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.
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Q1. Assertion (A): In a right-angled triangle $ABC$ right-angled at $B$, if $\sin A = \frac{3}{5}$, then the length of the opposite side to angle $A$ can be $3k$ and the hypotenuse $5k$ for any positive integer $k$.
Reason (R): For any acute angle $\theta$ in a right triangle, the sine of $\theta$ is defined as the ratio of the side adjacent to $\theta$ to the hypotenuse. -
Q2. Assertion (A): The value of $\sin \theta$ can never be greater than $1$.
Reason (R): The hypotenuse is the longest side in any right-angled triangle, making the ratio of the opposite side to the hypotenuse always less than or equal to $1$. -
Q3. Assertion (A): $\sec A$ is the abbreviation used for the cosecant of angle $A$.
Reason (R): $\sec A$ is the reciprocal of $\cos A$, whereas cosecant is abbreviated as $\csc A$ (or $\operatorname{cosec} A$) and is the reciprocal of $\sin A$. -
Q4. Assertion (A): For any acute angle $\theta$, $\sin^2 \theta + \cos^2 \theta = 1$.
Reason (R): This identity is a direct consequence of the Pythagorean theorem applied to a right-angled triangle where the sides satisfy $p^2 + b^2 = h^2$. -
Q5. Assertion (A): As the angle $\theta$ increases from $0^\circ$ to $90^\circ$, the value of $\tan \theta$ increases continuously and becomes undefined at $90^\circ$
Reason (R): $\tan 90^\circ = \frac{\sin 90^\circ}{\cos 90^\circ} = \frac{1}{0}$, which is undefined. -
Q6. Assertion (A): $\sin 42^\circ = \cos 48^\circ$
Reason (R): For any acute angle $\theta$, $\sin(90^\circ - \theta) = \cos \theta$ and $90^\circ - 42^\circ = 48^\circ$. -
Q7. Assertion (A): The value of $\sin 30^\circ \cdot \cos 60^\circ + \cos 30^\circ \cdot \sin 60^\circ$ is equal to $1$
Reason (R): This expression is the expansion of $\sin(A + B)$ where $A = 30^\circ$ and $B = 60^\circ$, yielding $\sin(90^\circ) = 1$. -
Q8. Assertion (A): If $\tan A = \frac{4}{3}$, then $\cot A = \frac{3}{4}$.
Reason (R): The cotangent of an angle is the reciprocal of its tangent. -
Q9. Assertion (A): $1 + \tan^2 A = \sec^2 A$ is valid for all $0^\circ \le A < 90^\circ$.
Reason (R): Dividing every term of the fundamental identity $\sin^2 A + \cos^2 A = 1$ by $\cos^2 A$ yields this relation. -
Q10. Assertion (A): $\csc^2 \theta - 1 = \cot^2 \theta$ for $0^\circ < \theta \le 90^\circ$.
Reason (R): This identity can be derived by dividing the fundamental trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ by $\sin^2 \theta$. -
Q11. Assertion (A): As the angle $\theta$ increases from $0^\circ$ to $90^\circ$, the value of $\cos \theta$ increases from $0$ to $1$.
Reason (R): $\cos 0^\circ = 1$ and $\cos 90^\circ = 0$, meaning the cosine value decreases as the angle increases. -
Q12. 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 leve
Reason (R): When looking upwards at an object, the observer must elevate their line of sight from the horizontal reference line. -
Q13. 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 from the observer and the horizontal ground level are parallel lines cut by a transversal line of sight. -
Q14. Assertion (A): The minimum value of $\sec \theta$ for an acute angle $\theta$ is $1$.
Reason (R): Since $\cos \theta \le 1$ for all real angles, its reciprocal $\sec \theta \ge 1$. -
Q15. Assertion (A): $\csc 0^\circ$ is not defined.
Reason (R): $\csc 0^\circ = \frac{1}{\sin 0^\circ} = \frac{1}{0}$, which is undefined -
Q16. Assertion (A): The value of $\sin \theta$ increases as $\theta$ increases from $0^\circ$ to $90^\circ$.
Reason (R): $\sin 0^\circ = 0$ and $\sin 90^\circ = 1$, and the sine function is strictly increasing in the first quadrant. -
Q17. Assertion (A): $\sec 50^\circ = \csc 40^\circ$
Reason (R): The secant of an acute angle is equal to the cosecant of its complementary angle, and $90^\circ - 50^\circ = 40^\circ$. -
Q18. Assertion (A): $\tan 20^\circ \cdot \tan 70^\circ = 1$.
Reason (R): Since $\tan(90^\circ - \theta) = \cot \theta$, $\tan 70^\circ = \cot 20^\circ$, and the product of a trigonometric ratio and its reciprocal is always $1$. -
Q19. Assertion (A): The maximum possible value for $(\sin \theta + \cos \theta)$ is $\sqrt{2}$.
Reason (R): Squaring the expression yields $\sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = 1 + \sin(2\theta)$, whose maximum value occurs when $\sin(2\theta) = 1$, giving $\sqrt{1 + 1} = \sqrt{2}$. -
Q20. Assertion (A): If $\sin \theta = \frac{4}{3}$, a valid right triangle can be constructed to find $\cos \theta$.
Reason (R): The sine ratio can never exceed $1$ for real angles because the opposite side can never be longer than the hypotenuse in a right triangle.
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.
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