Assertion-Reason MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. ASSERTION (A): In a right-angled triangle, the sine of an acute angle is the ratio of the length of the opposite side to the length of the hypotenuse.

REASON (R): The hypotenuse is always the longest side in a right-angled triangle, and the opposite side is one of the legs.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): If $\tan \theta = \frac{3}{4}$ in a right triangle, then $\sin \theta = \frac{3}{5}$.

REASON (R): For any acute angle $\theta$, $1 + \tan^2 \theta = \sec^2 \theta$ and $\sec \theta = \frac{1}{\cos \theta}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): In a right triangle ABC, right-angled at B, $\tan A \cdot \tan C = 1$.

REASON (R): A and C are complementary angles in a right-angled triangle ABC, right-angled at B.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): If $\sin \theta = \frac{5}{13}$, then $\text{cosec } \theta = \frac{13}{5}$.

REASON (R): $\text{cosec } \theta$ is the reciprocal of $\sin \theta$, i.e., $\text{cosec } \theta = \frac{1}{\sin \theta}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): If the sides of a right triangle are 6, 8, and 10, then the sine of the smallest acute angle is $\frac{6}{10}$.

REASON (R): In a right triangle, the smallest acute angle is opposite the shortest side.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The value of $\sin \theta$ for an acute angle $\theta$ cannot be greater than 1.

REASON (R): The definition of sine is the ratio of the opposite side to the hypotenuse, and the hypotenuse is always greater than the opposite side in a right triangle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The value of $\sin 60^\circ$ is equal to the value of $\cos 30^\circ$.

REASON (R): $60^\circ$ and $30^\circ$ are complementary angles, and $\sin(90^\circ - \theta) = \cos \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): $\tan 45^\circ = 1$.

REASON (R): In an isosceles right-angled triangle, the two acute angles are $45^\circ$, and the opposite side is equal to the adjacent side for $45^\circ$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): $\tan 0^\circ = 0$ and $\cot 90^\circ = 0$.

REASON (R): $\cot \theta = \tan (90^\circ - \theta)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): The value of $\sin 90^\circ$ is 1.

REASON (R): In a right triangle, as one acute angle approaches $90^\circ$, the opposite side approaches the hypotenuse in length.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): $\sec 0^\circ$ is undefined.

REASON (R): $\sec \theta = \frac{1}{\cos \theta}$ and $\cos 0^\circ = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): $\sin 1^\circ > \sin 0^\circ$.

REASON (R): The value of $\sin \theta$ increases as $\theta$ increases from $0^\circ$ to $90^\circ$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): $\sin^2 \theta + \cos^2 \theta = 1$ for all real values of $\theta$.

REASON (R): This identity is derived from the Pythagorean theorem in a unit circle or a right-angled triangle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): $\sec^2 \theta - \tan^2 \theta = 1$ for all values of $\theta$ where $\tan \theta$ and $\sec \theta$ are defined.

REASON (R): Dividing the identity $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$ gives $\tan^2 \theta + 1 = \sec^2 \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): $\tan \theta \cdot \cot \phi = 1$ if $\theta + \phi = 90^\circ$.

REASON (R): $\tan \theta$ and $\cot \theta$ are reciprocal ratios, and $\cot \phi = \tan (90^\circ - \phi)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): The expression $(\sin \theta + \cos \theta)^2$ simplifies to $1 + \sin 2\theta$.

REASON (R): $(\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta$, and $\sin^2 \theta + \cos^2 \theta = 1$ and $2 \sin \theta \cos \theta = \sin 2\theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): The expression $\frac{1}{\text{cosec } A - \cot A}$ is equal to $\text{cosec } A + \cot A$.

REASON (R): $\text{cosec}^2 A - \cot^2 A = 1$, which is a difference of squares formula.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The identity $\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \sin^2 \theta$ is true.

REASON (R): $\tan^2 \theta - \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta = \sin^2 \theta \left( \frac{1}{\cos^2 \theta} - 1 \right) = \sin^2 \theta (\sec^2 \theta - 1) = \sin^2 \theta \tan^2 \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): To convert $120^\circ$ to radians, multiply by $\frac{\pi}{180}$.

REASON (R): The relationship between degree and radian measure is $\pi \text{ radians} = 180^\circ$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): An arc of length 10 cm in a circle of radius 5 cm subtends an angle of 2 radians at the centre.

REASON (R): The length of an arc $l$ in a circle of radius $r$ is given by $l = r\theta$, where $\theta$ is the angle in radians.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): The area of a sector with radius 6 cm and central angle $30^\circ$ is $3\pi$ cm$^2$.

REASON (R): The area of a sector is given by $\frac{1}{2} r^2 \theta$, where $\theta$ is in degrees.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): An angle of 1 radian is approximately $57.3^\circ$.

REASON (R): 1 radian is defined as the angle subtended at the center by an arc equal in length to the radius of the circle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): The angle measure of $2\pi$ radians is equivalent to $360^\circ$.

REASON (R): The circumference of a circle is $2\pi r$, and a complete revolution subtends an angle proportional to the circumference.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): A minute hand of a clock rotating through 30 minutes covers an angle of $\pi$ radians.

REASON (R): In 60 minutes, the minute hand completes a full revolution ($360^\circ$ or $2\pi$ radians).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The value of $\sin \frac{3\pi}{2}$ is -1.

REASON (R): The point on the unit circle corresponding to $\frac{3\pi}{2}$ radians is $(0, -1)$, and $\sin \theta$ is the y-coordinate.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): The range of the function $f(x) = \sin x$ is $[-1, 1]$.

REASON (R): In the unit circle, the maximum value of the y-coordinate is 1 and the minimum value is -1.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): The function $f(x) = \tan x$ is positive in the third quadrant.

REASON (R): In the third quadrant, both the x and y coordinates are negative, and $\tan \theta = y/x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): The period of $\sin x$ is $2\pi$.

REASON (R): The values of $\sin x$ repeat every $2\pi$ rotation on the unit circle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): The domain of $\tan x$ is all real numbers.

REASON (R): $\tan x$ is defined as $\frac{\sin x}{\cos x}$, and $\cos x$ is zero at odd multiples of $\frac{\pi}{2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): If the terminal side of an angle passes through $(-1, 0)$, the value of $\cos \theta$ is -1.

REASON (R): On the unit circle, $\cos \theta$ is the x-coordinate of the point of intersection of the terminal side with the circle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The graph of $y = \sin x$ passes through the origin $(0, 0)$.

REASON (R): $\sin 0^\circ = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): The graph of $y = \cos x$ is symmetric about the y-axis.

REASON (R): $\cos x$ is an even function, i.e., $\cos(-x) = \cos x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): The graph of $y = \tan x$ has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

REASON (R): $\tan x = \frac{\sin x}{\cos x}$, and the function is undefined when $\cos x = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): The amplitude of $y = -2 \sin x$ is 2.

REASON (R): The amplitude of $y = A \sin(Bx)$ or $y = A \cos(Bx)$ is $|A|$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): The graph of $y = \sin(2x)$ has a period of $\pi$.

REASON (R): The period of $y = \sin(Bx)$ is $\frac{2\pi}{|B|}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The range of $y = \sec x$ is the set of all real numbers.

REASON (R): $\sec x = \frac{1}{\cos x}$, and the range of $\cos x$ is $[-1, 1]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): $\sin(A+B) = \sin A \cos B + \cos A \sin B$.

REASON (R): This is the formula for the sine of the sum of two angles, derived using geometric constructions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): $\cos 2A = 1 - 2 \sin^2 A$.

REASON (R): This formula is derived from the identity $\cos 2A = \cos^2 A - \sin^2 A$ and the Pythagorean identity $\cos^2 A = 1 - \sin^2 A$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): The value of $\tan 15^\circ$ is $2 - \sqrt{3}$.

REASON (R): $\tan 15^\circ = \tan (45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): $\sin 3A = 3 \sin A - 4 \sin^3 A$.

REASON (R): This is the standard triple angle formula for sine, derived from sum and double angle formulas.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): $\frac{1 - \cos 2\theta}{1 + \cos 2\theta} = \tan^2 \theta$.

REASON (R): $1 - \cos 2\theta = 2 \sin^2 \theta$ and $1 + \cos 2\theta = 2 \cos^2 \theta$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): $\sin(A+B) \sin(A-B) = \sin^2 A - \sin^2 B$.

REASON (R): This identity can be derived using the sum and difference formulas for sine.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$.

REASON (R): Adding the expansions of $\sin(A+B)$ and $\sin(A-B)$ gives this result.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): $\sin C + \sin D = 2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$.

REASON (R): Let $A = \frac{C+D}{2}$ and $B = \frac{C-D}{2}$, then $A+B = C$ and $A-B = D$. Substitute these into the product-to-sum formula for $2 \sin A \cos B$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): $\cos 4\theta + \cos 2\theta = 2 \cos 3\theta \cos \theta$.

REASON (R): This is an application of the sum-to-product formula $\cos C + \cos D = 2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): $2 \sin 5x \sin 3x = \cos 2x - \cos 8x$.

REASON (R): The product-to-sum formula for $2 \sin A \sin B$ is $\cos(A-B) - \cos(A+B)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): $\frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x$.

REASON (R): Using sum-to-product formulas, the numerator becomes $2 \sin 4x \cos x$ and the denominator becomes $2 \cos 4x \cos x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The formula $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ is always true.

REASON (R): This formula allows us to convert a product of cosine and sine into a sum or difference of sine functions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The equation $\sin x = 1/2$ has exactly two solutions in the interval $[0, 2\pi)$.

REASON (R): $\sin x$ is positive in the first and second quadrants, and there is one angle in each quadrant in $[0, 2\pi)$ where $\sin x = 1/2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): The general solution of $\tan x = 1$ is $x = n\pi + \frac{\pi}{4}$, where $n \in \mathbb{Z}$.

REASON (R): The tangent function has a period of $\pi$, and $\tan \frac{\pi}{4} = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): The equation $\cos x = 2$ has no real solutions.

REASON (R): The range of the cosine function is $[-1, 1]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): The principal solution of $\cos x = -1/\sqrt{2}$ is $\frac{3\pi}{4}$.

REASON (R): The principal value of $\cos^{-1} k$ lies in the interval $[0, \pi]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): The general solution of $\sin x = 0$ is $x = n\pi$, where $n \in \mathbb{Z}$.

REASON (R): $\sin x = 0$ at angles $0, \pm \pi, \pm 2\pi, ...$ which can be represented as $n\pi$ for any integer $n$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): To solve $2 \sin^2 x - \sin x - 1 = 0$, one can treat it as a quadratic equation in $\sin x$.

REASON (R): The equation can be written as $2y^2 - y - 1 = 0$ by substituting $y = \sin x$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The domain of $\sin^{-1} x$ is $[-1, 1]$.

REASON (R): The range of the sine function is $[-1, 1]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): The principal value branch of $\cos^{-1} x$ is $[0, \pi]$.

REASON (R): The cosine function restricted to $[0, \pi]$ is bijective (one-to-one and onto).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): $\sin^{-1} (\frac{1}{2}) + \cos^{-1} (\frac{1}{2}) = \frac{\pi}{2}$.

REASON (R): The identity $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$ holds for all $x \in [-1, 1]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): $\tan^{-1} (-1) = \frac{3\pi}{4}$.

REASON (R): The principal value branch of $\tan^{-1} x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): $\tan (\tan^{-1} x) = x$ for all $x \in \mathbb{R}$.

REASON (R): $\tan^{-1} x$ is the inverse function of $\tan x$, and the composition of a function and its inverse is the identity function on the appropriate domain.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The value of $\sin^{-1} (\sin \frac{7\pi}{6})$ is $\frac{\pi}{6}$.

REASON (R): $\sin^{-1} (\sin x) = x$ for all $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. ASSERTION (A): The angle of elevation from point A to the top of a tower is $30^\circ$. The angle of depression from the top of the tower to point A is also $30^\circ$.

REASON (R): The angle of elevation and the angle of depression between two points are always equal because they are alternate interior angles formed by parallel horizontal lines and the line of sight.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. ASSERTION (A): If the angle of elevation of a tower from a distance of 100 m is $45^\circ$, the height of the tower is 100 m.

REASON (R): In a right triangle, $\tan 45^\circ = 1$, and tangent relates the opposite side (height) and adjacent side (distance).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. ASSERTION (A): From the top of a cliff, the angle of depression of a boat is $60^\circ$. If the boat is 50 m from the base of the cliff, the height of the cliff is $50\sqrt{3}$ m.

REASON (R): The angle of elevation from the boat to the top of the cliff is equal to the angle of depression from the top of the cliff to the boat.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. ASSERTION (A): In a problem involving two points of observation in a line with the base of a tower, two right triangles are formed, and their sides can be related using trigonometric ratios.

REASON (R): Heights and distances problems often involve solving right triangles using SOH CAH TOA.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. ASSERTION (A): If the angle of elevation of a kite is $30^\circ$ and the length of the string is 100 m, the height of the kite is 50 m (assuming no slack).

REASON (R): In a right triangle formed by the string, height, and horizontal distance, the height is the opposite side and the string is the hypotenuse, so $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 6. ASSERTION (A): The angle of depression is measured from the line of sight upwards to the horizontal line.

REASON (R): The angle of depression is used when the observer is looking at an object above the horizontal level.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.