Negative Questions MCQs for Sub-Topics of Topic 8: Trigonometry

Question 1. In a right-angled triangle ABC, right-angled at B, which of the following statements is NOT correct?

(A) The side opposite to angle A is BC.

(B) The side adjacent to angle C is BC.

(C) The hypotenuse is AB.

(D) $\angle A + \angle C = 90^\circ$.

Question 2. Which of the following statements about the trigonometric ratio sine (sin) is NOT true for an acute angle $\theta$ in a right triangle?

(A) It is the ratio of the opposite side to the hypotenuse.

(B) Its value is always less than or equal to 1.

(C) Its value is always positive.

(D) Its value increases as the angle increases from $0^\circ$ to $90^\circ$.

Question 3. If $\cos \theta = \frac{5}{13}$ in a right triangle, where $\theta$ is an acute angle, which of the following statements is INCORRECT?

(A) The adjacent side is 5 and the hypotenuse is 13 (in some units).

(B) The opposite side is 12 (in the same units).

(C) $\sin \theta = \frac{12}{13}$.

(D) $\tan \theta = \frac{13}{5}$.

Question 4. Which of the following is NOT a correct relationship between trigonometric ratios?

(A) $\sec \theta = \frac{1}{\cos \theta}$

(B) $\cot \theta = \frac{\sin \theta}{\cos \theta}$

(C) $\tan \theta = \frac{\sin \theta}{\cos \theta}$

(D) $\text{cosec } \theta = \frac{1}{\sin \theta}$

Question 5. In a right triangle ABC, right-angled at B, if AB = 7 cm and BC = 24 cm, which of the following statements is FALSE?

(A) AC = 25 cm.

(B) $\sin A = \frac{24}{25}$.

(C) $\tan C = \frac{7}{24}$.

(D) $\cos A = \frac{24}{25}$.

Question 6. Which of the following values for a trigonometric ratio of an acute angle is NOT possible?

(A) $\sin \theta = \frac{1}{3}$

(B) $\cos \theta = 0.9$

(C) $\tan \theta = 100$

(D) $\sec \theta = 0.5$

Question 7. If $\cot A = \frac{p}{q}$, where A is an acute angle, which of the following statements is INCORRECT?

(A) The adjacent side is $p$ and the opposite side is $q$ (in some units).

(B) The hypotenuse is $\sqrt{p^2 + q^2}$.

(C) $\tan A = \frac{q}{p}$.

(D) $\sin A = \frac{p}{\sqrt{p^2 + q^2}}$.

Question 8. Which of the following pairs of trigonometric ratios are NOT reciprocals of each other?

(A) $\sin \theta$ and $\text{cosec } \theta$

(B) $\cos \theta$ and $\sec \theta$

(C) $\tan \theta$ and $\cot \theta$

(D) $\sin \theta$ and $\cos \theta$

Question 9. In a right triangle, which side is NOT a leg?

(A) The side opposite the right angle.

(B) The hypotenuse.

(C) The longest side.

(D) The adjacent side to an acute angle.

Question 10. Which statement about the angle in a right triangle is NOT correct?

(A) There is one right angle.

(B) There are two acute angles.

(C) The sum of the two acute angles is $90^\circ$.

(D) The sum of all three angles is $360^\circ$.

Question 1. Which of the following trigonometric values is NOT correct?

(A) $\sin 30^\circ = \frac{1}{2}$

(B) $\cos 45^\circ = \frac{1}{\sqrt{2}}$

(C) $\tan 60^\circ = \sqrt{3}$

(D) $\cot 0^\circ = 0$

Question 2. Which statement about complementary angles is INCORRECT?

(A) Two angles are complementary if their sum is $90^\circ$.

(B) $\sin \theta = \cos (90^\circ - \theta)$

(C) $\tan \theta = \cot (90^\circ - \theta)$

(D) $\sec \theta = \text{cosec } (90^\circ - \theta)$

Question 3. Evaluate the following expressions. Which evaluation is INCORRECT?

(A) $\sin 90^\circ = 1$

(B) $\cos 0^\circ = 1$

(C) $\tan 45^\circ = 1$

(D) $\sec 0^\circ = 0$

Question 4. Which of the following relationships based on special angles is FALSE?

(A) $\sin 60^\circ = \cos 30^\circ$

(B) $\tan 30^\circ = \frac{1}{\cot 60^\circ}$

(C) $\sin 45^\circ = \cos 45^\circ$

(D) $\sec 30^\circ = \frac{1}{\sin 60^\circ}$

Question 5. Simplify the following expressions using complementary angle properties. Which simplification is NOT correct?

(A) $\frac{\sin 20^\circ}{\cos 70^\circ} = 1$

(B) $\tan 55^\circ \cot 35^\circ = 1$

(C) $\sec 15^\circ - \text{cosec } 75^\circ = 0$

(D) $\sin^2 10^\circ + \sin^2 80^\circ = 2$

Question 6. Which of the following statements about the values of trigonometric ratios for special angles in the range $0^\circ \leq \theta \leq 90^\circ$ is INCORRECT?

(A) As $\theta$ increases, $\sin \theta$ increases.

(B) As $\theta$ increases, $\cos \theta$ decreases.

(C) As $\theta$ increases, $\tan \theta$ increases.

(D) As $\theta$ increases, $\cot \theta$ increases.

Question 7. Which trigonometric value is NOT correctly matched with the angle?

(A) $\sin 0^\circ = 0$

(B) $\cos 90^\circ = 0$

(C) $\tan 90^\circ = \text{undefined}$

(D) $\text{cosec } 90^\circ = 0$

Question 8. If $\sin A = \cos (A+10^\circ)$, where A is acute, which of the following is INCORRECT?

(A) $A + (A+10^\circ) = 90^\circ$

(B) $2A + 10^\circ = 90^\circ$

(C) $2A = 80^\circ$

(D) $A = 50^\circ$

Question 9. Which statement about trigonometric ratios of special angles is FALSE?

(A) $\cos 30^\circ = \frac{\sqrt{3}}{2}$

(B) $\sin 45^\circ = \frac{1}{\sqrt{2}}$

(C) $\tan 30^\circ = \sqrt{3}$

(D) $\text{cosec } 60^\circ = \frac{2}{\sqrt{3}}$

Question 10. Which of the following is NOT a correct application of complementary angles?

(A) $\cos 50^\circ = \sin 40^\circ$

(B) $\cot 70^\circ = \tan 20^\circ$

(C) $\text{cosec } 30^\circ = \sec 60^\circ$

(D) $\sin^2 \theta + \cos^2 (90^\circ - \theta) = 1$

Question 1. Which of the following is NOT a fundamental trigonometric identity?

(A) $\sin^2 \theta + \cos^2 \theta = 1$

(B) $1 + \tan^2 \theta = \sec^2 \theta$

(C) $\text{cosec}^2 \theta - \cot^2 \theta = 1$

(D) $\sin \theta + \cos \theta = 1$

Question 2. Which simplification using identities is INCORRECT?

(A) $\sin \theta \cdot \cot \theta = \cos \theta$

(B) $\cos \theta \cdot \tan \theta = \sin \theta$

(C) $\sec \theta \cdot \sin \theta = \tan \theta$

(D) $\text{cosec } \theta \cdot \cos \theta = \tan \theta$

Question 3. If $\tan \theta = \frac{12}{5}$, which statement using identities is FALSE?

(A) $\cot \theta = \frac{5}{12}$.

(B) $\sec^2 \theta = 1 + (\frac{12}{5})^2$.

(C) $\sin^2 \theta = 1 - (\frac{5}{13})^2$ (assuming $\theta$ is acute).

(D) $\text{cosec}^2 \theta = 1 - (\frac{5}{12})^2$.

Question 4. Which of the following expressions is NOT equivalent to $\cos^2 \theta$?

(A) $1 - \sin^2 \theta$

(B) $\frac{1}{\sec^2 \theta}$

(C) $\cot^2 \theta ( \sec^2 \theta - 1 )$

(D) $\frac{\cot^2 \theta}{\text{cosec}^2 \theta - 1}$

Question 5. Which of the following identities is NOT correct?

(A) $(\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta$

(B) $(\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1$

(C) $\frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A$

(D) $\sin^4 \theta - \cos^4 \theta = \sin^2 \theta + \cos^2 \theta$

Question 6. If $\sin \theta = x$, which expression is NOT necessarily equivalent to $\cos \theta$ (for acute $\theta$)?

(A) $\sqrt{1 - x^2}$

(B) $\frac{1}{\sec \theta}$

(C) $\sqrt{\sec^2 \theta - \tan^2 \theta} \cdot \sqrt{1-x^2}$

(D) $\frac{x}{\tan \theta}$

Question 7. Which transformation of the expression is INCORRECT?

(A) $\frac{\sin A}{1 + \cos A} = \tan \frac{A}{2}$

(B) $\frac{1 - \cos A}{\sin A} = \tan \frac{A}{2}$

(C) $\frac{1 - \sin A}{1 + \sin A} = \left(\tan(\frac{\pi}{4} - \frac{A}{2})\right)^2$

(D) $\frac{1 + \sin A}{1 - \sin A} = \left(\tan(\frac{\pi}{4} + \frac{A}{2})\right)^2$

Question 8. Which statement about trigonometric identities is FALSE?

(A) Identities are equations that are true for all values of the variables for which the expressions are defined.

(B) Pythagorean identities relate the squares of different trigonometric functions.

(C) Quotient identities express tangent and cotangent in terms of sine and cosine.

(D) Reciprocal identities hold only for acute angles.

Question 9. Which simplification is NOT correct?

(A) $(\sin \theta + \cos \theta)^2 = 1 + \sin 2\theta$

(B) $(\sin \theta - \cos \theta)^2 = 1 - \sin 2\theta$

(C) $(\sec \theta + \tan \theta)^2 = \sec^2 \theta + \tan^2 \theta + 2 \sec \theta \tan \theta$

(D) $(\text{cosec } \theta + \cot \theta)^2 = \text{cosec}^2 \theta + \cot^2 \theta + 2 \text{cosec } \theta \cot \theta$

Question 10. Which equation is NOT an identity?

(A) $\sin^2 x - \cos^2 x = -\cos 2x$

(B) $\tan^2 x \cos^2 x = \sin^2 x$

(C) $\sec^2 x \cot^2 x = \text{cosec}^2 x$

(D) $\sin x + \cos x = \sqrt{2}$

Question 1. Which of the following statements about angle measurement is NOT correct?

(A) One degree is divided into 60 minutes.

(B) One minute is divided into 60 seconds.

(C) A complete revolution is $360^\circ$.

(D) A straight angle is $90^\circ$.

Question 2. Which of the following conversions between degree and radian measure is INCORRECT?

(A) $45^\circ = \frac{\pi}{4}$ radians

(B) $180^\circ = \pi$ radians

(C) $\frac{\pi}{6}$ radians $= 60^\circ$

(D) $2\pi$ radians $= 360^\circ$

Question 3. If an arc of length $l$ subtends a central angle $\theta$ (in radians) in a circle of radius $r$, which formula is NOT correct?

(A) $l = r\theta$

(B) $\theta = l/r$

(C) $r = l/\theta$

(D) Area of sector $= r^2 \theta$

Question 4. Which statement about a radian is FALSE?

(A) It is a unit of angle measurement.

(B) One radian is the angle subtended by an arc whose length is equal to the radius of the circle.

(C) One radian is approximately $180/\pi$ degrees.

(D) One radian is exactly $57.3^\circ$.

Question 5. A wheel rotates at 60 revolutions per minute. Which statement is INCORRECT?

(A) It makes 1 revolution per second.

(B) It turns through $2\pi$ radians in one second.

(C) Its angular speed is $2\pi$ radians per second.

(D) It turns through $360^\circ$ in one second.

Question 6. Convert $210^\circ$ into radians. Which of the following is NOT equivalent to $210^\circ$ in radians?

(A) $\frac{210}{180} \pi$ radians

(B) $\frac{7\pi}{6}$ radians

(C) $\frac{11\pi}{6}$ radians

(D) $210 \times \frac{\pi}{180}$ radians

Question 7. Convert $\frac{3\pi}{2}$ radians into degrees. Which of the following is NOT equivalent to $\frac{3\pi}{2}$ radians in degrees?

(A) $270^\circ$

(B) $\frac{3 \times 180}{2}$ degrees

(C) $3 \times 90^\circ$

(D) $3 \times 60^\circ$

Question 8. Given a circle with radius $10\ \text{cm}$, find the length of the arc that subtends a central angle of $1.5$ radians. Which calculation is INCORRECT?

(A) Use $l = r\theta$ where $\theta$ is in radians.

(B) $l = 10 \times 1.5$ cm.

(C) $l = 15$ cm.

(D) If the angle was $1.5^\circ$, convert it to radians first: $1.5 \times \frac{\pi}{180}$.

Question 9. Find the area of a sector of a circle with radius $8\ \text{cm}$ and central angle $\frac{\pi}{4}$ radians. Which calculation is FALSE?

(A) Area $= \frac{1}{2} r^2 \theta$ where $\theta$ is in radians.

(B) Area $= \frac{1}{2} \times 8^2 \times \frac{\pi}{4}$ cm$^2$.

(C) Area $= \frac{1}{2} \times 64 \times \frac{\pi}{4}$ cm$^2$.

(D) Area $= 16\pi$ cm$^2$.

Question 10. Which of the following statements comparing degree and radian measure is INCORRECT?

(A) Degrees are commonly used in geometry and everyday life.

(B) Radians are more natural for calculus and theoretical work.

(C) Converting from radians to degrees involves multiplying by $180/\pi$.

(D) $1^\circ > 1$ radian.

Question 1. Which statement about the unit circle definition of trigonometric functions is NOT correct?

(A) The unit circle has a radius of 1.

(B) For an angle $\theta$ in standard position, the terminal side intersects the unit circle at $(x, y)$.

(C) $\cos \theta = y$ and $\sin \theta = x$.

(D) The coordinates $(x, y)$ satisfy $x^2 + y^2 = 1$.

Question 2. Which of the following trigonometric functions does NOT have a range of $[-1, 1]$?

(A) $\sin x$

(B) $\cos x$

(C) $\tan x$

(D) All of the above

Question 3. Which statement about the signs of trigonometric functions in different quadrants is FALSE?

(A) In Quadrant I, all functions are positive.

(B) In Quadrant II, only $\sin \theta$ and $\text{cosec } \theta$ are positive.

(C) In Quadrant III, only $\tan \theta$ and $\cot \theta$ are positive.

(D) In Quadrant IV, only $\sin \theta$ and $\tan \theta$ are negative.

Question 4. Which of the following statements about the periodicity of trigonometric functions is INCORRECT?

(A) The period of $\sin x$ is $2\pi$.

(B) The period of $\cos x$ is $2\pi$.

(C) The period of $\tan x$ is $2\pi$.

(D) The period of $\cot x$ is $\pi$.

Question 5. If the terminal side of an angle $\theta$ passes through the point $(-5, -12)$, which of the following statements is FALSE?

(A) The distance from the origin to the point is 13.

(B) $\sin \theta = -\frac{12}{13}$.

(C) $\cos \theta = -\frac{5}{13}$.

(D) $\tan \theta = \frac{12}{5}$.

Question 6. Which of the following values is NOT correct for trigonometric functions of quadrantal angles?

(A) $\cos \pi = -1$

(B) $\sin \frac{3\pi}{2} = -1$

(C) $\tan \frac{\pi}{2} = 0$

(D) $\cot 2\pi =$ undefined

Question 7. Which of the following functions does NOT have a domain of all real numbers?

(A) $\sin x$

(B) $\cos x$

(C) $\tan x$

(D) All of the above

Question 8. Which statement about the range of reciprocal trigonometric functions is FALSE?

(A) The range of $\sec x$ is $(-\infty, -1] \cup [1, \infty)$.

(B) The range of $\text{cosec } x$ is $(-\infty, -1] \cup [1, \infty)$.

(C) The range of $\cot x$ is $\mathbb{R}$.

(D) The range of $\sec x$ is the same as the range of $\cos x$.

Question 9. If $\sin \theta > 0$ and $\cos \theta < 0$, which quadrant does $\theta$ lie in? Which statement is FALSE?

(A) $\theta$ lies in Quadrant II.

(B) $\tan \theta < 0$.

(C) $\sec \theta > 0$.

(D) $\text{cosec } \theta > 0$.

Question 10. Which statement about the period of trigonometric functions is INCORRECT?

(A) The period of $y = \sin(x/2)$ is $4\pi$.

(B) The period of $y = \cos(3x)$ is $2\pi/3$.

(C) The period of $y = \tan(2x)$ is $\pi/2$.

(D) The period of $y = \cot(x/3)$ is $3\pi/2$.

Question 1. Which statement about the graph of $y = \sin x$ is NOT true?

(A) It passes through $(0, 0)$.

(B) It has a maximum value of 1.

(C) It is symmetric about the y-axis.

(D) Its period is $2\pi$.

Question 2. Which statement about the graph of $y = \cos x$ is FALSE?

(A) It passes through $(0, 1)$.

(B) It has a minimum value of -1.

(C) It has vertical asymptotes.

(D) Its period is $2\pi$.

Question 3. Which statement about the graph of $y = \tan x$ is INCORRECT?

(A) It has a period of $\pi$.

(B) It has vertical asymptotes at $x = n\pi, n \in \mathbb{Z}$.

(C) Its range is all real numbers.

(D) It passes through $(0, 0)$.

Question 4. Which function does NOT have a period of $2\pi$?

(A) $y = \sin x$

(B) $y = \cos x$

(C) $y = \sec x$

(D) $y = \tan x$

Question 5. Which statement about the graphs of reciprocal trigonometric functions is FALSE?

(A) The graph of $y = \sec x$ has vertical asymptotes where $\cos x = 0$.

(B) The graph of $y = \text{cosec } x$ has vertical asymptotes where $\sin x = 0$.

(C) The graph of $y = \cot x$ has vertical asymptotes where $\tan x = 0$.

(D) The range of $y = \sec x$ is $(-\infty, -1] \cup [1, \infty)$.

Question 6. Which statement about the amplitude of trigonometric functions is INCORRECT?

(A) The amplitude of $y = A \sin(Bx)$ is $|A|$.

(B) The amplitude of $y = \cos x$ is 1.

(C) The amplitude of $y = \tan x$ is 1.

(D) The amplitude represents half the distance between the maximum and minimum values.

Question 7. Which statement about the symmetry of trigonometric graphs is FALSE?

(A) The graph of $y = \sin x$ is symmetric about the origin.

(B) The graph of $y = \cos x$ is symmetric about the y-axis.

(C) The graph of $y = \tan x$ is symmetric about the y-axis.

(D) The graph of $y = \cot x$ is symmetric about the origin.

Question 8. The graph of $y = 2 \cos (3x)$. Which statement about this graph is INCORRECT?

(A) The amplitude is 2.

(B) The period is $2\pi/3$.

(C) The maximum value is 2.

(D) It passes through $(0, 0)$.

Question 9. Which of the following graphs has a period of $\pi/2$?

(A) $y = \sin(x/2)$

(B) $y = \cos(2x)$

(C) $y = \tan(2x)$

(D) $y = \cot(2x)$

Question 10. Which statement about the relationship between $y = \sin x$ and $y = \cos x$ graphs is FALSE?

(A) They have the same period.

(B) They have the same amplitude.

(C) The graph of $\cos x$ is a horizontal shift of the graph of $\sin x$.

(D) The graph of $\cos x$ is a vertical shift of the graph of $\sin x$.

Question 1. Which of the following is NOT a correct formula for $\cos(A+B)$?

(A) $\cos A \cos B - \sin A \sin B$

(B) $\cos A \cos B + \sin A \sin B$

(C) $\sin (90^\circ - (A+B))$

(D) $\cos A \cos B - \sqrt{1-\cos^2 A} \sqrt{1-\cos^2 B}$ (for appropriate A, B)

Question 2. Which of the following is NOT a correct formula for $\sin 2A$?

(A) $2 \sin A \cos A$

(B) $\frac{2 \tan A}{1 + \tan^2 A}$

(C) $\sin(A+A)$

(D) $2 \sin A$

Question 3. Which statement about the value of $\tan 75^\circ$ is FALSE?

(A) $\tan 75^\circ = \tan(45^\circ + 30^\circ)$

(B) $\tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}$

(C) $\tan 75^\circ = \frac{1 + 1/\sqrt{3}}{1 - 1/\sqrt{3}}$

(D) $\tan 75^\circ = 2 - \sqrt{3}$

Question 4. Which of the following is NOT a correct formula for $\cos 2A$?

(A) $\cos^2 A - \sin^2 A$

(B) $2 \cos^2 A - 1$

(C) $1 - 2 \sin^2 A$

(D) $\frac{2 \tan A}{1 + \tan^2 A}$

Question 5. Which expression is NOT equivalent to $\frac{\sin 2\theta}{1 + \cos 2\theta}$?

(A) $\frac{2 \sin \theta \cos \theta}{2 \cos^2 \theta}$

(B) $\tan \theta$

(C) $\frac{1 - \cos 2\theta}{\sin 2\theta}$

(D) $\frac{\sin \theta}{\cos \theta}$

Question 6. Which of the following is NOT a correct formula for $\tan(A-B)$?

(A) $\frac{\tan A - \tan B}{1 + \tan A \tan B}$

(B) $\frac{\sin(A-B)}{\cos(A-B)}$

(C) $\frac{\tan A - \tan B}{1 - \tan A \tan B}$

(D) $\frac{(\sin A / \cos A) - (\sin B / \cos B)}{1 + (\sin A / \cos A) (\sin B / \cos B)}$

Question 7. If $\sin A = 3/5$ and $\cos B = 5/13$, where A and B are acute angles, which statement is FALSE about $\sin(A+B)$?

(A) $\cos A = 4/5$ and $\sin B = 12/13$.

(B) $\sin(A+B) = \sin A \cos B + \cos A \sin B$.

(C) $\sin(A+B) = (3/5)(5/13) + (4/5)(12/13)$.

(D) $\sin(A+B) = \frac{15+48}{65} = \frac{63}{65}$.

Question 8. Which identity is NOT related to triple angle formulas?

(A) $\sin 3A = 3 \sin A - 4 \sin^3 A$

(B) $\cos 3A = 4 \cos^3 A - 3 \cos A$

(C) $\tan 3A = \frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

(D) $\sin^2 3A = 1 - \cos^2 3A$

Question 9. Which expression is NOT equivalent to $\cos^2 \theta$?

(A) $\frac{1 + \cos 2\theta}{2}$

(B) $1 - \sin^2 \theta$

(C) $\frac{1}{\sec^2 \theta}$

(D) $1 + \tan^2 \theta$

Question 10. Which statement about compound and multiple angle identities is FALSE?

(A) They are derived from basic geometric definitions and algebraic manipulation.

(B) They are useful for simplifying expressions and solving equations.

(C) The formula for $\sin(A+B)$ can be used to find $\sin 90^\circ = \sin(60^\circ+30^\circ)$.

(D) $\cos(A-B) = \cos A \cos B - \sin A \sin B$ for all A and B.

Question 1. Which of the following is NOT a correct product-to-sum formula?

(A) $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$

(B) $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$

(C) $2 \cos A \cos B = \cos(A+B) - \cos(A-B)$

(D) $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

Question 2. Which of the following is NOT a correct sum-to-product formula?

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

(B) $\cos C + \cos D = 2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$

(C) $\sin C - \sin D = 2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

(D) $\cos C - \cos D = 2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)$

Question 3. Which transformation is INCORRECT?

(A) $\sin 7\theta + \sin 3\theta = 2 \sin 5\theta \cos 2\theta$

(B) $\cos 5x - \cos 3x = -2 \sin 4x \sin x$

(C) $2 \cos 6A \sin 2A = \sin 8A - \sin 4A$

(D) $2 \sin 4B \sin B = \cos 3B - \cos 5B$

Question 4. Which expression is NOT equivalent to $\frac{\sin x + \sin y}{\cos x - \cos y}$ (where defined)?

(A) $\frac{2 \sin((x+y)/2) \cos((x-y)/2)}{-2 \sin((x+y)/2) \sin((x-y)/2)}$

(B) $-\frac{\cos((x-y)/2)}{\sin((x-y)/2)}$

(C) $-\cot((x-y)/2)$

(D) $\cot((y-x)/2)$

Question 5. Which statement about product-to-sum and sum-to-product formulas is FALSE?

(A) They are useful for simplifying trigonometric expressions.

(B) They can be used to solve certain types of trigonometric equations.

(C) They are derived from the compound angle formulas.

(D) They apply only to acute angles.

Question 6. Which expression is NOT equivalent to $\sin 20^\circ + \sin 40^\circ$?

(A) $2 \sin 30^\circ \cos 10^\circ$

(B) $2 \cdot \frac{1}{2} \cos 10^\circ$

(C) $\cos 10^\circ$

(D) $\sin 60^\circ$

Question 7. Which transformation is INCORRECT?

(A) $\cos x \cos y = \frac{1}{2}[\cos(x+y) + \cos(x-y)]$

(B) $\sin x \sin y = \frac{1}{2}[\cos(x-y) - \cos(x+y)]$

(C) $\sin x \cos y = \frac{1}{2}[\sin(x+y) + \sin(x-y)]$

(D) $\cos x \sin y = \frac{1}{2}[\sin(x+y) + \sin(y-x)]$

Question 8. Which expression is NOT equivalent to $\frac{\cos A + \cos B}{\sin A + \sin B}$ (where defined)?

(A) $\frac{2 \cos((A+B)/2) \cos((A-B)/2)}{2 \sin((A+B)/2) \cos((A-B)/2)}$

(B) $\cot((A+B)/2)$

(C) $\tan((A+B)/2)$

(D) $\frac{1}{\tan((A+B)/2)}$

Question 9. Which statement about using sum-to-product formulas to simplify expressions is FALSE?

(A) $\sin \theta + \cos \theta = \sqrt{2} \sin(\theta + \pi/4)$

(B) $\sin \theta - \cos \theta = -\sqrt{2} \cos(\theta + \pi/4)$

(C) $\sin \theta + \sin \phi$ can be simplified if $\theta + \phi$ is a simple angle.

(D) $\cos \theta + \cos \phi$ can be simplified if $\theta - \phi$ is a simple angle.

Question 10. Which transformation is INCORRECT?

(A) $\sin A = 2 \sin (A/2) \cos (A/2)$

(B) $\cos A = \cos^2 (A/2) - \sin^2 (A/2)$

(C) $1 + \cos A = 2 \sin^2 (A/2)$

(D) $1 - \cos A = 2 \sin^2 (A/2)$

Question 1. Which of the following is NOT a principal solution for the equation $\sin x = -\frac{1}{2}$?

(A) $-\frac{\pi}{6}$

(B) $\frac{7\pi}{6}$

(C) $\frac{11\pi}{6}$

(D) $\frac{5\pi}{6}$

Question 2. Which of the following is NOT part of the general solution for $\tan x = \frac{1}{\sqrt{3}}$?

(A) $x = n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

(B) $x = n\pi + 30^\circ, n \in \mathbb{Z}$

(C) $x = n\pi - \frac{5\pi}{6}, n \in \mathbb{Z}$

(D) $x = 2n\pi + \frac{\pi}{6}, n \in \mathbb{Z}$

Question 3. Which statement about solving the equation $\cos x = -1$ is FALSE?

(A) The principal value is $\pi$.

(B) The general solution is $x = 2n\pi + \pi, n \in \mathbb{Z}$.

(C) The general solution can be written as $x = (2n+1)\pi, n \in \mathbb{Z}$.

(D) The equation has no real solutions.

Question 4. Which of the following is NOT part of the general solution for $\sin x = \sin \alpha$?

(A) $x = n\pi + (-1)^n \alpha, n \in \mathbb{Z}$

(B) If $\alpha$ is in the first quadrant, $x = n\pi + \alpha$ (for odd n).

(C) If $\alpha$ is in the first quadrant, $x = (2n)\pi + \alpha$ (for even n).

(D) $x = 2n\pi + \alpha$ or $x = (2n+1)\pi - \alpha, n \in \mathbb{Z}$.

Question 5. Which statement about solving trigonometric equations is FALSE?

(A) Equations like $\sin x = k$ where $|k| > 1$ have no real solutions.

(B) Equations involving $\sin^2 x, \cos^2 x$, etc., can sometimes be reduced to quadratic equations.

(C) Using identities is often useful to simplify complex equations.

(D) The equation $\cos x = x$ can be solved algebraically using standard methods.

Question 6. Which is NOT a correct step in solving $\tan 2x = \cot x$?

(A) Write $\cot x = \tan (\frac{\pi}{2} - x)$.

(B) Set $2x = n\pi + (\frac{\pi}{2} - x)$, where $n \in \mathbb{Z}$.

(C) $3x = n\pi + \frac{\pi}{2}$.

(D) $x = \frac{n\pi}{2} + \frac{\pi}{4}$.

Question 7. Which statement about the principal solution of trigonometric equations is INCORRECT?

(A) The principal solution for $\sin x = k$ lies in $[-\pi/2, \pi/2]$.

(B) The principal solution for $\cos x = k$ lies in $[0, \pi]$.

(C) The principal solution for $\tan x = k$ lies in $[-\pi/2, \pi/2]$.

(D) The principal solution is the smallest positive value of the angle satisfying the equation.

Question 8. Which equation is NOT equivalent to $\cos 2x = \sin x$?

(A) $1 - 2\sin^2 x = \sin x$

(B) $2\sin^2 x + \sin x - 1 = 0$

(C) $(\sin x + 1)(2\sin x - 1) = 0$

(D) $\cos 2x = \cos (\frac{\pi}{2} - x)$

Question 9. Which of the following statements about the number of solutions in a specific interval is FALSE?

(A) $\sin x = 1/2$ has two solutions in $[0, 2\pi)$.

(B) $\cos x = 0$ has two solutions in $[0, 2\pi)$.

(C) $\tan x = 1$ has two solutions in $[0, 2\pi)$.

(D) $\sin x = -1$ has two solutions in $[0, 2\pi)$.

Question 10. Which statement about the general solution of $\cos^2 x = \cos^2 \alpha$ is INCORRECT?

(A) The equation is equivalent to $\cos x = \pm \cos \alpha$.

(B) The general solution is $x = n\pi \pm \alpha$, where $n \in \mathbb{Z}$.

(C) The general solution for $\sin^2 x = \sin^2 \alpha$ is also $x = n\pi \pm \alpha$.

(D) The general solution for $\tan^2 x = \tan^2 \alpha$ is $x = 2n\pi \pm \alpha$.

Question 1. Which statement about the domain of inverse trigonometric functions is NOT correct?

(A) Domain of $\sin^{-1} x$ is $[-1, 1]$.

(B) Domain of $\cos^{-1} x$ is $[-1, 1]$.

(C) Domain of $\tan^{-1} x$ is $\mathbb{R}$.

(D) Domain of $\sec^{-1} x$ is $[-1, 1]$.

Question 2. Which statement about the principal value branch (range) of inverse trigonometric functions is FALSE?

(A) Range of $\sin^{-1} x$ is $[-\pi/2, \pi/2]$.

(B) Range of $\cos^{-1} x$ is $[0, \pi]$.

(C) Range of $\tan^{-1} x$ is $(0, \pi)$.

(D) Range of $\text{cosec}^{-1} x$ is $[-\pi/2, \pi/2] - \{0\}$.

Question 3. Which principal value is INCORRECT?

(A) $\sin^{-1} (1/\sqrt{2}) = \pi/4$

(B) $\cos^{-1} (-1) = \pi$

(C) $\tan^{-1} (-\sqrt{3}) = -\pi/3$

(D) $\cot^{-1} (-1) = -\pi/4$

Question 4. Which of the following identities is NOT correct?

(A) $\sin^{-1} x + \cos^{-1} x = \pi/2$ for $x \in [-1, 1]$.

(B) $\tan^{-1} x + \cot^{-1} x = \pi/2$ for $x \in \mathbb{R}$.

(C) $\sec^{-1} x + \text{cosec}^{-1} x = \pi/2$ for $x \in \mathbb{R} - (-1, 1)$.

(D) $\cos^{-1} (-x) = -\cos^{-1} x$ for $x \in [-1, 1]$.

Question 5. Which simplification is FALSE?

(A) $\sin (\sin^{-1} x) = x$ for $x \in [-1, 1]$.

(B) $\cos (\cos^{-1} x) = x$ for $x \in [-1, 1]$.

(C) $\tan (\tan^{-1} x) = x$ for $x \in \mathbb{R}$.

(D) $\sin^{-1} (\sin x) = x$ for all $x \in \mathbb{R}$.

Question 6. Which of the following is NOT a correct formula for the sum/difference of inverse tangents?

(A) $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\frac{x+y}{1-xy}\right)$, if $xy < 1$.

(B) $\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1} \left(\frac{x+y}{1-xy}\right)$, if $xy > 1$ and $x, y > 0$.

(C) $\tan^{-1} x - \tan^{-1} y = \tan^{-1} \left(\frac{x-y}{1+xy}\right)$, if $xy > -1$.

(D) $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\frac{x+y}{1+xy}\right)$.

Question 7. Which statement about the relationship between trigonometric functions and their inverses is INCORRECT?

(A) The domain of the inverse function is the range of the original function.

(B) The range of the inverse function is a restricted domain of the original function where it is bijective.

(C) $\sin (\cos^{-1} x) = \sqrt{1-x^2}$ for $x \in [-1, 1]$.

(D) $\cos (\sin^{-1} x) = x$ for $x \in [-1, 1]$.

Question 8. Which principal value is INCORRECT?

(A) $\sin^{-1} (-1/2) = -\pi/6$

(B) $\cos^{-1} (\sqrt{3}/2) = \pi/6$

(C) $\tan^{-1} (1) = \pi/4$

(D) $\cot^{-1} (-1/\sqrt{3}) = -\pi/3$

Question 9. Which statement is FALSE?

(A) The domain of $\cot^{-1} x$ is $\mathbb{R}$.

(B) The range of $\cot^{-1} x$ is $(0, \pi)$.

(C) The range of $\sec^{-1} x$ is $[0, \pi] - \{\pi/2\}$.

(D) The range of $\text{cosec}^{-1} x$ is $[0, \pi] - \{\pi/2\}$.

Question 10. Which identity is NOT correct?

(A) $\sin^{-1} (-x) = -\sin^{-1} x$

(B) $\cos^{-1} (-x) = \pi - \cos^{-1} x$

(C) $\tan^{-1} (-x) = -\tan^{-1} x$

(D) $\cot^{-1} (-x) = -\cot^{-1} x$

Question 1. Which statement about angles in heights and distances is NOT correct?

(A) The angle of elevation is measured upwards from the horizontal line.

(B) The angle of depression is measured downwards from the horizontal line.

(C) The angle of elevation from point A to point B is equal to the angle of depression from point B to point A.

(D) The line of sight is always horizontal in these problems.

Question 2. A ladder leans against a wall making an angle of $60^\circ$ with the ground. If the foot of the ladder is 5 m from the wall, which statement is FALSE?

(A) A right triangle is formed by the ladder, the wall, and the ground.

(B) The length of the ladder is the hypotenuse.

(C) The height the ladder reaches on the wall is $5 \tan 60^\circ$ m.

(D) The length of the ladder is $5 \sin 60^\circ$ m.

Question 3. From the top of a $40\ \text{m}$ high tower, the angle of depression of a car on the ground is $45^\circ$. Which statement is INCORRECT?

(A) The angle of elevation from the car to the top of the tower is $45^\circ$.

(B) The horizontal distance of the car from the foot of the tower is 40 m.

(C) $\tan 45^\circ = \frac{40}{\text{distance}}$.

(D) The distance of the car from the foot of the tower is $40\sqrt{2}$ m.

Question 4. A tree breaks due to wind. The broken part is inclined at $30^\circ$ to the ground. The top of the tree touches the ground at a distance of 10 m from the foot of the tree. Which statement is FALSE?

(A) The part of the tree still standing is the adjacent side in the right triangle formed.

(B) The length of the broken part is the hypotenuse.

(C) The height of the standing part is $10 \tan 30^\circ$ m.

(D) The length of the broken part is $10 \sin 30^\circ$ m.

Question 5. From a point on the ground, the angles of elevation of the top of a building and the top of a flagstaff on top of the building are $45^\circ$ and $60^\circ$ respectively. If the distance from the observation point to the building is 50 m, which statement is INCORRECT?

(A) The height of the building is 50 m.

(B) Let the height of the flagstaff be h.

(C) $\tan 60^\circ = \frac{50 + h}{50}$.

(D) The height of the flagstaff is $50(\sqrt{3}-1)$ m.

Question 6. Two poles of equal height are standing opposite to each other on either side of a road which is 80 m wide. From a point between them on the road, the angles of elevation of the tops of the poles are $60^\circ$ and $30^\circ$. Which statement is FALSE?

(A) Let the height of the poles be h.

(B) Let the distance of the point from one pole be x, then the distance from the other pole is $80-x$.

(C) $\tan 60^\circ = h/x$ and $\tan 30^\circ = h/(80-x)$.

(D) The height of the poles is $40\sqrt{3}$ m.

Question 7. Which of the following trigonometric ratios is NOT typically used in basic heights and distances problems involving a single right triangle?

(A) Sine

(B) Cosine

(C) Tangent

(D) Cosecant

Question 8. From the top of a lighthouse, the angle of depression of two ships on the same side of the lighthouse are observed to be $45^\circ$ and $60^\circ$. If the height of the lighthouse is 100 m, which statement is FALSE about the distances of the ships from the base?

(A) The distance of the ship with angle of depression $45^\circ$ is 100 m.

(B) The distance of the ship with angle of depression $60^\circ$ is $100/\sqrt{3}$ m.

(C) The distance between the two ships is $100(1 - 1/\sqrt{3})$ m.

(D) The distance between the two ships is $100(\sqrt{3} - 1)$ m.

Question 9. A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is $60^\circ$. When he moves 30 m away from the bank, the angle of elevation is $30^\circ$. Which statement is INCORRECT about the width of the river (w) and height of the tree (h)?

(A) $\tan 60^\circ = h/w$

(B) $\tan 30^\circ = h/(w+30)$

(C) $h = w\sqrt{3}$ and $h = (w+30)/\sqrt{3}$.

(D) The width of the river is $30\sqrt{3}$ m.

Question 10. Which type of triangle is primarily used in the basic concepts of heights and distances?

(A) Equilateral triangle

(B) Isosceles triangle

(C) Right-angled triangle

(D) Scalene triangle