std xi sci. triumph maths - target...

STD. XI Sci. Triumph Maths

Based on Maharashtra Board Syllabus

Printed at: Jasmine Art Printers Pvt. Ltd., Navi Mumbai

Useful for all Engineering Entrance Examinations held across India.

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

www.targetpublications.org/tp10144

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) 2014, 15, 16.

• Evaluation test provided at the end of each chapter.

10144_10940_JUP

P.O. No. 28827

© Target Publications Pvt. Ltd. No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical

including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

Preface

“Std. XI: Sci. Triumph Maths” 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 (Main), CET and various other competitive examinations.

The content of this book is based on the Maharashtra State Board Syllabus. Formulae that form a key part for solving MCQs have been provided in each chapter. Shortcuts provide easy and less tedious solving methods.

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 (Main), CET, 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. 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 Angle and It’s Measurement 1

2 Trigonometric Functions 15

3 Trigonometric Functions of

Compound Angles 35

4 Factorization Formulae 76

5 Locus 98

6 Straight Line 113

7 Circle and Conics 151

8 Vectors 205

9 Linear Inequations 239

10 Determinants 251

Sr. No. Topic Name Page No.

11 Matrices 292

12 Sets, Relations and Functions 324

13 Logarithms 371

14 Complex Numbers 386

15 Sequence and Series 434

16 Permutations and Combinations 493

17 Method of Induction and Binomial Theorem

521

18 Limits 556

19 Differentiation 601

20 Integration 619

21 Statistics (Measures of Dispersion) 643

22 Probability 660

1

Chapter 01: Angle and It’s Measurement

Subtopics

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

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!

Angle and It’s Measurement 01

2

Std. XI : Triumph Maths

P

Q

O lc

r

r

r

1. Sexagesimal system (Degree measure): i. 1 right angle = 90 degree (= 90) ii. 1 = 60 minutes (= 60) iii. 1 = 60 seconds (= 60) 2. Circular system (Radian measure): i. If r is radius of circle

with centre O, and P and Q are two points such that l(arc PQ) = r, then mPOQ is defined to be l radian. It is denoted by lc.

ii. A radian is a constant angle iii. Radian measure is independent of the

radius of the circle. 3. Relation between degree measure and

radian measure:

i. 1 = c

π

180

= 0.01745c (approx.)

ii. lc = 180

= 57 17 48(approx.)

iii. x = c

π

180

xand yc =

180

y

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

subtends an angle of c at the centre, then i. Length of arc (S) = r

ii. Area of corresponding sector = 21r θ

2

i.e., Area = 1

2 r S

1. If the difference between measures of two

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

2. The measure of quadrantal angles are integral

multiples of 90. 3. The angle between two consecutive digits of a

clock = 30 = c

6

4. Angle moved by hour hand in one hour = 30.

5. Angle moved by hour hand in one minute

= 1

2

.

6. Angle moved by minute hand in one minute = 6. 7. Sum of the measures of angles of triangle is c

and of quadrilateral is 2c. 8. The sum of the measure of interior angles of a

polygon of n sides = (n 2) 180 = (n 2) c 9. Each interior angle of a regular polygon of n

sides = 180 2

1n

=

π n 2

n

radian

10. In a regular polygon: i. All the sides are equal ii. All the interior angles are equal iii. All the exterior angles 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

Shortcuts

Chapter at a glance

The angle between two numbers on the clock is

c 12

.

24-hour clock

3


1.1 Directed Angles and Systems of

Measurement of Angles 1. 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 2. The measure of co-terminal angles always

differ by an integral multiple of (A) 90 (B) 180 (C) 270 (D) 360 3. Angles with measure 45 and 315 are (A) zero angles. (B) straight angles. (C) co-terminal angles. (D) standard angles. 4. 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 5. 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 6. The measure of quadrantal angles is an

integral multiple of (A) 360 (B) 180 (C) 90 (D) 60 7. _______ is the largest unit in Sexagesimal

system. (A) Degree (B) Radian (C) Minute (D) Second 8. _____ part of one degree is called one minute.

(A) 60th (B) th

1

6

(C) th

1

30

(D) th

1

60

9. The minute hand rotates through an angle of

_______ in one minute. (A) 6 (B) 30 (C) 60 (D) 1

10. An hour hand rotates through _______ in one minute.

(A) 1

3

(B)

1

2

(C) 30 (D) 6 11. Minute hand of a clock gains _______ on hour

hand in one minute. (A) 530 (B) 59 (C) 550 (D) 360 12. (74.87) = (A) 745252 (B) 745212 (C) 741252 (D) 74052 13. 45 30 is equal to

(A) 95 (B) o

46

2

(C) o

91

2

(D) 50

14. 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 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 16. In circular system, the unit of measurement of

an angle is a (A) degree (B) radian (C) minute (D) second 17. A radian is a (A) terminal angle (B) co-terminal angle (C) quadrantal angle (D) constant angle 1.2 Relation between degree measure and

radian measure 18. The radian measure of an angle of 75 is

(A) c5π

12 (B)

cπ

12

(C) c4π

3 (D)

c7π

12

Classical Thinking

4


19. 240º is equal to

(A) c

4π

3

æ ö÷ç ÷ç ÷çè ø (B)

c3π

4

æ ö÷ç ÷ç ÷çè ø

(C) 4π

3

¢æ ö÷ç ÷ç ÷çè ø (D)

3π

4

¢¢æ ö÷ç ÷ç ÷çè ø

20. The radian measure of an angle of –260 is

(A)

c13

12

(B)

c13

9

(C)

c12

9

(D)

c26

9

21. – 37 30 =

(A) c5π

24 (B) –

c5π

24

(C) c7π

24 (D) –

c7π

24

22. radians = ______ right angles

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

2 (D) 2

23. c19π

9

- is equal to

(A) 360 (B) 380 (C) 340 (D) 300 24. Taking c = 3.14159, 1c = (A) 60 (B) 180 (C) 57.3 (D) 0 25. The sum of two angles is 5c and their

difference is 60. The angles in degrees are (A) 400, 480 (B) 340, 420 (C) 480, 420 (D) 440, 460 26. The measures of angles of a triangle are in the

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

(A) cπ

5,

c3π

10,

cπ

2

(B) cπ

5,

c3π

10,

cπ

3

(C) cπ

6,

c5π

12,

c3π

4

(D) cπ

4,

c3π

10,

cπ

2

27. If the radian measures of two angles of a

triangle are 3π

5,

4π

15, then the radian measure

of third angle is

(A) cπ

15 (B)

c2π

15

(C) cπ

5 (D)

c4π

15

28. If the measures of angles of a quadrilateral are

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

(A) c

π

6,

c5π

12,

c3π

2,

c3π

4 (B)

cπ

3,

c5π

12,

c2π

3,

c2π

5

(C) c

π

6,

c5π

12,

c2π

3,

c4π

3(D)

cπ

6,

c5π

12,

c2π

3,

c3π

4 29. The exterior angle of a regular pentagon in

radian measure is

(A) cπ

5 (B)

c2π

5

(C) c3π

5 (D)

c4π

5

30. The radian measure of exterior angle of

octagon is

(A) c

4

(B)

c

3

(C) c

6

(D)

c

2

31. If each exterior angle of a polygon is 24 then

the number of sides of polygon are (A) 12 (B) 15 (C) 10 (D) 24 1.3 Length of an arc and area of sector 32. The length of the arc subtended by an angle

of 7π

4radians on a circle of radius 20 cm is

(A) 80π

7cm (B) 35 cm

(C) 20 cm (D) 7 cm 33. The length of an arc of a circle of radius 5 cm

subtending an angle of measure 45 is

(A) 4

cm (B)

5

4

cm

(C) 5

cm (D)

4

5

cm

5


34. An arc of a circle of radius 77 cm subtends an angle of 10 at the centre. The length of the arc is

(A) 121

9 cm (B) 88 cm

(C) 111 cm (D) 77 cm 35. If a pendulum 18 cm long oscillates through

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

(A) 5

16

cm (B)

16

5

cm

(C) 8π

45cm (D)

6

5

cm

36. 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 37. The radius of circle is 5 cm and the length of

arc is 5

12

. The angle subtended by the arc at

the centre is (A) 20 (B) 15 (C) 25 (D) 45 38. 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) 3

4 (B)

4

3

(C) 3

2 (D)

9

16

39. 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 40. The area of a sector, whose arc length is 25

cm and the angle of the sector is 60, will be (A) 1925.5 sq.cm (B) 1875 sq.cm (C) 937.5 sq.cm (D) 75 sq.cm 41. Area of the sector is 25 and the length of arc

is 10. The angle subtended by the arc at the centre is

(A) c

2

(B) c

5

(C) c

2

(D) c

5


Measurement of Angles 1. Which of the following pairs of angles are not

coterminal?

(A) 330, 60 (B) 405, 675 (C) 1230, 930 (D) 450, 630 2. The angle of measure 1560 lies in

(A) 1st quadrant (B) 2nd quadrant

(C) 3rd quadrant (D) 4th quadrant 3. The angle between minute hand and hour hand

of a clock at 8:30 is

(A) 80 (B) 75 C) 60 (D) 105 4. The angle between two hands of a clock at

quarter past one is

(A) 60 (B) 1

522

(C) c

π

3

(D) 1

72

5. The angles of a triangle are in AP. If the

smallest angle is 36, then the measure of the other angles are

(A) 60, 84 (B) 54, 90 (C) 36, 108 (D) 72, 108 6. A wheel makes 3600 rotations in 1 hour.

Through how many radians does it turn in 1 minute?

(A) 12c (B) 10c

(C) 60c (D) 120c 1.2 Relation between degree measure and

radian measure 7. The radian measure of an angle of 19 30 is

equal to

(A) c

12π

130


13π

120


c

(C) c

4π

3

æ ö÷ç ÷ç ÷çè ø (D)

c13π

12


Critical Thinking

6


8. 53730 =

(A) c

π

4

(B) c

π

8

(C) c

π

16

(D) c

π

32

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

clock are inclined at

(A) c

13π

18


cπ

9


(C) c

3π

8

æ ö÷ç ÷ç ÷÷çè ø (D)

c5π

6


10. 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

11. If the sum of two angles is 1 radian and the

difference between them is 1, then the smaller angle is

(A) ο

90 1

π 2

(B) ο

90 1

π 2

(C) ο

1801

π

(D) ο

1801

π

12. 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

13. The difference between two acute angles of a

right angled triangle is π

9


c

. The angles in

degrees are (A) 50º, 30º (B) 25º, 45º (C) 20º, 40º (D) 55º, 35º 14. The radian measure of the interior angle of a

regular heptagon is

(A) cπ

7 (B)

c3π

7

(C) c5π

7 (D)

c7π

5

15. The radian measure of the interior angle of a

regular dodecagon is

(A) c5π

6 (B)

c3π

2

(C) cπ

4 (D)

c4π

3

1.3 Length of an arc and area of sector 16. 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) 33 cm (B) 11 cm

(C) 22 cm (D) 5.5 cm 17. 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 18. 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) c7

15

(C) c90

11

(D) c11

90

7


19. If Kalyan is 48 km from Mumbai and the earth being regarded as a sphere of radius 6400 km, then the angle subtended at the centre of the earth by the arc joining them is

(Take = 22/7)

(A) 2264

(B) 2465

(C) 2362

(D) 2546 20. 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) ( + 2) radians

(B) ( 2) radians

(C) ( + 1) radians

(D) ( 1) radians 21. The perimeter of a sector of a circle, of area

36 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 22. The perimeter of a sector of a circle of area

64 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 23. Two circles each of radius 14 cm intersect

each other. If the distance between their

centres is 14 2 cm, the area common to both is

(A) 140 sq.cm

(B) 112 sq.cm

(C) 154 sq.cm

(D) 308 sq.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) 1

222

(C) 28 (D) 30 1.3 Length of an arc and area of sector 2. 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) 111

4cm (D) 22

1

2cm

3. 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) 1

3 radian (D) 3 radians

4. 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° 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) 35

2 cm (B) 18 cm

(C) 37

2 cm (D) 19 cm

Competitive Thinking

8


Classical Thinking 1. (B) 2. (D) 3. (C) 4. (A) 5. (B) 6. (C) 7. (A) 8. (D) 9. (A) 10. (B) 11. (A) 12. (B) 13. (C) 14. (B) 15. (C) 16. (B) 17. (D) 18. (A) 19. (A) 20. (B) 21. (B) 22. (D) 23. (B) 24. (C) 25. (C) 26. (A) 27. (B) 28. (D) 29. (B) 30. (A) 31. (B) 32. (B) 33. (B) 34. (A) 35. (B) 36. (D) 37. (B) 38. (B) 39. (A) 40. (C) 41. (A) Critical Thinking 1. (A) 2. (C) 3. (B) 4. (B) 5. (A) 6. (D) 7. (B) 8. (D) 9. (A) 10. (B) 11. (A) 12. (A) 13. (D) 14. (C) 15. (A) 16. (B) 17. (D) 18. (D) 19. (D) 20. (B) 21. (C) 22. (C) 23. (B) Competitive Thinking 1. (B) 2. (B) 3. (C) 4. (B) 5. (C) Classical Thinking 3. 45 (315) = 45 + 315 = 360 which is integral multiple of 360 the angles are co-terminal. 4. 1105 = 3 360 + 25 Since, 0 < 25 < 90 it lies in 1st quadrant. 9. In 60 minute, minute hand covers 360

In 1 minute hand covers 360

60

= 6

10. In 60 minutes, hour hand covers 30

in 1 minute, hour hand covers 30

60

=

1

2

11. In one minute, minute hand covers 6, and

hour hand covers 1

2

Minute hand gains = 6 1

2

= 5 + 1

2

= 530

12. 74.87 = 74 + (0.87) = 74 + (0.87 60) = 74 + 52.2 = 74 + 52 + (0.2 60) = 745212

13. 30 = o

1

2

4530 = 45 + 1

2

=

91

2

14. Let the measures of the angles be x, 2x and 3x

in degrees. Now, x + 2x + 3x = 180

….[ sum of the measures of the angles of a

triangle = 180] 6x = 180 x = 30, 2x = 60 and 3x = 90 15. Let the measures of the angles of the

quadrilateral be 2k, 3k, 7k and 6k in degrees. Now, 2k + 3k + 7k + 6k = 360

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

18k = 360 k = 20 The measures of the angles of quadrilateral are 2k = 2 20 = 40 3k = 3 20 = 60 7k = 7 20 = 140 6k = 6 20 = 120

Hints

Answer Key

9


18. 1 = c

π

180


75 = c

π75 ×

180

æ ö÷ç ÷ç ÷çè ø=

c5π

12

19. 240º =c

π240×

180


c4π

3


20. 260 = c

260180

= c13

9

21. –37 30 = – ο

137

2

= ο

75

2

= – c

75 π×

2 180

= –c

5π

24

22. c 180

= 180 = 2 90

23. c19π

9

-=

o19π 180

×9 π

= 380

24. 1c = o

1801×

π

= 57.3 25. Let the measures of two angles be x and y

x + y = 5c

x + y = o

1805π ×

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. 26. Let the measures of the angles 2k, 3k, 5k in

radians. 2k + 3k + 5k =

…. [∵ the sun of measures of the angles of triangle is c]

10k =

k = 10

The angles are c c3

, ,5 10 2

27. Let the measure of third angle of the triangle be x

3 4

5 15

+ x =

x = 13

15

x =

c2

15

28. Let the measure of the angles be 2k, 5k, 8k

and 9k in radians.

2k + 5k + 8k + 9k = 2

….[ the sum of the measures of the angles

of a quadrilateral is 2c]

24k = 2

k = 12

Measures of angles of the quadrilateral are

c

π

6,

c5π

12,

c2π

3,

c3π

4 29. Number of sides = 5

Exterior angle = 360

no. of sides


5

=

c360 π

×5 180


c2π

5

30. Number of sides = 8


8

=

c

45180

= c

4

31. Exterior angle = 360

n

where n is the number of sides of polygon

24 = 360

n

n = 15

32. S = r = 20 7π

4 = 35 cm

33. S = r = 5 45 = 5 45180

= 5

4

cm

10


34. = 10 =c

π10×

180

= cπ

18

S = r = 77 π

18

= 77× 22

18×7

= 121

9 cm

35. = 32 = c

32180

= c8

45

S = r = 18 8

45

S = 16

5

cm

36. S = r

= 14 c

π18×

180

= 14 1

10

22

7

= 44

10 = 4.4 cm

37. Since S = r

= S

r

=

5

125

= c

12

= 180

2

= 15

38. Given that, S1 = S2 If the radii are r1 and r2, then r1 1 = r2 2

r1 60π

180

= r2 80π

180

….[ S = r ]

1

2

r

r=

80

60 =

4

3

39. S1 = S2 r1 1 = r2 2

r175

180

= r2140

180

1

2

r

r =

140

75

1

2

r

r =

28

15

40. = 60 =c

π60×

180

= cπ

3

and S = 25 cm S = r

25 = r 3

r = 75 cm

Area of sector = 2

1 r S =

2

1 75 25

= 937.5 sq.cm

41. Area of sector = 1

2 r S

25 = 1

r 102

r = 5 Now S = r

= S

r

= 10

5 =

c2

Critical Thinking 1. Here, 405 ( 675) = 1080 = 3(360), 1230 ( 930) = 2160 = 6(360), and 450 ( 630) = 1080 = 3(360) the above paris of angles are co-terminal. Now, 330 ( 60) = 390 Since 390 is not a multiple of 360. This pair of angles is not co-terminal. 2. 1560 = 4 360 120 Since, 180 < 120 < 90 it lies in 3rd quadrant 3. Angle between two 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 Angle between minute hand and hour hand =

60 + 15 = 75

12

23

5 6 8 4

7

9

1011 1

11


4. If hour hand is at 1 and minute hand is at 3,

the angle between the two hands is 60. In 15 minutes, hour hand covers

o360×15

720

= o

17

2

….[ In 12 hours, i.e., 720 min, hours

hand revolves through 360]

Required angle between the hands of clock

= 60 o

17

2

= 1

522

5. Let the angles be x, x + d and x + 2d

Given that x = 36

the angles are 36, (36 + d) and (36 + 2d)

Now, 36 + 36 + d + 36 + 2d = 180

3d = 72

d = 24

The angels are (36 + 24) and (36 + 48)

i.e. 60 and 84 6. No. of rotations in 60 minutes = 3600

No. of rotations in 1 minute = 3600

60 = 60

Angle traced in 1 rotations = 2c

Angle traced in 60 rotations = 60 2c

= 120 c

7. 19 30 = o

3019 +

60

= o

39

2

= c

39 π×

2 180


= c

13π

120


8. 30 = 1

2

3730 = 1

372

= 75

2

= 75 1

×2 60

= 5

8

53730 = ο

55

8

= 45

8

= c

45 π×

8 180

= c

π

32

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

numbers = 30 angle traced by hour hand in 40 minutes

=2

3(30) = 20

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

= c

π130×

180

= c

13π

18


10. xc = c

π340×

180

= c17

9

y = 2 180

5

= 72

11. Let the measure of two angles be x and y,

where x > y.

x + y = 1c = ο

180

π

and x – y = 1 Subtracting, we get

2y = ο

1801

π

y = ο

90 1

π 2

12

23

5 6 8 4

7

9

10 11 1

12


12. Let the measure of the two acute angles be x and y in degrees

x + y = 90 ….(i)

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 13. Let the mesures of the two acute angles be x

and y in degrees

x + y = 90 ….(i)

x y = π

9


c

= 20º ….(ii)

Solving (i) and (ii), we get x = 55º, y = 35º 14. Here, n = 7

Each interior angle = c

(n 2)

n

=

c(7 2)

7

=

c5π

7

15. Here, n = 12

Each interior angle = c

π(12 2)

12

= c

10π

12


= c5π

6

16. Let ‘O’ be the centre of the circle d = 66 cm

r = 33 cm

OAB is an equilateral triangle

= 60 = cπ

3

l(minor arc AB) = r

= 33 π

3

= 11 cm

17. S = 96 cm, r = 180 cm Since, S = r

= S

r=

96

180

=

96 180×

180 π

= 30.5 = 30 30

18. Speed = 66 km/hr = 66 5

18m/s

Speed = 55

3m/s

Since, Distance = speed time

S = 55

3 10 =

550

3

Also, S = r

550

3 = 1500

= 550

3 1500 =

c11

90

19. 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

= S

r =

48

6400 =

c3

400

MOK = c

3

400

= 3

400

180°

π

= 27

20π =

27

20

7

22

= 189

440

=

189

440 60

= 2546 nearly

20. Perimeter of sector = 1

2(Perimeter of circle)

r + r + S = 1

2 (2r)

S = r 2r S = ( 2) r ….(i) Since, S = r ….(ii) = 2 ….[From (i) and (ii)]

O

A B

O M

K

48 6400

6400

13


21. Area of circle = r2 = 36 sq.cm

r = 6 cm

Since, perimeter of sector

= 2r + S

28 = 12 + S

S = 16 cm

Area of sector = 1

2 r S =

1

2 6 16

= 48 sq.cm 22. Area of circle = r2 = 64

r = 8 cm

Since, perimeter of sector = 2r + S

S + r + r = 56

S + 16 = 56

S = 40

Area of sector = 1

2 r S

= 1

2 8 40

= 160 sq.cm. 23.

Let P and Q be the centres of the two circles.

PQ = 14 2 cm

PR = RQ = SQ = SP = 14

PQRS is square

= c

2

Now A(sector PRS) = A(sector QRS) = 1

2r2

= 1

2 142

2

= 154 sq.cm.

A(shaded region) = A(sector PRS) +

A(sector QRS) A( PRQS)

= 154 + 154 142

= 112 sq. cm

Competitive Thinking 1. Angle between 2 and 3 on the clock = 30 Angle covered by hour hand in 15 minutes

= 360

15720

= 1

72

required angle = 30 – 1

72

= 1

222

2. Since S = r

r = S

r = 1534

= 20 cm

3. Since S = r

= S

r=

1

3radian

4. Radius of circular wire = 7 cm

length of circular wire = 2r = 14 cm

Now, S = 14, R = 12

= arc

radius=

14π

12

= 7π

6 =

7 180×

6 π

= 210°

5. Angle covered from 6 A.M. to 3.15 P.M. ()

= o

1270 7

2

= 277 1

2

= 555

2

c

180

= c37

24

Length of hour hand = 12 cm

i.e., r = 12 cm

Now, S = r = 12 37

24

=

37

2

cm

Hence, required distance = 37

2

cm

r r

S

P

S

R

Q

14 14

14 14

14


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 176 metres when it has traced out 54 at the centre, then the length of the rope is

(A) 100 m (B) 186.67 m (C) 176 m (D) 280.7 m 3. If the circular measures of two angles of a

triangle are 1

2 and

1

3, then the measure of

third angle in degrees is 22

Take7

(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) radian4

(B) radian

3

(C) radian5

(D) radian

10

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 : . 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) 49

2( + 2) sq.cm

(B) 49

2( 4) sq.cm

(C) 49

2( 2) sq.cm

(D) 49

2( + 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) 652716 (B) 652710 (C) 652727 (D) 652712 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)

Evaluation Test

Answers to Evaluation Test

std xi sci. triumph maths - target...

Documents