) away from a \(30\text{ m}\) high tower, the angle of elevation of top of tower from his eye is \(45^\circ\).
Reason (R)
Height above eye level is \(30 - 1.5 = 28.5\text{ m}\). \(\tan \theta = 28.5/28.5 = 1 \implies \theta = 45^\circ\).
Correct Option: (B)
Both A and R are true calculations.
Both A and R are true calculations.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Angle of depression is measured from the horizontal line downward.
Reason (R)
Angle of elevation is measured from the horizontal line upward.
Correct Option: (B)
Both A and R are true measurement definitions.
Both A and R are true measurement definitions.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If a tower of height \(h\) subtends angle \(\theta\) at a point on ground \(d\) meters away, then \(h = d \tan \theta\).
Reason (R)
\(\tan \theta = \frac{\text{Adjacent}}{\text{Opposite}}\).
Correct Option: (C)
Assertion (A) is true (\(h/d = \tan\theta \implies h = d \tan\theta\)). Reason (R) is false because \(\tan\theta = \text{Opposite}/\text{Adjacent}\), not \(\text{Adjacent}/\text{Opposite}\).
Assertion (A) is true (\(h/d = \tan\theta \implies h = d \tan\theta\)). Reason (R) is false because \(\tan\theta = \text{Opposite}/\text{Adjacent}\), not \(\text{Adjacent}/\text{Opposite}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
As sun's elevation decreases from \(60^\circ\) to \(30^\circ\), shadow length decreases.
Reason (R)
Lower angle of elevation produces a longer shadow length.
Correct Option: (D)
Assertion (A) is false (shadow increases). Reason (R) is true.
Assertion (A) is false (shadow increases). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If angle of elevation of a kite is \(60^\circ\) and string length is \(100\text{ m}\), height of kite is \(50\sqrt{3}\text{ m}\).
Reason (R)
Height \(h = 100 \sin 60^\circ = 100 \times \frac{\sqrt{3}}{2} = 50\sqrt{3}\text{ m}\).
Correct Option: (B)
Both A and R are true and R provides the trigonometry calculation.
Both A and R are true and R provides the trigonometry calculation.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If shadow of pole is equal to height, sun's elevation is \(60^\circ\).
Reason (R)
When height = shadow, \(\tan \theta = 1 \implies \theta = 45^\circ\).
Correct Option: (D)
Assertion (A) is false (it is \(45^\circ\)). Reason (R) is true.
Assertion (A) is false (it is \(45^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Height of tower with \(30^\circ\) elevation at \(100\text{ m}\) distance is \(100\sqrt{3}\text{ m}\).
Reason (R)
Height \(h = 100 \tan 30^\circ = 100 \times \frac{1}{\sqrt{3}} = \frac{100}{\sqrt{3}}\text{ m}\).
Correct Option: (D)
Assertion (A) is false (it is \(100/\sqrt{3}\), not \(100\sqrt{3}\)). Reason (R) is true.
Assertion (A) is false (it is \(100/\sqrt{3}\), not \(100\sqrt{3}\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The angle of elevation of top of a tower from a point on ground is \(30^\circ\). On moving \(20\text{ m}\) towards tower, angle becomes \(60^\circ\). Height of tower is \(10\sqrt{3}\text{ m}\).
Reason (R)
\(\tan 60^\circ = \sqrt{3}\) and \(\tan 30^\circ = 1/\sqrt{3}\).
Correct Option: (B)
Both A and R are true.
Both A and R are true.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If shadow of a tower is \(\sqrt{3}\) times its height, angle of elevation of sun is \(30^\circ\).
Reason (R)
\(\tan \theta = \frac{\text{Shadow}}{\text{Height}} = \sqrt{3} \implies \theta = 60^\circ\).
Correct Option: (C)
Assertion (A) is true (\(\tan\theta = h/(h\sqrt{3}) = 1/\sqrt{3} \implies \theta = 30^\circ\)). Reason (R) is false because formula is \(\tan\theta = \text{Height}/\text{Shadow}\), not \(\text{Shadow}/\text{Height}\).
Assertion (A) is true (\(\tan\theta = h/(h\sqrt{3}) = 1/\sqrt{3} \implies \theta = 30^\circ\)). Reason (R) is false because formula is \(\tan\theta = \text{Height}/\text{Shadow}\), not \(\text{Shadow}/\text{Height}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If two towers of heights \(h_1\) and \(h_2\) subtend angles of \(60^\circ\) and \(30^\circ\) at the midpoint of line joining their feet, then \(h_1 : h_2 = 3 : 1\).
Reason (R)
Let midpoint distance be \(x\). \(\tan 60^\circ = h_1/x = \sqrt{3} \implies h_1 = x\sqrt{3}\). \(\tan 30^\circ = h_2/x = 1/\sqrt{3} \implies h_2 = x/\sqrt{3}\). Ratio \(h_1/h_2 = x\sqrt{3} / (x/\sqrt{3}) = 3/1\).
Correct Option: (A)
Ratio \(h_1/h_2 = \sqrt{3} / (1/\sqrt{3}) = 3:1\). Reason explains Assertion.
Ratio \(h_1/h_2 = \sqrt{3} / (1/\sqrt{3}) = 3:1\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If a tower \(30\text{ m}\) high casts a shadow \(30\sqrt{3}\text{ m}\) long, the angle of elevation of the sun is \(30^\circ\).
Reason (R)
\(\tan \theta = \frac{\text{Height}}{\text{Shadow}} = \frac{30}{30\sqrt{3}} = \frac{1}{\sqrt{3}} \implies \theta = 30^\circ\).
Correct Option: (A)
\(\tan\theta = 1/\sqrt{3} \implies \theta = 30^\circ\). Reason explains Assertion.
\(\tan\theta = 1/\sqrt{3} \implies \theta = 30^\circ\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
A ladder \(15\text{ m}\) long makes an angle of \(60^\circ\) with a wall. The height of the point on the wall reached by the ladder is \(15/2\text{ m}\).
Reason (R)
If ladder makes \(60^\circ\) with the wall, angle with ground is \(30^\circ\). Height \(h = 15 \sin 30^\circ = 15 \times (1/2) = 15/2\text{ m}\).
Correct Option: (A)
Angle with ground \(= 90 - 60 = 30^\circ\). Height \(h = 15 \sin 30^\circ = 15/2\text{ m}\). Reason explains Assertion.
Angle with ground \(= 90 - 60 = 30^\circ\). Height \(h = 15 \sin 30^\circ = 15/2\text{ m}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If the height of a tower and distance of point of observation from its foot are both doubled, the angle of elevation remains unchanged.
Reason (R)
\(\tan \theta = \frac{h}{d} = \frac{2h}{2d} = \frac{h}{d}\), so \(\theta\) remains the same.
Correct Option: (A)
Ratio \((2h)/(2d) = h/d\), so \(\theta\) does not change. Reason explains Assertion.
Ratio \((2h)/(2d) = h/d\), so \(\theta\) does not change. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If angle of elevation of top of a tower from two points at distances \(a\) and \(b\) (\(a > b\)) from base and in same straight line are complementary, height of tower is \(h = \sqrt{ab}\).
Reason (R)
Complementary angles means \(\theta_1 + \theta_2 = 180^\circ\).
Correct Option: (C)
Assertion (A) is true (\(h/a = \tan\theta, h/b = \cot\theta \implies h^2/ab = 1 \implies h = \sqrt{ab}\)). Reason (R) is false because complementary angles sum to \(90^\circ\), not \(180^\circ\).
Assertion (A) is true (\(h/a = \tan\theta, h/b = \cot\theta \implies h^2/ab = 1 \implies h = \sqrt{ab}\)). Reason (R) is false because complementary angles sum to \(90^\circ\), not \(180^\circ\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If string of kite is horizontal, its height from ground is equal to string length.
Reason (R)
Height \(h = L \sin \theta\). If string is horizontal (\(\theta = 0^\circ\), \(h = 0\)).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
A tree breaks due to storm and the broken part bends so that top touches ground at \(30^\circ\). If distance from foot to top is \(8\text{ m}\), tree height is \(8\sqrt{3}\text{ m}\).
Reason (R)
Total height = standing part + hypotenuse \(= 8\tan 30^\circ + 8\sec 30^\circ = 8/\sqrt{3} + 16/\sqrt{3} = 24/\sqrt{3} = 8\sqrt{3}\text{ m}\).
Correct Option: (B)
Both A and R are true tree problem calculations.
Both A and R are true tree problem calculations.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.
Reason (R)
The angle of elevation is formed by line of sight with the horizontal line when the object is above horizontal level.
Correct Option: (B)
Both A and R are true definitions of line of sight and angle of elevation.
Both A and R are true definitions of line of sight and angle of elevation.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If height of tower is doubled and distance is halved, angle of elevation remains same.
Reason (R)
\(\tan \theta' = \frac{2h}{d/2} = 4 \frac{h}{d} = 4 \tan \theta\), so angle changes significantly.
Correct Option: (D)
Assertion (A) is false (tangent becomes 4 times). Reason (R) is true.
Assertion (A) is false (tangent becomes 4 times). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
When the sun's altitude increases from \(30^\circ\) to \(60^\circ\), the length of shadow of a tower decreases.
Reason (R)
Higher angle of elevation corresponds to shorter shadow length.
Correct Option: (B)
Both A and R are true shadow behavior properties.
Both A and R are true shadow behavior properties.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Angle of elevation can be greater than \(90^\circ\).
Reason (R)
Angle of elevation is measured from horizontal line upward and ranges between \(0^\circ\) and \(90^\circ\).
Correct Option: (D)
Assertion (A) is false (\(0 \le \theta < 90^\circ\)). Reason (R) is true.
Assertion (A) is false (\(0 \le \theta < 90^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Angle of depression is measured from vertical line.
Reason (R)
Angle of depression is measured from horizontal line drawn through eye level of observer.
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Angle of elevation is always equal to \(45^\circ\) regardless of tower height.
Reason (R)
Angle of elevation depends on ratio of height to distance (\(\tan \theta = h/d\)).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If a pole \(6\text{ m}\) high casts a shadow \(2\sqrt{3}\text{ m}\) long, sun's elevation is \(60^\circ\).
Reason (R)
\(\tan 60^\circ = 1/\sqrt{3}\).
Correct Option: (C)
Assertion (A) is true (\(\tan\theta = 6/(2\sqrt{3}) = 3/\sqrt{3} = \sqrt{3} \implies \theta = 60^\circ\)). Reason (R) is false because \(\tan 60^\circ = \sqrt{3}\), not \(1/\sqrt{3}\).
Assertion (A) is true (\(\tan\theta = 6/(2\sqrt{3}) = 3/\sqrt{3} = \sqrt{3} \implies \theta = 60^\circ\)). Reason (R) is false because \(\tan 60^\circ = \sqrt{3}\), not \(1/\sqrt{3}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The angle of depression of an object viewed is equal to the angle of elevation of the observer from the object.
Reason (R)
The horizontal line through the observer's eye and horizontal line through the object are parallel, making the angle of depression and angle of elevation alternate interior angles.
Correct Option: (A)
Alternate interior angles formed by parallel horizontal lines are equal. Reason explains Assertion.
Alternate interior angles formed by parallel horizontal lines are equal. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Clinometer is an instrument used for measuring angles of elevation or depression.
Reason (R)
Trigonometry was originally used by astronomers to find distances from Earth to stars.
Correct Option: (B)
Both are true historical/instrument facts.
Both are true historical/instrument facts.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The height of a building whose top has angle of elevation \(45^\circ\) from \(50\text{ m}\) away is \(50\text{ m}\).
Reason (R)
At \(\theta = 45^\circ\), \(\sin 45^\circ = 1\).
Correct Option: (C)
Assertion (A) is true (\(h = 50 \tan 45^\circ = 50\text{ m}\)). Reason (R) is false because \(\sin 45^\circ = 1/\sqrt{2}\), not \(1\).
Assertion (A) is true (\(h = 50 \tan 45^\circ = 50\text{ m}\)). Reason (R) is false because \(\sin 45^\circ = 1/\sqrt{2}\), not \(1\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
As the observer moves closer to a tower, the angle of elevation of its top increases.
Reason (R)
In \(\tan \theta = \frac{h}{d}\), as distance \(d\) decreases for constant height \(h\), \(\tan \theta\) increases, so \(\theta\) increases.
Correct Option: (A)
Decreasing denominator \(d\) increases \(\tan\theta\) and hence angle \(\theta\). Reason explains Assertion.
Decreasing denominator \(d\) increases \(\tan\theta\) and hence angle \(\theta\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The angle of elevation of the sun when the length of the shadow of a pole is equal to its height is \(45^\circ\).
Reason (R)
If height \(h\) equals shadow length \(s\), then \(\tan \theta = h/s = 1 \implies \theta = 45^\circ\).
Correct Option: (A)
\(\tan\theta = h/h = 1 \implies \theta = 45^\circ\). Reason provides the proof and explains Assertion.
\(\tan\theta = h/h = 1 \implies \theta = 45^\circ\). Reason provides the proof and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The angle of elevation of a cloud from a point \(h\) meters above a lake is \(\alpha\) and angle of depression of its reflection in lake is \(\beta\). Then height of cloud is \(h\frac{\tan\beta + \tan\alpha}{\tan\beta - \tan\alpha}\).
Reason (R)
Angle of elevation of cloud and angle of depression of reflection are always equal (\(\alpha = \beta\)).
Correct Option: (C)
Assertion (A) is true standard result. Reason (R) is false because observer is \(h\) meters ABOVE lake, so \(\beta > \alpha\) always.
Assertion (A) is true standard result. Reason (R) is false because observer is \(h\) meters ABOVE lake, so \(\beta > \alpha\) always.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If two objects are viewed from a tower, the object farther from the tower has a smaller angle of depression.
Reason (R)
Angle of depression increases as distance from foot of tower increases.
Correct Option: (C)
Assertion (A) is true (farther object has smaller angle). Reason (R) is false because angle of depression DECREASES as distance increases.
Assertion (A) is true (farther object has smaller angle). Reason (R) is false because angle of depression DECREASES as distance increases.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.