std xi sci. - target publications · mcq solving are provided in each chapter. notes provide...

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STD. XI Sci. Triumph Physics

Based on Maharashtra Board Syllabus

Printed at: India Printing Works Mumbai

Useful for all Agricultural, Medical, Pharmacy and Engineering Entrance Examinations held across India.

TEID : 770

Solutions/hints to Evaluation Test available in downloadable PDF format at

www. targetpublications.org

Salient Features • Exhaustive subtopic wise coverage of MCQs

• Important formulae provided in each chapter

• Hints included for relevant questions

• Various competitive exam questions updated till the latest year

• Includes solved MCQs from JEE (Main), AIPMT, CET 2014

• Evaluation test provided at the end of each chapter

First Edition: July 2014Fourth Edition: October 2014

Preface

“Std. XI: Sci. Triumph Physics” is a complete and thorough guide to prepare students for a competitive level examination. The book will not only assist students with MCQs of Std. XI but will also help them to prepare for JEE, AIPMT, CET and various other competitive examinations.

The content of this book is based on the Maharashtra State Board Syllabus. Formulae that form a vital part of MCQ solving are provided in each chapter. Notes provide important information about the topic. Shortcuts provide easy and less tedious solving methods. Mindbenders have been introduced to bridge the gap between a text book topic and the student’s understanding of the same. A quick reference to the notes, shortcuts and mindbenders has been provided wherever possible.

MCQs in each chapter are divided into three sections: Classical Thinking: consists of straight forward questions including knowledge based questions. Critical Thinking: consists of questions that require some understanding of the concept. Competitive Thinking: consists of questions from various competitive examinations like JEE, AIPMT, CET, CPMT etc.

Hints have been provided to the MCQs which are broken down to the simplest form possible. An Evaluation Test has been provided at the end of each chapter to assess the level of preparation of the

student on a competitive level. An additional feature of pictorial representation of a topic is added to give the student a glimpse of various

interesting physics concept. The journey to create a complete book is strewn with triumphs, failures and near misses. If you think we’ve

nearly missed something or want to applaud us for our triumphs, we’d love to hear from you. Please write to us on : [email protected]

Best of luck to all the aspirants!

Yours faithfully Authors

Sr. No. Topic Name Page No. 1 Measurements 1 2 Scalars and Vectors 28 3 Projectile Motion 55 4 Force 91 5 Friction in Solids and Liquids 126 6 Sound Waves 161 7 Thermal Expansion 184 8 Refraction of Light 218 9 Ray Optics 256

10 Electrostatics 284 11 Current Electricity 323 12 Magnetic Effect of Electric Current 356 13 Magnetism 382 14 Electromagnetic Waves 403

Publications Pvt. Ltd. Target Chapter 01: Angle and It’s Measurement

1

Syllabus

1.1 Directed angles and systems of measurement of angles

1.2 Relation between degree

measure and radian measure 1.3 Length of an arc and area of

sector

Angle and It’s Measurement01

Roller coasters are the best example, when we look at the real life situation for measuring and drawing the angles. It involves reading the angles of rises and falls on roller coasters.

Roller coasters, all about the angles!

Publications Pvt. Ltd. Target Std. XI : Triumph Maths

2

1. Sexagesimal system (Degree measure): i. 1 right angle = 90 degree (= 90°) ii. 1° = 60 minutes (= 60′) iii. 1′ = 60 seconds (= 60′′) 2. Relation between degree measure and

radian measure:

i. 1° = c π

180⎛ ⎞⎜ ⎟⎝ ⎠

= 0.01746 radian

ii. lc = 180⎛ ⎞⎜ ⎟π⎝ ⎠

= 57° 17′ 48′′(approx)

iii. x° = cπ

180⎛ ⎞⎜ ⎟⎝ ⎠

x and yc = 180⎛ ⎞⎜ ⎟π⎝ ⎠

y

3. Length of an arc and area of sector: If in a circle of radius r an arc of length S

subtends an angle of θ radian at the centre, then S = r θ and

i. Angle in radian, θc = Sr

= arcradius

ii. Area of corresponding sector = 21 r θ2

.

i.e., Area = 12

× r × s 4. The sum of interior angles of a polygon of n

sides = (n − 2) × 180° = (n − 2) × π 5. Each interior angle of a regular polygon of n

sides = 180 21n

⎛ ⎞−⎜ ⎟⎝ ⎠

= ( )π n 2

n−

radian

6. In a regular polygon: i. All the interior angles are equal ii. All the exterior angles are equal iii. All the sides are equal iv. Sum of all the exterior angles is 360° v. Each exterior angle

= 360°number of exterior angles

vi. Each interior angle =180°− exterior angle

1. The angle between two consecutive digits of a

clock = 30° = 6π radians.

2. Angle moved by hour hand in one hour = 30°. 3. Angle moved by hour hand in one minute

= 12

°⎛ ⎞⎜ ⎟⎝ ⎠

.

4. Angle moved by minute hand in one minute = 6°. 5. If the difference between measures of two

directed angles is an integral multiple of 360°, then the two directed angles are co−terminal angles.

Formulae Shortcuts

The angle between two numbers on the clock is c

⎛ ⎞⎜ ⎟⎝ ⎠12

π.

24-hour clock


3

1.1 Directed angles and systems of

measurement of angles 1. A radian is a (A) terminal angle (B) co-terminal angle (C) quadrantal angle (D) constant angle 2. In circular system, the unit of measurement of

an angle is a (A) degree (B) radian (C) minute (D) second 3. If the initial ray and directed ray are opposite

rays, then directed angle formed is called as (A) zero angle (B) straight angle (C) co-terminal angle (D) standard angle 4. The measure of quadrantal angles is an

integral multiple of (A) 360° (B) 180° (C) 90° (D) 60° 5. _____ part of one degree is called one minute.

(A) 60th (B) th1

6⎛ ⎞⎜ ⎟⎝ ⎠

(C) th1

30⎛ ⎞⎜ ⎟⎝ ⎠

(D) th1

60⎛ ⎞⎜ ⎟⎝ ⎠

6. If the terminal arm of a directed standard

angle lies along any one of the co-ordinate axes, then it is called

(A) co-terminal angle (B) quadrantal angle (C) zero angle (D) constant angle 7. (74.87)° = (A) 74°52′52′′ (B) 74°52′12′′ (C) 74°12′52′′ (D) 74°0′52′′ 8. If the angles of a triangle are in the ratio

1 : 2 : 3, then the angles in degrees are (A) 40°, 50°, 90° (B) 30°, 60°, 90° (C) 35°, 45°, 90° (D) 20°, 70°, 90° 9. An hour hand rotates through _______ in one

minute.

(A) 13

⎛ ⎞⎜ ⎟⎝ ⎠

(B) 12

⎛ ⎞⎜ ⎟⎝ ⎠

(C) 30° (D) 6°

10. The minute hand rotates through an angle of _______ in one minute.

(A) 6° (B) 30° (C) 60° (D) 1° 11. 45° 30′ is equal to

(A) 95° (B) o46

2⎛ ⎞⎜ ⎟⎝ ⎠

(C) o91

2⎛ ⎞⎜ ⎟⎝ ⎠

(D) 50° 12. Minute hand of a clock gains _______ on hour

hand in one minute. (A) 5°30′ (B) 59° (C) 5°50′ (D) 360° 13. Which of the following pairs of angles are not

coterminal? (A) 330°, − 60° (B) 405°, − 675° (C) 1230°, − 930° (D) 450°, − 630° 14. If the measure of an angle is 1105°, then it

will lie in (A) 1st quadrant (B) 2nd quadrant (C) 3rd quadrant (D) 4th quadrant 15. If the measures of angles of a quadrilateral are

in the ratio 2 : 3 : 7 : 6, then their measures in degrees will be

(A) 20°, 40°, 60°, 80° (B) 40°, 60°, 80°, 100° (C) 40°, 60°, 140°, 120° (D) 40°, 60°, 160°, 120° 1.2 Relation between degree measure and

radian measure 16. 240º is equal to

(A) c4π

3⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(B) c3π

4⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(C) 4π3

′⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ (D) 3π

4

′′⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

17. The radian measure of an angle of –260° is

(A) c13

12− π⎛ ⎞

⎜ ⎟⎝ ⎠

(B) c13

9− π⎛ ⎞

⎜ ⎟⎝ ⎠

(C) c12

9− π⎛ ⎞

⎜ ⎟⎝ ⎠

(D) c26

9− π⎛ ⎞

⎜ ⎟⎝ ⎠

18. Taking πc = 3.14159, 1c = (A) 60° (B) 180° (C) 57.3° (D) 0°

Classical Thinking


4

19. If xc = 340° and y° = −c2π

5, then x and y is

equal to

(A) x = c7π

9 , y = 72°

(B) x =c17π

9 , y = −72°

(C) x =c9π

7 , y = −72°

(D) x =c17π

9 , y = −27°

20. – 37° 30′ =

(A) c5π

24 (B) –

c5π24

⎛ ⎞⎜ ⎟⎝ ⎠

(C) c7π

24 (D) –

c7π24

⎛ ⎞⎜ ⎟⎝ ⎠

21. The radian measure of an angle of 75° is

(A) c5π

12 (B)

cπ12

(C) c4π

3 (D)

c7π12

22. c19π

9− is equal to

(A) −360° (B) −380° (C) −340° (D) −300° 23. The exterior angle of a regular pentagon in

radian measure is

(A) cπ

5 (B)

c2π5

(C) c3π

5 (D)

c4π5

24. If the difference between two acute angles of a

right angled triangle is c2π

5, then the angles in

degrees are (A) 81°, 9° (B) 35°, 55° (C) 20°, 40° (D) 50°, 30° 25. The measures of angles of a triangle are in the

ratio 2 : 3 : 5. Their measures in radians are

(A) cπ

5,

c3π10

, cπ

2 (B)

cπ5

, c3π

10,

cπ3

(C) cπ

6,

c5π12

, c3π

4 (D)

cπ4

, c3π

10,

cπ2

26. If the radian measures of two angles of a

triangle are 3π5

, 4π5

, then the radian measure

of third angle is

(A) cπ

15 (B)

c2π15

(C) cπ

5 (D)

c4π15

27. The sum of two angles is 5πc and their

difference is 60°. The angles in degrees are (A) 400°, 480° (B) 340°, 420° (C) 480°, 420° (D) 440°, 460° 1.3 Length of an arc and area of sector 28. The length of the arc subtended by an angle

of 7π4

radians on a circle of radius 20 cm is

(A) 80π7

cm (B) 35π cm

(C) 20π cm (D) 7π cm 29. If two circular arcs of the same length subtend

angles of 60° and 80° at their respective centres, then the ratio of their radii is

(A) 34

(B) 43

(C) 32

(D) 916

30. If the arcs of the same length of two circles

subtend 75° and 140° at the centre, then the ratio of the radii of the circles is

(A) 28:15 (B) 11:13 (C) 22:15 (D) 21:13 31. An arc of a circle of radius 77 cm subtends an

angle of 10° at the centre. The length of the arc is

(A) 1219

cm (B) 88 cm

(C) 111 cm (D) 77 cm 32. The perimeter of a sector of a circle, of area

36π sq.cm., is 28 cm. The area of sector is equal to

(A) 12 sq.cm (B) 16 sq.cm (C) 48 sq.cm (D) 96 sq.cm 33. A pendulum 14 cm long oscillates through an

angle of 18°. The length of path described by its extremity is

(A) 4.6 cm (B) 4.4 mm (C) 4.8 cm (D) 4.4 cm


5


measurement of angles 1. π radians = ______ right angles

(A) 0 (B) 1 (C) 12

(D) 2 2. Angles with measure 45° and −315° are (A) zero angles. (B) straight angles. (C) co-terminal angles. (D) standard angles. 3. ____ is the largest unit in Sexagesimal system. (A) Degree (B) Radian (C) Minute (D) Second 4. The measure of co-terminal angles always

differ by an integral multiple of (A) 90° (B) 180° (C) 270° (D) 360° 5. The angle between minute hand and hour hand

of a clock at 8:30 is (A) 80° (B) 75° (C) 60° (D) 105° 6. The angle of measure −1560° lies in (A) 1st quadrant (B) 2nd quadrant (C) 3rd quadrant (D) 4th quadrant 7. The angle between two hands of a clock at

quarter past one is

(A) 60° (B) 1522

ο⎛ ⎞⎜ ⎟⎝ ⎠

(C) cπ

3⎛ ⎞⎜ ⎟⎝ ⎠

(D) 172

ο⎛ ⎞⎜ ⎟⎝ ⎠

1.2 Relation between degree measure and

radian measure 8. The radian measure of an angle of 19° 30′ is

equal to

(A) c12π

130⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(B) 13π120⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

c

(C) c4π

3⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(D) c13π

12⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

9. At 3:40, the hour hand and minute hands of a

clock are inclined at

(A) c13π

18⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(B) cπ

9⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(C) c3π

8⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

(D) c5π

6⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

10. The radian measure of the interior angle of a regular dodecagon is

(A) c5π

6 (B)

c3π2

(C) cπ

4 (D)

c4π3

11. The radian measure of the interior angle of a

regular heptagon is

(A) cπ

7 (B)

c3π7

(C) c5π

7 (D)

c7π5

12. If the measures of angles of a quadrilateral are

in the ratio 2 : 5 : 8 : 9, then their measures in radians, will be

(A) cπ

6,

c5π12

,c3π

2,

c3π4

(B) cπ

3,

c5π12

,c2π

3,

c2π5

(C) cπ

6,

c5π12

,c2π

3,

c4π3

(D) cπ

6,

c5π12

,c2π

3,

c3π4

13. The difference between two acute angles of a

right angled triangle is π9

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

c

. The angles in

degrees are (A) 50º, 30º (B) 25º, 45º (C) 20º, 40º (D) 55º, 35º 14. If the sum of two angles is 1 radian and the

difference between them is 1°, then the smaller angle is

(A) ο90 1

π 2⎛ ⎞−⎜ ⎟⎝ ⎠

(B) ο90 1

π 2⎛ ⎞+⎜ ⎟⎝ ⎠

(C) ο180 1

π⎛ ⎞−⎜ ⎟⎝ ⎠

(D) ο180 1

π⎛ ⎞+⎜ ⎟⎝ ⎠

15. 5°37′30″ =

(A) cπ

4⎛ ⎞⎜ ⎟⎝ ⎠

(B) cπ

8⎛ ⎞⎜ ⎟⎝ ⎠

(C) cπ

16⎛ ⎞⎜ ⎟⎝ ⎠

(D) cπ

32⎛ ⎞⎜ ⎟⎝ ⎠

1.3 Length of an arc and area of sector 16. The length of an arc of a circle of radius 5 cm

subtending a central angle measuring 15º is

(A) 3π12

cm (B) 7π12

cm

(C) 5π12

cm (D) π4

cm 17. The area of a sector, whose arc length is 25π

cm and the angle of the sector is 60°, will be (A) 1925.5π sq.cm (B) 1875π sq.cm (C) 937.5π sq.cm (D) 75π sq.cm

Critical Thinking


6

18. In a circle of diameter 66 cm, the length of a chord is 33 cm. The length of minor arc of the chord is

(A) 33π cm (B) 11π cm (C) 22π cm (D) 5.5π cm 19. A railway engine is travelling along a circular

railway track of radius 1500 meters with a speed of 66 km/ hour. The angle turned by the engine in 10 seconds is

(A) c15

7 (B)

c715

(C) c90

11 (D)

c1190

20. If a pendulum 18 cm long oscillates through

an angle of 32°, then length of the path described by its extremity is

(A) 516

π cm (B) 165π cm

(C) 8π45

cm (D) 65π cm

21. If Kalyan is 48 km from Mumbai and the earth

being regarded as a sphere of radius 6400 km, then the nearest second an angle subtended at the centre of the earth by the arc joining them is (Take π = 22/7)

(A) 22°64′ (B) 24°65′ (C) 23′62′′ (D) 25′46′′ 22. The perimeter of a certain sector of a circle is

equal to half that of the circle of which it is a sector. Then the circular measure of sector is

(A) (π + 2) radians (B) (π − 2) radians (C) (π + 1) radians (D) (π − 1) radians 23. A wire 96 cm long is bent, so as to lie along

the arc of a circle of 180 cm radius. The angle subtended at the centre of the arc in degree is

(A) 30° (B) 29° 30′ (C) 28° 30′ (D) 30° 30′ 24. The perimeter of a sector of a circle of area 64 π sq. cm is 56 cm, then area of sector is (A) 140 sq.cm (B) 150 sq.cm (C) 160 sq.cm (D) 170 sq.cm 25. The length of an arc of a circle of radius 5 cm

subtending an angle of measure 45° is

(A) 4π cm (B) 5

4π cm

(C) 5π cm (D) 4

5π cm


measurement of angles 1. At 2.15 O’clock, the hour and the minute

hands of a clock form an angle of [AMU 1992]

(A) 5° (B) 1222

ο

(C) 28° (D) 30° 1.3 Length of an arc and area of sector 2. The angle subtended at the centre of a circle of

radius 3 metre by an arc of length 1 metre is equal to

[MNR 1973] (A) 20° (B) 60°

(C) 13

radian

(D) 3 radians 3. A circular wire of radius 7 cm is cut and bend

again into an arc of a circle of radius 12 cm. The angle subtended by the arc at the centre is

[Kerala (Engg.) 2002] (A) 50° (B) 210° (C) 100° (D) 60° 4. The radius of the circle whose arc of length 15

cm makes an angle of 3/4 radian at the centre is

[Karnataka CET 2002] (A) 10 cm (B) 20 cm

(C) 11 14

cm (D) 22 12

cm 5. The distance between 6.00 A. M. and

3.15 P. M. by the tip of the 12 cm long hour hand in a clock is [SCRA 1999]

(A) 352

π cm

(B) 18 π cm

(C) 372

π cm

(D) 19 π cm

Competitive Thinking


7

Classical Thinking

1. (D) 2. (B) 3. (B) 4. (C) 5. (D) 6. (B) 7. (B) 8. (B) 9. (B) 10. (A)

11. (C) 12. (A) 13. (A) 14. (A) 15. (C) 16. (A) 17. (B) 18. (C) 19. (B) 20. (B)

21. (A) 22. (B) 23. (B) 24. (A) 25. (A) 26. (B) 27. (C) 28. (B) 29. (B) 30. (A)

31. (A) 32. (C) 33. (D) Critical Thinking

1. (D) 2. (C) 3. (A) 4. (D) 5. (B) 6. (C) 7. (B) 8. (B) 9. (A) 10. (A)

11. (C) 12. (D) 13. (D) 14. (A) 15. (D) 16. (C) 17. (C) 18. (B) 19. (D) 20. (B)

21. (D) 22. (B) 23. (D) 24. (C) 25. (B) Competitive Thinking

1. (B) 2. (C) 3. (B) 4. (B) 5. (C) Classical Thinking 7. 74.87° = 74° + (0.87)° = 74° + (0.87 × 60)′ = 74° + 52.2′ = 74° + 52′ + (0.2 × 60)′′ = 74°52′12′′ 8. Let the angles be x, 2x and 3x. Then, x + 2x + 3x = 180°

….[∵ sum of the angles of a triangle = 180°]

∴ 6x = 180° ∴ x = 30°, 2x = 60° and 3x = 90° 10. In 1 minute, minute hand covers 360°

i.e., in 60 mins, minute hand covers 36060

° = 6°

11. 30′ = o1

2⎛ ⎞⎜ ⎟⎝ ⎠

∴ 45°30′ = 45° + 12

⎛ ⎞⎜ ⎟⎝ ⎠

= 912

⎛ ⎞⎜ ⎟⎝ ⎠

12. In one minute, minute hand covers 6°

∴ Hour hand covers 12

⎛ ⎞⎜ ⎟⎝ ⎠

∴ Minute hand gains = 6° − 12

⎛ ⎞⎜ ⎟⎝ ⎠

= 5° + 12

⎛ ⎞⎜ ⎟⎝ ⎠

= 5°30′ 13. Here, 405° − (− 675°) = 1080° = 3(360°), 1230° − (− 930°) = 2160° = 6(360°) and 450° − (− 630°) = 1080° = 3(360°) are a multiple of 360°. ∴ these angles are co-terminal Now, 330° − (− 60°) = 390° Which is not a multiple of 360°. ∴ these pair of angles are not co-terminal. 14. 1105° = 3 × 360° + 25° ∵ 0° < 25° < 90°

∴ it lies in 1st quadrant

Hints

Answer Key


8

15. Let the angles of a quadrilateral be 2k, 3k, 7k and 6k in degrees.

….[∵ the sum of angles of a quadrilateral is 360°]

∴ 2k + 3k + 7k + 6k = 360° ∴ 18k = 360° ∴ k = 20° ∴ The measures of four angles are 2k = 2 × 20° = 40° 3k = 3 × 20° = 60° 7k = 7 × 20° = 140° 6k = 6 × 20° = 120° 16. 180º = πc

⇒ 1° = cπ

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

⇒ 240º =cπ240×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= c4π

3⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

17. −260° = c

260180

π⎛ ⎞− ×⎜ ⎟⎝ ⎠

= c13

9− π

18. 1c = o1801×

π⎛ ⎞⎜ ⎟⎝ ⎠

= 57.3°

19. xc = cπ340×

180⎛ ⎞⎜ ⎟⎝ ⎠

= c17

9π

Also y° = −c2π

5

∴ y° = − 2 1805π⎛ ⎞×⎜ ⎟π⎝ ⎠

= −72°

∴ y = −72°

20. 30′ = 12

ο⎛ ⎞⎜ ⎟⎝ ⎠

∴ –37° 30′ = – ο137

2⎛ ⎞⎜ ⎟⎝ ⎠

= ο75

2⎛ ⎞−⎜ ⎟⎝ ⎠

= – c75 π×

2 180⎛ ⎞⎜ ⎟⎝ ⎠

= –c5π

24⎛ ⎞⎜ ⎟⎝ ⎠

21. 180° = πc

∴ 1° = cπ

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

∴ 75° = cπ75 ×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= c5π

12

22. c19π

9− =

o19π 180×9 π

−⎛ ⎞⎜ ⎟⎝ ⎠

= −380°

23. Number of sides = 5

Exterior angle = 360no.of sides

°

∴ Exterior angle = 3605° =

c360 π×5 180

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠=

c2π5

24. Let the two acute angles measured in degrees

be x and y, then x + y = 90° ….(i)

and x − y = c2π

5=

o2π 180×5 π

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

∴ x − y = 72° ….(ii) Adding (i) and (ii), we get 2x = 162° ∴ x = 81° Putting the value of x in (i), we get y = 9° 25. Let the measures of angles of the triangle be

2k, 3k, 5k in degrees. ∴ 2k + 3k + 5k = 180°

….[∵ sum of the angles of a triangle is 180°]

∴ k = 18° ∴ Angles are of measure 36°, 54°, 90° ∴ Three angles in radians are

36° = cπ36×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= cπ

5

54° = cπ54×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= c3π

10

90° = cπ90×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= cπ

2


9

26. The measures of two angles of a triangle are c3π

5,

c4π5

i.e. o3π 180×

5 π⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

, o4π 180×

15 π⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

i.e. 108°, 48° Let the measure of third angle of the triangle

be x ∴ 108° + 48° + x = 180°

∴ x = 24° = cπ24×

180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= c2π

15

27. Let the measures of two acute angles be x and

y ∴ x + y = 5πc

∴ x + y = o1805π×

π⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

∴ x + y = 900° ….(i) and x − y = 60° ….(ii) Adding (i) and (ii), we get x = 480° Putting x = 480° in (i), we get y = 420° ∴ the two angles are 480° and 420°. 28. S = r θ = 20 × 7π

4 = 35π cm

29. Given that, S1 = S2 If the radii are r1 and r2, then r1 × θ1 = r2 × θ2

∴ r1 60π180

⎛ ⎞⎜ ⎟⎝ ⎠

= r2 80π180

⎛ ⎞⎜ ⎟⎝ ⎠

….[∵ S = r θ]

∴ 1

2

rr

= 80π180

× 18060π

= 43

30. S1 = S2 ∴ r1 × θ1 = r2 × θ2

∴ r175180

π⎛ ⎞⎜ ⎟⎝ ⎠

= r2140180

π⎛ ⎞⎜ ⎟⎝ ⎠

∴ 1

2

rr

= 14075

∴ 1

2

rr

= 2815

31. We have, r = 77 cm

and θ = 10° =cπ10×

180⎛ ⎞⎜ ⎟⎝ ⎠

= cπ

18

Since, S = r θ = 77 π18

⎛ ⎞⎜ ⎟⎝ ⎠

= 77× 2218×7

= 1219

cm 32. Area of circle = πr2 = 36π sq.cm ∴ r = 6 cm Now, perimeter of sector = 2r + S But, perimeter is given to

be 28 cm. ∴ 28 = 12 + S ∴ S = 16 cm

Area of sector = 12

× r × S = 12

× 6 × 16

= 48 sq.cm 33. Since, S = r θ

= 14 × cπ18×

180⎛ ⎞⎜ ⎟⎝ ⎠

= 14 × 110

× 227

∴ S = 4410

= 4.4 cm Critical Thinking 5. Angle between the consecutive digits on a

clock is 30°. ∴ Angle between 6 and 8 is 60°. Also at 8:30, hour hand is between 8 and 9. ∴ It must have covered 15° ∴ 60° + 15° = 75° 6. −1560° = −4 × 360° − 120° ∵ −180 < − 120° < − 90°

∴ it lies in 3rd quadrant

r r

S

12 23

568 4

7

910

11 1


10

7. If hours hand was at 1 and minutes hand at 3, the angle between the two hands would have been 60°.

In 15 minutes, hours hand revolves through o360×15

720⎛ ⎞⎜ ⎟⎝ ⎠

= o17

2⎛ ⎞⎜ ⎟⎝ ⎠

….[∵ In 12 hours, i.e., 720 min, hours

hand revolves through 360°] ∴ Required angle between the hands of clock

= 60° − o17

2⎛ ⎞⎜ ⎟⎝ ⎠

= 1522

ο⎛ ⎞⎜ ⎟⎝ ⎠

8. 19°30′ = o3019+

60⎛ ⎞⎜ ⎟⎝ ⎠

= o39

2⎛ ⎞⎜ ⎟⎝ ⎠

= c39 π×

2 180⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

= c13π

120⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

9. At 3:40, the minute−hand is at mark 8 and

hour hand has crossed rd2

3of the angle

between 3 and 4 Now, angle between any two consecutive

marks = 360°12

= 30°

Angle traced by hour hand in 40 minutes

= 23

(30°) = 20°

Angle between mark 3 and 8 = 5 × 30° = 150° Angle between two hands = 150° − 20° = 130°

= cπ130×

180⎛ ⎞⎜ ⎟⎝ ⎠

= c13π

18⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

10. Number of sides = 12 Number of exterior angles = 12 Sum of exterior angles = 360°

∴ Each exterior angle = 360°12

∴ Each interior angle = 36018012

°⎛ ⎞⎟⎜ − ⎟⎜ ⎟⎜⎝ ⎠

= 150° = cπ150

180⎛ ⎞⎟⎜ × ⎟⎜ ⎟⎜⎝ ⎠

= c5π

6

11. Interior angle = 360180n

⎛ ⎞−⎜ ⎟⎝ ⎠

∴ Each interior angle = 3601807

⎛ ⎞−⎜ ⎟⎝ ⎠

=

9007

⎛ ⎞⎜ ⎟⎝ ⎠

= c900 π×

7 180⎛ ⎞⎜ ⎟⎝ ⎠

=

c5π7

12. Let the angles of the quadrilateral be 2k, 5k, 8k and 9k in degrees. ∴ 2k + 5k + 8k + 9k = 360° ∴ 24k = 360° ∴ k = 15° ∴ Measures of angles of the quadrilateral are 30°, 75°, 120°, 135°

∴ c

30180

π⎛ ⎞×⎜ ⎟⎝ ⎠

, cπ75×

180⎛ ⎞⎜ ⎟⎝ ⎠

, c

120180

π⎛ ⎞×⎜ ⎟⎝ ⎠

,

c

135180

π⎛ ⎞×⎜ ⎟⎝ ⎠

i.e., cπ

6,

c5π12

,c2π

3,

c3π4

13. In triangle ABC, let ∠ C = 90º

So, ∠ A – ∠ B = π9

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

c

= 20º …. (i)

and sum of all the angles in ∆ABC is 180° ∴ ∠A +∠B +∠C = 180º Since, ∠C = 90º ∴ ∠A + ∠B = 90º ..... (ii) Solving (i) and (ii), we get ∠A = 55º, ∠B = 35º

12 23

5 6 8 4

7

910

11 1


11

14. Let the two angles be A and B, where A > B.

Then A + B = 1 radian = ο180

π⎛ ⎞⎜ ⎟⎝ ⎠

and A – B = 1° Subtracting, we get

2B = ο180 1

π⎛ ⎞−⎜ ⎟⎝ ⎠

∴ B = ο90 1

π 2⎛ ⎞−⎜ ⎟⎝ ⎠

15. 30″ = 12

′⎛ ⎞⎜ ⎟⎝ ⎠

∴ 37′30″ = 1372

′⎛ ⎞⎜ ⎟⎝ ⎠

= 752

′⎛ ⎞⎜ ⎟⎝ ⎠

= 75 1×2 60

ο⎛ ⎞⎜ ⎟⎝ ⎠

= 58

ο⎛ ⎞⎜ ⎟⎝ ⎠

Then, 5°37′30″ = ο55

8⎛ ⎞⎜ ⎟⎝ ⎠

= 458

ο⎛ ⎞⎜ ⎟⎝ ⎠

= c45 π×

8 180⎛ ⎞⎜ ⎟⎝ ⎠

= cπ

32⎛ ⎞⎜ ⎟⎝ ⎠

16. Given that, r = 5 cm

and θ = 15° = π15×180

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

c

= cπ

12⎛ ⎞⎜ ⎟⎝ ⎠

∴ S = r θ = 5 × π12

= 5π12

cm

17. θ = 60° =cπ60×

180⎛ ⎞⎜ ⎟⎝ ⎠

=cπ

3

and S = 25π cm But, S = r θ

∴ 25π = r ×3π

∴ r = 75 cm

∴ Area of sector = 21

× r × S = 21

× 75 × 25π

= 937.5π sq.cm

18. Let ‘O’ be the centre of the circle d = 66 cm ∴ r = 33 cm ∴ ∆OAB is an equilateral triangle

∴ θ = 60° = cπ

3

∴ l(minor arc AB) = r θ

= 33 × π3

= 11π cm

19. Speed = 66 km/hr = 66 × 518

m/s

∴ Speed = 553

m/s

∵ Distance = speed × time

∴ S = 553

× 10 = 5503

Also, S = r θ

∴ 5503

= 1500 × θ

∴ θ = 5503 1500×

= c11

90

20. Given, r = 18 cm,

θ = 32° =c

32180

π⎛ ⎞×⎜ ⎟⎝ ⎠

=c8

45π

Since, S = r θ = 18 × 845π⎛ ⎞

⎜ ⎟⎝ ⎠

∴ S = 165π cm

21. Let O be the centre of the earth. Let K and M be the positions of Kalyan and Mumbai. KM = 48 km, r = radius of earth = 6400 km

∴ θ = Sr

= 486400

= c3

400⎛ ⎞⎜ ⎟⎝ ⎠

O

θ

A B

O M

K48 6400

θ 6400


12

∠MOK = c3

400⎛ ⎞⎜ ⎟⎝ ⎠

= 3400

× 180°π

= 2720π

= 2720

× 722

= 189440

° = 189440

× 60′

= 25′46′′ nearly

22. Perimeter of sector = 12

(Perimeter of circle)

∴ r + r + S = 12

(2πr)

∴ S = πr − 2r ∴ S = (π − 2) r ….(i) Since, S = r θ ….(ii) ∴ θ = π − 2 ….[From (i) and (ii)] 23. S = 96 cm, r = 180 cm Since, S = r θ

∴ θ = Sr

=

96180

=

96 180×180 π

ο⎛ ⎞⎜ ⎟⎝ ⎠

= 30.5° = 30° 30′ 24. A = 64 π ∴ πr2 = 64π ∴ r = 8 cm Since, perimeter of sector = 2r + S ∴ S + r + r = 56 ∴ S + 8 + 8 = 56 ∴ S = 40 Since, S = r θ ∴ 40 = 8 × θ ∴ θ = 5c

∴ Area of sector = 12

r2 θ

= 12

× 82 × 5

= 160 sq.cm.

25. S = r θ = 5 × 45° = 5 × 45180

π⎛ ⎞×⎜ ⎟⎝ ⎠

= 54π cm

Competitive Thinking 1. Angle covered by hour hand in 60 minutes = 30° ∴ angle covered by hour hand in 15 minutes

= 30 1560

°⎛ ⎞×⎜ ⎟⎝ ⎠

= 172

ο

∴ reqd. angle = 30° – 172

ο

= 1222

ο

2. radius (r) = 3m and arc (S) = 1m

∴ Angle = arcradius

= 13

radian

3. Diameter of circular wire = 14 cm ∴ length of circular wire = 14π cm

∴ Required angle = arcradius

= 14π12

= 7π6

= 7 180×6 π

οπ⎛ ⎞⎜ ⎟⎝ ⎠

= 210°

4. Angle = arcradius

∴ Radius = arcangle

∴ Radius = 1534

⎛ ⎞⎜ ⎟⎝ ⎠

∴ Radius = 20 cm 5. Angle covered from 6 A.M. to 3.15 P.M.

= 277 12

ο

= 5552

× 180

π

∴ θ = 3724

π radians

Length of hour hand = 12 cm i.e., r = 12 cm

Now, S = r θ = 12 3724× π = 37

2π

Hence, required distance = 372π cm


13

1. The angle subtended at the centre of a circle of

diameter 50 cm by an arc of 11 cm is (A) 25° 10′ (B) 20° 12′ (C) 25° 12′ (D) 20° 10′ 2. A horse is tied to a post by a rope. If the horse

moves along a circular path always keeping the rope tight and describes 88 metres when it has traced out 72° at the centre, then the length of the rope is

(A) 60 m (B) 70 m (C) 80 m (D) 90 m 3. If the circular measures of two angles of a

triangle are 12

and 13

, then the measure of

third angle in degrees is 22Take7

⎛ ⎞π =⎜ ⎟⎝ ⎠

(A) 145° 15′22′′ (B) 132° 16′22′′ (C) 132° 3′22′′ (D) 123° 16′21′′ 4. The moon’s distance from the earth is 360000

kms and its diameter subtends an angle of 31′ at the eye of observer, then the diameter of the moon is

(A) 3247.62 km (B) 3246.62 km (C) 3245.62 km (D) 3244.62 km 5. A circular wire of diameter 10 cm is cut and

placed along the circumference of a circle of diameter 1 metre. The angle subtended by the wire at the centre of the circle is equal to

(A) radian4π (B) radian

3π

(C) radian5π (D) radian

10π

6. If a person of normal sight can read print at

such a distance that the letters subtend an angle of 5′ at his eye, then the height of the letters that he can read at a distance of 12 metres is

(A) 1.6 cm (B) 1.5 cm (C) 1.9 cm (D) 1.7 cm 7. The angles of a triangle are in A.P. and ratio

of the number of degrees in the least to the number of radians in the greatest is 60 : π. The angles of the triangle in degrees are

(A) 24°, 60°, 96° (B) 30°, 60°, 90° (C) 20°, 60°, 100° (D) 32°, 60°, 88°

8. Two circles each of radius 7 cm intersect each other. If the distance between their centres is 7 2 cm, then the area common to both the circles is

(A) 492

(π + 2) sq.cm

(B) 492

(π − 4) sq.cm

(C) 492

(π − 2) sq.cm

(D) 492

(π + 4) sq.cm 9. The perimeter of a certain sector of a circle is

equal to the length of the arc of a semicircle having the same radius. The angle of the sector in degrees is

(A) 65°27′16′′ (B) 65°27′10′′ (C) 65°27′27′′ (D) 65°27′12′′ 10. The ratio of the interior angle of first polygon

to that of the second polygon is 3 : 2 and the number of sides in first are twice that in the second. The number of sides of the two polygons are

(A) 3, 6 (B) 8, 4 (C) 2, 4 (D) 6, 12

1. (C) 2. (B) 3. (B) 4. (A) 5. (C) 6. (D) 7. (B) 8. (C) 9. (A) 10. (B)

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