SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c) Assertion (A) is true, but Reason (R) is false.
- (d) Assertion (A) is false, but Reason (R) is true.
-
Q1. 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 line of sight is always perpendicular to the horizontal ground level regardless of the position of the object. -
Q2. Assertion (A): The angle of elevation of an object viewed is the angle formed by the line of sight with the horizontal when the object is above the horizontal level
Reason (R): An observer has to tilt their head upwards to look at an object located higher than their eye level. -
Q3. Assertion (A): When an observer looks downwards at an object, the angle which the line of sight makes with the horizontal is called the angle of depression.
Reason (R): The angle of depression is always measured from the vertical plumb line down towards the object. -
Q4. Assertion (A): The angle of depression of an object from a point is equal to the angle of elevation of that point from the object.
Reason (R): Alternate interior angles are equal because the horizontal line drawn from the observer's eye is parallel to the horizontal ground. -
Q5. Assertion (A): As the sun's altitude (angle of elevation) increases from $30^\circ$ to $60^\circ$, the length of the shadow of a vertical tower decreases.
Reason (R): The tangent ratio $\tan \theta$ decreases as the angle $\theta$ increases in the first quadrant. -
Q6. Assertion (A): If a vertical pole of height $10\text{ m}$ casts a shadow of length $10\text{ m}$ on the ground, the angle of elevation of the sun is $45^\circ$.
Reason (R): In a right-angled triangle, when the base and perpendicular are equal, the angle opposite to the perpendicular is $45^\circ$ because $\tan 45^\circ = 1$. -
Q7. Assertion (A): In a right-angled triangle formed for a height and distance problem, the hypotenuse is always shorter than the sum of the base and the perpendicular height.
Reason (R): The hypotenuse is the longest side of a right-angled triangle, satisfying the triangle inequality theorem where its length is strictly less than the sum of the other two sides. -
Q8. Assertion (A): If the angles of elevation of the top of a tower from two points at distances $4\text{ m}$ and $9\text{ m}$ from the base in the same straight line are complementary, the height of the tower is $6\text{ m}$.
Reason (R): If two angles are complementary, their sum is $180^\circ$. -
Q9. Assertion (A): A man moves towards a vertical tower on level ground. As he moves closer, the angle of elevation of the top of the tower increases.
Reason (R): As the distance $x$ from the base of the tower decreases while the height $h$ remains constant, the ratio $\frac{h}{x}$ increases, leading to a larger tangent value and hence a greater angle of elevation. -
Q10. Assertion (A): When an airplane flies horizontally at a constant altitude away from an observer, the angle of elevation of the airplane continuously decreases.
Reason (R): As the horizontal distance between the observer and the airplane increases, the line of sight drops closer to the horizontal plane. -
Q11. Assertion (A): The depth of the reflection of a cloud below the surface of a calm lake is equal to the height of the cloud above the surface of the lake.
Reason (R): Plane mirrors and calm water surfaces form virtual images at an equal perpendicular distance behind the reflecting surface as the object is in front of it. -
Q12. Assertion (A): If a straight bridge of length $L$ crosses a river at an angle $\theta$ to the bank, the actual breadth of the river perpendicular to the banks is given by $L \sin \theta$.
Reason (R): The breadth of the river forms the perpendicular side of a right-angled triangle where the bridge serves as the hypotenuse and the bank angle is $\theta$. -
Q13. Assertion (A): If two observers on opposite sides of a vertical tower observe the top at the same angle of elevation $\theta$, they must be equidistant from the base of the tower.
Reason (R): $\tan \theta = \frac{h}{x_1}$ and $\tan \theta = \frac{h}{x_2}$ implies $x_1 = x_2$ for equal heights and angles -
Q14. Assertion (A): When a tree breaks due to a storm and touches the ground, the original height of the tree is equal to the sum of the height of the remaining vertical part and the length of the broken part touching the ground.
Reason (R): The total length of a rigid body remains conserved before and after it breaks and bends. -
Q15. Assertion (A): If a car takes $t$ seconds to move between two points where angles of depression change from $60^\circ$ to $30^\circ$, its speed can be determined if the height of the tower is known
Reason (R): Speed is calculated as distance divided by time, and the distance traveled along the ground can be found using standard trigonometric ratios corresponding to the angles of depression. -
Q16. Assertion (A): An object approaching the base of a tower at a uniform speed covers unequal horizontal distances in equal intervals of time when measured by angles of elevation.
Reason (R): The trigonometric function $\cot \theta$ (or $\tan \theta$) is non-linear, meaning equal increments in angles do not correspond to equal linear displacements along the ground. -
Q17. Assertion (A): To find the height of an inaccessible tower using a single observation point on level ground, we need to know at least the distance from the base and the angle of elevation.
Reason (R): A single right-angled triangle can be completely solved if one side and one acute angle are known -
Q18. Assertion (A): When an observer of height $h_0$ stands at a distance $d$ from a vertical tower and measures the angle of elevation of the top, the height of the tower above the observer's eye level is given by $d \tan \theta$
Reason (R): The line of sight originates from the ground level rather than the observer's eyes when calculating standard heights and distances -
Q19. Assertion (A): The angle of elevation of the top of a vertical tower from any point on horizontal ground can equal $90^\circ$.
Reason (R): An angle of elevation of $90^\circ$ would require the observer to stand at the exact point directly underneath the top of the tower (at the base), where the horizontal distance becomes zero and the ratio becomes undefined. -
Q20. Assertion (A): If a flagstaff of length $h$ is mounted on top of a building, the angle of elevation of the top of the flagstaff from a point on the ground is always smaller than the angle of elevation of the bottom of the flagstaff.
Reason (R): The top of the flagstaff is located at a greater vertical height above the ground than the bottom of the flagstaff, which increases the angle of elevation from any point on the horizontal ground.
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
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.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
Copyright © 2026 Deepa Maths Academy | Global Examination & Academic Portal. All Rights Reserved.