Vardaan Learning Institute

Chapter Practice Sheet: Introduction to Trigonometry

Class: 10 (CBSE) Subject: Mathematics Max. Marks: 50
SECTION A: OBJECTIVE TYPE QUESTIONS (1 Mark Each)
1. If $\sin A = 3/4$, then $\cos A$ is:
  1. $4/3$
  2. $\sqrt{7}/4$
  3. $3/\sqrt{7}$
  4. $\sqrt{7}/3$
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2. The value of $\tan 45^\circ + \cot 45^\circ$ is:
  1. 0
  2. 1
  3. 2
  4. $1/2$
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3. If $\sin(A+B) = 1$ and $\cos(A-B) = \sqrt{3}/2$, $0^\circ < A+B \le 90^\circ, A>B$, then A is:
  1. $30^\circ$
  2. $45^\circ$
  3. $60^\circ$
  4. $90^\circ$
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4. The value of $(\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ)$ is:
  1. -1
  2. 0
  3. 1
  4. 2
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5. The value of $(\sec A + \tan A)(1 - \sin A)$ is:
  1. $\sec A$
  2. $\sin A$
  3. $\csc A$
  4. $\cos A$
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6. The value of $9 \sec^2 A - 9 \tan^2 A$ is:
  1. 1
  2. 9
  3. 8
  4. 0
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7. If $x = a \cos \theta$ and $y = b \sin \theta$, then the value of $b^2 x^2 + a^2 y^2$ is:
  1. $a^2 + b^2$
  2. $a^2 b^2$
  3. $ab$
  4. 1
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8. The value of $(1 + \tan^2 A)(1 - \sin A)(1 + \sin A)$ is:
  1. 0
  2. 1
  3. 2
  4. -1
[1]
9. If $\sin \alpha = 1/2$ and $\cos \beta = 1/2$, find the value of $\alpha + \beta$.
  1. $30^\circ$
  2. $60^\circ$
  3. $90^\circ$
  4. $45^\circ$
[1]
10. Assertion (A): The value of $\sin \theta$ increases as $\theta$ increases from 0 to 90 degrees.
Reason (R): $\sin 0^\circ = 0$ and $\sin 90^\circ = 1$.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
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SECTION B: SHORT ANSWER TYPE QUESTIONS (2 Marks Each)
11. Prove that: $(\sec A + \tan A)(1 - \sin A) = \cos A$.
[2]
12. If $\sin \theta = \cos \theta$, find the value of $2 \tan \theta + \cos^2 \theta$.
[2]
13. Evaluate: $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ$.
[2]
14. Prove that $\sqrt{\frac{1+\sin A}{1-\sin A}} = \sec A + \tan A$.
[2]
SECTION C: SHORT ANSWER TYPE II QUESTIONS (3 Marks Each)
15. If $\tan A + \sin A = m$ and $\tan A - \sin A = n$, show that $m^2 - n^2 = 4\sqrt{mn}$.
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16. If $\sec \theta + \tan \theta = p$, prove that $\sin \theta = \frac{p^2-1}{p^2+1}$.
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17. Prove that $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1-\cos A}$.
[3]
18. If $7 \sin^2 \theta + 3 \cos^2 \theta = 4$, show that $\tan \theta = \frac{1}{\sqrt{3}}$.
[3]
19. Prove that $(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$.
[3]
20. If $\cos \theta + \sin \theta = \sqrt{2} \cos \theta$, show that $\cos \theta - \sin \theta = \sqrt{2} \sin \theta$.
[3]
SECTION D: LONG ANSWER TYPE QUESTIONS (5 Marks Each)
21. Prove that: $\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$.
[5]
22. Prove that: $\frac{\tan \theta}{1-\cot \theta} + \frac{\cot \theta}{1-\tan \theta} = 1 + \sec \theta \csc \theta$.
[5]
SECTION E: CASE STUDY (4 Marks)
23. Case Study: Right Triangle
In a right-angled triangle ABC, right-angled at B, if $\tan A = 1$ and AC = 10 cm.

(i) Find the measure of angle A. (1 Mark)
(ii) Find the length of AB. (2 Marks)
(iii) Evaluate $\sin A \cos C + \cos A \sin C$. (1 Mark)
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