CBSE Class 10 Mathematics Introduction to Trigonometry MCQs Set 08

Question. If \(x = p\sec\theta\) and \(y = q\tan\theta\), then
(a) \(x^2 - y^2 = p^2q^2\)
(b) \(x^2q^2 - y^2p^2 = pq\)
(c) \(x^2q^2 - y^2p^2 = \frac{1}{p^2q^2}\)
(d) \(x^2q^2 - y^2p^2 = p^2q^2\)
Answer: D
We know, \(\sec^2\theta - \tan^2\theta = 1\)
and \(\sec\theta = \frac{x}{p}\)
\(\tan\theta = \frac{y}{q}\)
\(\frac{x^2}{p^2} - \frac{y^2}{q^2} = 1\)
\(x^2q^2 - y^2p^2 = p^2q^2\)

Question. If \(b\tan\theta = a\), the value of \(\frac{a\sin\theta - b\cos\theta}{a\sin\theta + b\cos\theta}\) is
(a) \(\frac{a - b}{a^2 + b^2}\)
(b) \(\frac{a + b}{a^2 + b^2}\)
(c) \(\frac{a^2 + b^2}{a^2 - b^2}\)
(d) \(\frac{a^2 - b^2}{a^2 + b^2}\)
Answer: D
\(\tan\theta = \frac{a}{b}\)
\(\frac{a\sin\theta - b\cos\theta}{a\sin\theta + b\cos\theta} = \frac{a\tan\theta - b}{a\tan\theta + b} = \frac{a(\frac{a}{b}) - b}{a(\frac{a}{b}) + b} = \frac{a^2 - b^2}{a^2 + b^2}\)

Question. The value of \(\tan 1^\circ \tan 2^\circ \tan 3^\circ ... \tan 89^\circ\) is
(a) 0
(b) 1
(c) \(\infty\)
(d) None of the options
Answer: B
Given, \(\tan 1^\circ \tan 2^\circ \tan 3^\circ ... \tan 89^\circ\)
\(= \tan(90^\circ - 89^\circ) \tan(90^\circ - 88^\circ) ... \tan 88^\circ \tan 89^\circ\)
\(= \cot 89^\circ \cot 88^\circ \cot 87^\circ ... \tan 87^\circ \tan 88^\circ \tan 89^\circ\)
\(= (\cot 89^\circ \tan 89^\circ)(\cot 88^\circ \tan 88^\circ) ... (\cot 44^\circ \tan 44^\circ) \tan 45^\circ\)
\(= 1 \times 1 \times 1 ... \times 1 \times 1 = 1\)

Question. \((\cos^4 A - \sin^4 A)\) is equal to
(a) \(1 - 2\cos^2 A\)
(b) \(2\sin^2 A - 1\)
(c) \(\sin^2 A - \cos^2 A\)
(d) \(2\cos^2 A - 1\)
Answer: D
\((\cos^4 A - \sin^4 A) = (\cos^2 A - \sin^2 A)(\cos^2 A + \sin^2 A)\)
\(= (\cos^2 A - \sin^2 A)(1)\)
\(= \cos^2 A - (1 - \cos^2 A)\)
\(= 2\cos^2 A - 1\)

Question. If \(\sec 5A = \text{cosec}(A + 30^\circ)\), where \(5A\) is an acute angle, then the value of \(A\) is
(a) \(15^\circ\)
(b) \(5^\circ\)
(c) \(20^\circ\)
(d) \(10^\circ\)
Answer: D
We have, \(\sec 5A = \text{cosec}(A + 30^\circ)\)
\(\sec 5A = \sec[90^\circ - (A + 30^\circ)]\)
\(\sec 5A = \sec(60^\circ - A)\)
\(5A = 60^\circ - A\)
\(6A = 60^\circ\)
\(A = 10^\circ\)

Question. If \(x\sin^3\theta + y\cos^3\theta = \sin\theta\cos\theta\) and \(x\sin\theta = y\cos\theta\), than \(x^2 + y^2\) is equal to
(a) 0
(b) 1/2
(c) 1
(d) 3/2
Answer: C
We have, \(x\sin^3\theta + y\cos^3\theta = \sin\theta\cos\theta\)
\((x\sin\theta)\sin^2\theta + (y\cos\theta)\cos^2\theta = \sin\theta\cos\theta\)
\(x\sin\theta(\sin^2\theta + \cos^2\theta) = \sin\theta\cos\theta\) [Since \(x\sin\theta = y\cos\theta\)]
\(x\sin\theta(1) = \sin\theta\cos\theta \Rightarrow x = \cos\theta\)
Now, \(x\sin\theta = y\cos\theta\)
\(\cos\theta\sin\theta = y\cos\theta\)
\(y = \sin\theta\)
Hence, \(x^2 + y^2 = \cos^2\theta + \sin^2\theta = 1\)

Question. If \(\tan 2A = \cot(A - 18^\circ)\), where \(2A\) is an acute angle, then the value of \(A\) is
(a) \(12^\circ\)
(b) \(18^\circ\)
(c) \(36^\circ\)
(d) \(48^\circ\)
Answer: C
Given, \(\tan 2A = \cot(A - 18^\circ)\)
\(\cot(90^\circ - 2A) = \cot(A - 18^\circ)\)
\(90^\circ - 2A = A - 18^\circ\)
\(90^\circ + 18^\circ = A + 2A\)
\(3A = 108^\circ\)
\(A = \frac{108^\circ}{3} = 36^\circ\)

Question. If \(\tan\theta + \sin\theta = m\) and \(\tan\theta - \sin\theta = n\), then \(m^2 - n^2\) is equal to
(a) \(\sqrt{mn}\)
(b) \(\sqrt{\frac{m}{n}}\)
(c) \(4\sqrt{mn}\)
(d) None of the options
Answer: C
Given, \(\tan\theta + \sin\theta = m\) and \(\tan\theta - \sin\theta = n\)
\(m^2 - n^2 = (\tan\theta + \sin\theta)^2 - (\tan\theta - \sin\theta)^2\)
\(= 4\tan\theta\sin\theta\)
\(= 4\sqrt{\tan^2\theta\sin^2\theta}\)
\(= 4\sqrt{\frac{\sin^2\theta}{\cos^2\theta}\sin^2\theta}\)
\(= 4\sqrt{\frac{\sin^2\theta}{\cos^2\theta} - \sin^2\theta}\)
\(= 4\sqrt{\tan^2\theta - \sin^2\theta}\)
\(= 4\sqrt{(\tan\theta + \sin\theta)(\tan\theta - \sin\theta)}\)
\(= 4\sqrt{mn}\)

Question. If \(0 < \theta < \frac{\pi}{4}\), then the simplest form of \(\sqrt{1 - 2\sin\theta\cos\theta}\) is
(a) \(\sin\theta - \cos\theta\)
(b) \(\cos\theta - \sin\theta\)
(c) \(\cos\theta + \sin\theta\)
(d) \(\sin\theta\cos\theta\)
Answer: B

FILL IN THE BLANK

Question. \(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ = \dots\dots\dots\dots\)
Answer: 1

Question. \(\sin^2 \theta + \sin^2 (90^\circ - \theta) = \dots\dots\dots\dots\)
Answer: 1 [Hint : \(\sin^2 (90^\circ - \theta) = \cos^2 \theta\)]

Question. \(2 \tan^2 45^\circ + 3 \cos^2 30^\circ - \sin^2 60^\circ = \dots\dots\dots\dots\)
Answer: \(\frac{7}{2}\)

Question. Triangle in which we study trigonometric ratios is called \(\dots\dots\dots\dots\)
Answer: Right Triangle

Question. \(\frac{\cos 45^\circ}{\sec 30^\circ + \csc 30^\circ} = \dots\dots\dots\dots\)
Answer: \(\frac{3(\sqrt{3} - 1)}{4}\)

Question. \(\frac{\sin 18^\circ}{\cos 72^\circ} = \dots\dots\dots\dots\)
Answer: 1

Question. \(\cos 48^\circ - \sin 42^\circ = \dots\dots\dots\dots\)
Answer: 0

Question. Cosine of \(90^\circ\) is \(\dots\dots\dots\dots\)
Answer: Zero

Question. If \(15 \cot A = 8\), \(\sec A = \dots\dots\dots\dots\)
Answer: 17/8

Question. The value of \(\sin A\) or \(\cos A\) never exceeds \(\dots\dots\dots\dots\)
Answer: 1

TRUE/FALSE

Question. The value of \(\sin \theta\) increases as \(\theta\) increases.
Answer: True

Question. \(\sqrt{(1 - \cos^2 \theta) \sec^2 \theta} = \tan \theta\)
Answer: True

Question. \(\sec A = \frac{12}{5}\) for some value of angle \(A\).
Answer: True

Question. \(\sin (A + B) = \sin A + \sin B\).
Answer: False

Question. The value of \(\cos \theta\) increases as \(\theta\) increases.
Answer: False

Question. \(\sin \theta = \frac{5}{3}\) for some angle \(\theta\).
Answer: False

Question. The value of \(\tan A\) is always less than 1.
Answer: False

Question. The value of the expression \((\cos^2 23^\circ - \sin^2 67^\circ)\) is positive.
Answer: False

Question. \(\cot A\) is not defined for \(A = 0^\circ\).
Answer: True

Question. \(\sin(90^\circ - A) = \cos A\)
Answer: True

Question. If \(\angle B\) and \(\angle Q\) are acute angles such that \(\sin B = \sin Q\), then \(\angle B \neq \angle Q\).
Answer: False

Question. \((\tan \theta + 2)(2 \tan \theta + 1) = 5 \tan \theta + \sec^2 \theta\)
Answer: False

Question. \(\frac{\tan 65^\circ}{\tan 25^\circ} = 1\)
Answer: False

MATCHING QUESTIONS

Question. In \(\triangle ABC\), \(\angle B = 90^\circ\), \(AB = 3 \text{ cm}\) and \(BC = 4 \text{ cm}\) then match the column.
Column-I | Column-II
(A) \(\sin C\) | (p) 3/5
(B) \(\cos C\) | (q) 4/5
(C) \(\tan A\) | (r) 5/3
(D) \(\sec A\) | (s) 4/3
Answer: (A) - p, (B) - q, (C) - s, (D) - r

Question. Match the following:
Column-I | Column-II
(A) \(\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A}\) | (p) \(\csc A + \cot A\)
(B) \(\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1}\) | (q) \(2 \sec A\)
(C) \(\sqrt{\frac{1 + \sin A}{1 - \sin A}}\) | (r) \(\sec A + \tan A\)
(D) \(\frac{\sin^2 A}{1 - \cos A}\) | (s) \(\frac{1 + \sec A}{\sec A}\)
Answer: (A) - q, (B) - p, (C) - r, (D) - s

Question. If \(\sin A = \frac{7}{25}\), then match the following:
Column-I | Column-II
(A) \(\cos A\) | (p) 24/25
(B) \(\tan A\) | (q) 7/24
(C) \(\csc A\) | (r) 25/7
(D) \(\sec A\) | (s) 25/24
| (t) \(1 - 1/25\)
| (u) \(1 + 1/24\)
Answer: (A) - (p, t), (B) - q, (C) - r, (D) - (s, u)

Question. Match the following:
Column I | Column II
(A) \(\sin^2 37^\circ + \sin^2 53^\circ + \sin^2 90^\circ\) | (p) 0
(B) \(\tan 35^\circ \tan 45^\circ \tan 55^\circ\) | (q) 3
(C) \(\frac{\sec 72^\circ \sin 18^\circ + \tan 72^\circ \cot 18^\circ}{\cos 60^\circ}\) | (r) 1
(D) \(\frac{\tan 60^\circ}{\tan 30^\circ}\) | (s) 2
(E) \(\sin^2 30^\circ + \cos^2 30^\circ - \sin^2 60^\circ - \cos^2 60^\circ\) | (t) 4
Answer: (A) - s, (B) - r, (C) - t, (D) - q, (E) - p

ASSERTION AND REASON

Question. DIRECTION: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(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.

Question. Assertion: The value of \(\sin \theta = \frac{4}{3}\) is not possible.
Reason: Hypotenuse is the largest side in any right angled triangle.
(a) A
(b) B
(c) C
(d) D
Answer: A
\(\sin \theta = \frac{P}{H} = \frac{4}{3}\). Here, perpendicular is greater than the hypotenuse which is not possible in any right triangle.

Question. Assertion: \(\sin^2 67^\circ + \cos^2 67^\circ = 1\)
Reason: For any value of \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\)
(a) A
(b) B
(c) C
(d) D
Answer: A
\(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, \(\sin^2 67^\circ + \cos^2 67^\circ = 1\).

Question. Assertion: The value of \(\sec^2 10^\circ - \cot^2 80^\circ\) is 1
Reason: The value of \(\sin 30^\circ = \frac{1}{2}\)
(a) A
(b) B
(c) C
(d) D
Answer: B
We have, \(\sec^2 10^\circ - \cot^2 80^\circ = \sec^2 10^\circ - \cot^2 (90^\circ - 10^\circ) = \sec^2 10^\circ - \tan^2 10^\circ = 1\). Also, \(\sin 30^\circ = \frac{1}{2}\). Both are true, but Reason is not the explanation for Assertion.

Question. Assertion: \(\sin 47^\circ = \cos 43^\circ\)
Reason: \(\sin \theta = \cos(90 + \theta)\), where \(\theta\) is an acute angle.
(a) A
(b) B
(c) C
(d) D
Answer: C
Assertion is true, but reason is not correct. \(\sin \theta = \cos(90^\circ - \theta)\). So, \(\sin 47^\circ = \cos(90^\circ - 47^\circ) = \cos 43^\circ\).