STD. XI Sci. Perfect Mathematics - I

Transcription

1 Written as per the revised syllabus presribed by the Maharashtra State Board of Seondary and Higher Seondary Eduation, Pune. STD. XI Si. Perfet Mathematis - I Fifth Edition: May 0 Salient Features Exhaustive overage of entire syllabus. Covers answers to all textual and misellaneous exerises. Preise theory for every topi. Neat, labelled and authenti diagrams. Written in systemati manner. Self evaluative in nature. Pratie problems and multiple hoie questions for effetive preparation. Printed at: Dainik Saamna, Navi Mumbai No part of this book may be reprodued or transmitted in any form or by any means, C.D. RM/Audio Video Cassettes or eletroni, mehanial inluding photoopying; reording or by any information storage and retrieval system without permission in writing from the Publisher. TEID : 06

2 Prefae In the ase of good books, the point is not how many of them you an get through, but rather how many an get through to you. Std. XI Si. : PERFECT MATHEMATICS - I is a omplete and thorough guide ritially analysed and extensively drafted to boost the students onfidene. The book is prepared as per the Maharashtra State board syllabus and provides answers to all textual questions. At the beginning of every hapter, topi wise distribution of all textual questions inluding pratie problems has been provided for simpler understanding of different types of questions. Neatly labelled diagrams have been provided wherever required. Pratie Problems and Multiple Choie Questions help the students to test their range of preparation and the amount of knowledge of eah topi. Important theories and formulae are the highlights of this book. The steps are written in systemati manner for easy and effetive understanding. The journey to reate a omplete 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 : mail@targetpubliations.org Yours faithfully, Publisher Best of luk to all the aspirants! No. Topi Name Page No. Angle and It s Measurement Trigonometri Funtions Trigonometri Funtions of Compound Angles 6 Fatorization Formulae Lous 6 6 Straight Line 7 Cirle and Conis 0 8 Vetors 77 Linear Inequations 0 0 Determinants 67 Matries

3 Target Publiations Pvt. Ltd. Chapter 0: Angle and it s Measurement 0 Angle and it s Measurement Coterminal angles Type of Problems Exerise Q. Nos. Degree measure and radian measure Length of an ar Area of a setor Length of an ar and area of a setor. Q. (i. to iv.) Pratie Problems Q. (i., ) (Based on Exerise.) Q.. (i. to v) Q.. (i. to v) Q.. (i., ). Q., 6, 7 Q.8. (i., ) Q., 0,,,, Pratie Problems (Based on Exerise.) Misellaneous Pratie Problems (Based on Misellaneous) Q. (i. to v.) Q. (i. to iv.) Q. (i. to i) Q., 6, 7, 8 Q. (i., ) Q.0 Q.. (i., ) Q.. (i., ) Q.. (i. to i) Q.,,,,, 6, 7, 8, 0 Q. (i., ) Q. (i., ) Q.,,, 6,,, (i., ), 6,. Q.,,,, Pratie Problems Q.,,,, (Based on Exerise.) Misellaneous Q.7, 8,, 0, Pratie Problems (Based on Misellaneous) Q.8,, 0,, 7, 8. Q.7, 8,, 0 Pratie Problems (Based on Exerise.) Q.6, 8,, 0 Misellaneous Q.6 Pratie Problems (Based on Misellaneous) Q.7. Q.6 Pratie Problems (Based on Exerise.) Q.7 Misellaneous Q., Pratie Problems (Based on Misellaneous) Q.

4 Target Publiations Pvt. Ltd. Syllabus: Direted angles, zero angle, straight angle, oterminal angles, standard angles, angle in a quadrant and quadrantal angles. Systems of measurement of angles: Sexagesimal system (degree measure), Cirular system (radian measure), Relation between degree measure and radian measure, length of an ar of a irle and area of setor of a irle. Introdution In shool geometry we have studied the definition of angle and trigonometri ratios of some aute angles. In this hapter we will extend the onept for different angles. You are familiar with the definition of angle as the union of two non-ollinear rays having ommon end point. But aording to this definition measure of angle is always positive and it lies between 0 to 80. In order to study the onept of angle in broader manner, we will extend it for magnitude and sign. The measurement of angle and sides of a triangle and the inter-relation between them was first studied by Greek astronomers Hipparhus and Ptolemy and Indian mathematiians Aryabhatta and Brahmagupta. Direted angles Suppose X is the initial position of a ray. This ray rotates about from initial position X and takes a finite position along ray P. In suh a ase we say that rotating ray X desribes a direted angle XP. P Terminal ray Vertex Initial ray X In the above figure, the point is alled the vertex. The ray X is alled the initial ray and ray P is alled the terminal ray of an angle XP. The pair of rays are also alled the arms of angle XP. In general, an angle an be defined as the ordered pair of initial and terminal rays or arms rotating from initial position to terminal position. The direted angle inludes two things i. Amount of rotation (magnitude of angle). Diretion of rotation (sign of the angle). Positive angle: Std. XI Si.: Perfet Maths - I If a ray rotates about the vertex (the point) from initial position X in antilokwise diretion, then the angle desribed by the ray is positive angle. Terminal ray In the above figure, XP is obtained by the rotation of a ray in antilokwise diretion denoted by arrow. Hene XP is positive i.e., +XP. Negative angle: If a ray rotates about the vertex (the point), from initial position X in lokwise diretion, then the angle desribed by the ray is negative angle. Terminal ray P +ve angle Initial ray In the above figure, XP is obtained by the rotation of a ray in lokwise diretion denoted by arrow. Hene XP is negative angle i.e., XP. Angle of any magnitude: i. Suppose a ray starts from the initial position X in antilokwise sense and makes omplete rotation (revolution) about and takes the final position along X as shown in the figure (i), then the angle desribed by the ray is 60. Initial ray ve angle P In figure (ii) initial ray rotates about in antilokwise sense and ompletes two rotations (revolutions). Hene, the angle desribed by the ray is 60 = 70. Fig. (i) Fig. (ii) X X X X

5 Target Publiations Pvt. Ltd. In figure (iii) initial ray rotates about in lokwise sense and ompletes two rotations (revolutions). Hene, the angle desribed by the ray is 60 = 70 Fig. (iii) X Suppose a ray starting from the initial position X makes one omplete rotation in antilokwise sense and takes the position P as shown in figure, then the angle desribed by the revolving ray is 60 + XP. P Straight angle: Chapter 0: Angle and it s Measurement In figure, X is the initial position and P is the final position of rotating ray. The rays X and P lie along the same line but in opposite diretion. In this ase XP is alled a straight angle and m XP = 80. Coterminal angles: P X Two angles with different measures but having the same positions of initial and terminal ray are alled as oterminal angles. P i If XP =, then the traed angle is If the rotating ray ompletes two rotations, then the angle desribed is 60 + = 70 + and so on. Suppose the initial ray makes one omplete rotation about in lokwise sense and attains its terminal position P, then the desribed angle is (60 + XP). P If XP =, then the traed angle is (60 + ). If final position P is obtained after,,,. omplete rotations in lokwise sense, then angle desribed are ( 60 + ), ( 60 + ), ( 60 + ),.. Types of angles Zero angle: If the initial ray and the terminal ray lie along same line and same diretion i.e., they oinide, the angle so obtained is of measure zero and is alled zero angle. P X X X In figure, the direted angles having measures 0, 0, 0 have the same initial arm, ray X and the same terminal arm, ray P. Hene, these angles are oterminal angles. Note: If two direted angles are o-terminal angles, then the differene between measures of these two direted angles is an integral multiple of 60. Standard angle: An angle whih has vertex at origin and initial arm along positive Xaxis is alled standard angle. Y Q P X 0 0 R Y In figure XP, XQ, XR with vertex and initial ray along positive Xaxis are alled standard angles or angles in standard position. Angle in a Quadrant: 0 An angle is said to be in a partiular quadrant, if the terminal ray of the angle in standard position lies in that quadrant. X X

6 Target Publiations Pvt. Ltd. X Q R In figure XP, XQ and XR lie in first, seond and third quadrants respetively. Quadrantal Angles: If the terminal arm of an angle in standard position lie along any one of the o-ordinate axes, then it is alled as quadrantal angle. X Q R Y In figure XP, XQ, and XR are quadrantal angles. Note: The quadrantal angles are integral multiples of 0 i.e., n, where n N. Systems of measurement of angles Y Y Y P There are two systems of measurement of an angle: i. Sexagesimal system (Degree measure) Cirular system (Radian measure) i. Sexagesimal system (Degree Measure): In this system, the unit of measurement of an angle is a degree. Suppose a ray P starts rotating in the antilokwise sense about and attains the original position for the first time, then the amount of rotation aused is alled revolution. Divide revolution into 60 equal parts. Eah P part is alled as a one degree(). revolution i.e., revolution = 60 P X X Std. XI Si.: Perfet Maths - I Divide into 60 equal parts. Eah part is alled as a one minute ( ). i.e., = 60 Divide into 60 equal parts. Eah part is alled as a one seond () i.e., = 60 Note: The sexagesimal system is extensively used in engineering, astronomy, navigation and surveying. Cirular system (Radian measure): In this system, the unit of measurement of an angle is radian. Angle subtended at the entre of a irle by an ar whose length is equal to the radius is alled as one radian denoted by. Draw any irle with entre and radius r. Take the points P and Q on the irle suh that the length of ar PQ is equal to radius of the irle. Join P and Q. Q r r Then by the definition, the measure of PQ is radian ( ). Notes: i. This system of measuring an angle is used in all the higher branhes of mathematis. The radian is a onstant angle, therefore radian does not depend on the irle i.e., it does not depend on the radius of the irle as shown below. Q C r r P P r B In figure we draw two irles of different radii r and r and entres and B respetively. Then the angle at the entre of both irles is equal to. i.e., PQ = = ABC. r r A

7 Target Publiations Pvt. Ltd. Theorem: A radian is a onstant angle. R Angle subtended at the entre of a irle by an ar whose length is equal to the radius of the irle is always onstant. Proof: Let be the entre and r be the radius of the irle. Take points P, Q and R on the irle suh that ar PQ = r and PR = 0. By definition of radian, PQ = ar PR = irumferene of the irle = R r = r Q r By proportionality theorem PQ = arpq r P PR arpr PQ = arpq arpr PR r = right angle r i.e., = ( right angle).(i) = onstant R.H.S. of equation (i) is onstant and hene a radian is a onstant angle. Relation between degree measure and radian measure: i. = (. right angle) = 0 = 80 = 80 i = 80 = 0.07 (approx.) 80 = = 80. = x iv. In general x = and y 80y = 80 v. 80 = / 7 = 60 = 7 = 7. (approx). Exerise. Chapter 0: Angle and it s Measurement. Determine whih of the following pairs of angles are oterminal: i. 0, 0 0, 60 i 0, 67 iv. 0, 0 i. 0 ( 0) = = 60 = (60) whih is a multiple of 60. Hene, the given angles are oterminal. i iv. 0 ( 60) = = 0 whih is not a multiple of 60. Hene, the given angles are not oterminal. 0 ( 67) = = 080 = (60) whih is a multiple of 60. Hene, the given angles are oterminal. 0 ( 0) = = 60 = 6(60) whih is a multiple of 60. Hene, the given angles are oterminal.. Express the following angles in degrees: i. 7 i 8 iv. v. 7 vi. v 7 i. = 80 = 7 7 = 7 80 = 0 i 8 80 = 8 = 0 iv. = 80 = 60

8 Target Publiations Pvt. Ltd. v. vi. v 7 = 80 = (8.7) approx 7 = 80 = 0 7 = 7 80 = (.). Express the following angles in radians: i. 0 i iv. 600 v. vi. 08 v i. 0 = 0 80 = = 80 = i = 80 = iv. 600 = = 0 v. = vi. 08 = = v = 80 =. Express the following angles in degrees, minutes and seonds form: i. (.) (00.6) i. (.) = + 0. = + (0. 60) = + = (00.6) = 00 + (0.6) = 00 + (0.6 60) = = 00 6 Std. XI Si.: Perfet Maths - I. If x = 0 and y =, find x and y. x = 0 (given) x = 0 80 = x = Also, y = 80 y = = y = 6. If = and = 00, find and. =..(given) 80 = = 00 = 00 Also, = 00 = = = 7. In ABC, ma = and mb =. Find mc in both the systems. In ABC, 80 ma = = = 0 and mb = But, ma + mb + mc = 80...(sum of measures of angles of a triangle is 80) mc = 80 mc = 80 6 = = 80 = mc = = 6

9 Target Publiations Pvt. Ltd. 8. If the radian measures of two angles of a triangle are as given below. Find the radian measure and the degree measure of the third angle: i., 8, i. The measures of two angles of a triangle are, 8 80 i.e.,, 80 8 i.e., 00, 0 Let the measure of third angle of the triangle be x x = (sum of measures of angles of a triangle is 80) 0 + x = 80 x = 80 0 x = 0 = 0 80 = 6 Measure of third angle of the triangle is 0 or. 6 The measures of two angles of a triangle are, 80 i.e.,, 80 i.e., 08, 8 Let the measure of third angle of the triangle be x x = 80. (sum of measures of angles of a triangle is 80) 6 + x = 80 x = 80 6 x = = 80 = Measure of third angle of the triangle is or. Chapter 0: Angle and it s Measurement. The differene between two aute angles of a right angled triangle is. Find the 0 angles in degrees. Let the two aute angles measured in degrees be x and y. x + y = 0..(i) and x y = 0. (given) = 80 0 x y =..(ii) Adding (i) and (ii), we get x = x = 7 Putting the value of x in (i), we get 7 + y = 0 y = 8 Hene, the two aute angles are 7 and The sum of two angles is and their differene is 60. Find the angles in degrees. Let the two aute angles measured in degrees be x and y. x + y =.. (given) 80 x + y = x + y = 00..(i) and x y = 60..(ii) (given) Adding (i) and (ii), we get x = 60 x = 80 Putting the value of x in (i), we get 80 + y = 00 y = 0 Hene, the two angles are 80 and 0.. The measures of angles of a triangle are in the ratio : :. Find their measures in radians. Let the measures of angles of the triangle be k, k, k in degrees. k + k + k = 80.(sum of measures of angles of a triangle is 80) 7

10 Target Publiations Pvt. Ltd. 0k = 80 k = 8 the measures of three angles are k = 8 = 6 k = 8 = k = 8 = 0 These three angles in radians are 6 = 6 80 = = 80 = 0 0 = 0 80 =.. ne angle of a triangle has measure and the measures of other two angles are in the ratio :, find their measures in degrees and radians. 80 = = 0 Let the measures of other two angles of the triangle be k and k in degrees. 0 + k + k = 80 7k = 0 k = 0 the measures of two angles are k = 0 = 80 k = 0 = 60 These two angles in radians are 80 = = The measures of angles of a quadrilateral are in the ratio : : : 6. Find their measures in radians. Let the measures of angles of the quadrilateral be k, k, k, 6k in degrees. k + k + k + 6k = 60.(sum of measures of angles of a quadrilateral is 60) 8 8k = 60 k = 0 the measures of angles are k = 0 = 60 k = 0 = 80 k = 0 = 00 6k = 6 0 = 0 These angles in radians are 60 = = 80 = = 00 = = 0 = 0 80 = Std. XI Si.: Perfet Maths - I. ne angle of a quadrilateral has measure and the measures of other three angles are in the ratio : :. Find their measures in radians and in degrees. 80 = = 7 Let the measures of other three angles of the quadrilateral be k, k, k in degrees. 7 + k + k + k = 60 k = 88 k = the measures of angles are k = = 6 k = = 6 k = = 8 the angles of the quadrilateral in degrees are 7, 6, 6, 8. The angles in radians are 6 = 6 80 = 6 6 = 6 80 = 8 8 = 8 80 =

11 Target Publiations Pvt. Ltd. Length of an ar and area of setor of a irle Theorem: If S is the length of an ar of a irle of radius r whih subtends an angle at the entre of the irle, then S = r. Proof: Let be the entre and r be the radius of the irle. Let AB be an ar of the irle with length S units and mab =. Let AA be the diameter of the irle (Note that is measured in radians) Now, (ar AB) m AB and (ar ABA) m AA B arab = ar ABA ' S = A irumferene S r = S = r Length of an ar, S = r. Theorem: If is an angle between two radii of the irle of radius r, then the area of the orresponding setor is r. Proof: Let be the entre and r be the radius of the irle and m AB =. Let AA be the diameter of the irle Area of setor AB m AB and area of setor ABA m AA Area of setor AB mab Area of setor ABA' maa' B A r Area of setor AB = Area of setor ABA r A r r S A Chapter 0: Angle and it s Measurement Area of setor AB = r. Note: A(setor) = r = r r = r S Note: The above theorems are not asked in examination but are provided just for referene. Exerise.. Find the length of an ar of a irle whih subtends an angle of 08 at the entre, if the radius of the irle is m. Here, r = m and = 08 = = Sine, S = r. S = = m.. The radius of a irle is m. Find the length of an ar of this irle whih uts off a hord of length equal to length of radius. B Here, r = m Let the ar AB ut off a hord equal to the radius of the irle. A AB is an equilateral triangle. mab = 60 = 60 = = Sine, S = r. = (r ) = r S = = m.

12 Target Publiations Pvt. Ltd.. Find in radians and degrees the angle subtended at the entre of a irle by an ar whose length is m, if the radius of the irle is m. Here, r = m and S = m Sine, S = r. = = = = 80 = A pendulum of m long osillates through an angle of 8. Find the length of the path desribed by its extremity. Here, r = m and = 8 = 8 80 = 0 Sine, S = r. = 0 S = 7 m.. Two ars of the same length subtend angles of 60 and 7 at the entres of the irles. What is the ratio of radii of two irles? Let r and r be the radii of the given irles and let their ars of same length S subtend angles of 60 and 7 at their entres. Angle subtended at the entre of the first irle, = 60 = = S = r = r.(i) Angle subtended at the entre of the seond irle, = 7 = 7 80 = S = r = r.(ii) 8 From (i) and (ii), we get r = r r r r r r : r = :. Std. XI Si.: Perfet Maths - I 6. The area of the irle is sq.m. Find the length of its ar subtending an angle of at the entre. Also find the area of the orresponding setor. Area of irle = r But area is given to be sq.m = r r = r = m = = 80 = Sine, S = r = = m. A(setor) = r S = = 0 sq.m. 7. AB is a setor of the irle with entre and radius m. If mab = 60, find the differene between the areas of setor AB and AB. Here, A = B = r = m Given mab = 60 mab = mba A B AB is an equilateral triangle = 60 = = Now, A (setor AB) A(AB) = r.(side) = () A 60 B.()

13 Target Publiations Pvt. Ltd. =.() = 6 = ( ) sq.m. 8. PQ is a setor of a irle with entre and radius m. If mpq = 0, find the area enlosed by ar PQ and hord PQ. Here, r = m mpq = 0 = m M = 6 P Q Draw QM P sin 0 = QM QM = = Shaded portion indiates the area enlosed by ar PQ and hord PQ. A(shaded portion) = A(setor PQ) A(PQ) = r P QM = 6 = 7 = 7 ( ) sq.m.. The perimeter of a setor of a irle, of area sq.m, is 0 m. Find the area of setor. Area of irle = r But area is given to be sq.m = r r = m Perimeter of setor = r + S But perimeter is given to be 0 m 0 = 0 + S S = 0 m Area of setor = r S = 0 = sq.m. Chapter 0: Angle and it s Measurement 0. The perimeter of a setor of a irle, of area 6 sq.m, is 6 m. Find the area of setor. Area of irle = r But area is given to be 6 sq.m 6 = r r = 8 m Perimeter of setor = r + S But perimeter is given to be 6 m 6 = 6 + S S = 0 m Area of setor = r S = 8 0 = 60 sq.m. Misellaneous Exerise -. Express the following angles into radians: i i = = ( ) = 0.6 = = = ( ) = (0.67) = Express the following angles in degrees, minutes and seonds. i. (.0) (.66) i. (.0) = + (0.0) = + (0.0 60) = + (0.78) = + ( ) = + (7.88) = 8 (approx) (.66) = + (0.66) = + ( ) = + (0.6) = (0.6) = (0.6 60) = (.76) = 0 (approx)

14 Target Publiations Pvt. Ltd.. In LMN, ml = and mn = 0. Find the measure of M both in degrees and radians. In LMN, 80 ml = = and mn = 0 But ml + mm + mn = 80 + mm + 0 = 80 mm = 80 6 = = 80 mm = Measure of M = =. Find the radian measure of the interior angle of a regular i. Pentagon Hexagon i tagon. i. Pentagon: Number of sides = Number of exterior angles = Sum of exterior angles = Eah exterior angle = = 60 no.of sides 60 Eah interior angle = 80 = (80 7) = 08 = = Hexagon: Number of sides = 6 Number of exterior angles = 6 Sum of exterior angles = Eah exterior angle = = 60 no.of sides 6 60 Eah interior angle = 80 6 = (80 60) i tagon: Std. XI Si.: Perfet Maths - I = 0 = 0 80 = Number of sides = 8 Number of exterior angles = 8 Sum of exterior angles = 60 Eah exterior angle = 60 no.of sides = Eah interior angle = 80 8 = (80 ) = = 80 =. Find the number of sides of a regular polygon, if eah of its interior angles is. Eah interior angle of a regular polygon = 80 = = Exterior angle = 80 =. Let the number of sides of the regular polygon be n. But in a regular polygon, 60 exterior angle = no.of sides = 60 n n = 60 = 8 Number of sides of a regular polygon = 8.

15 Target Publiations Pvt. Ltd. 6. Two irles eah of radius 7 m, interset eah other. The distane between their entres is 7 m. Find the area ommon to both the irles. Let and be the entres of two irles interseting eah other at A and B. Then, A = B = A = B = 7 m and = 7 m = 8..(i) Sine, A + A = = 8 =.[From (i)] ma = 0 A B is a square. mab = ma B = 0 = 0 = 80 B Now, A(setor AB) = r = 7 = sq.m and A(setor AB) = r = 7 7 A 7 = sq.m A( A B) = (side) = sq.m required area = area of shaded portion = A(setor AB) + A(setor AB) A( A B) = + = = ( ) sq.m Chapter 0: Angle and it s Measurement 7. PQR is an equilateral triangle with side 8 m. A irle is drawn on segment QR as diameter. Find the length of the ar of this irle interepted within the triangle. Let be the entre of the irle drawn on QR as a diameter. Let the irle intersets seg PQ and PR at points M and N respetively. P Sine, l(q) = l(m) 60 mmq = mqm = 60 mmq = 60 M Similarly, mnr = 60 QR = 8 m r = m = 60 = = l(ar MN) = S = r = = m. 8. Find the radius of the irle in whih a entral angle of 60 interepts an ar of length 7. m. use 7 Let S be the length of the ar and r be the radius of the irle. = 60 = 60 = 80 S = 7. m Sine, S = r 7. = r 7. = r 7 r = 7. 7 r =.7 m. A wire of length 0 m is bent so as to form an ar of a irle of radius m. What is the angle subtended at the entre in degrees? S = 0 m and r = m.(given) Q N R

16 Target Publiations Pvt. Ltd. Sine, S = r 0 = = = 80 = 0 0 m 0. If the ars of the same lengths in two irles subtend angles 6 and 0 at the entre. Find the ratio of their rad Let r and r be the radii of the given irles and let their ars of same length S subtend angles of 6 and 0 at their entres. Angle subtended at the entre of the first irle, = 6 = 6 = 80 6 S = r = r 6.(i) Angle subtended at the entre of the seond irle, = 0 = 0 80 = 8 S = r = r 8.(ii) From (i) and (ii), we get r 6 = r 8 r = r r : r = :. Find the area of a setor whose ar length is 0 m and the angle of the setor is 0. Let S be the length of the ar. = 0 = 0 80 = and S = 0 m Sine, S = r 0 = r r = 0 r = m Area of setor = r S Std. XI Si.: Perfet Maths - I = 0 = 0 7 = 66.8 sq.m. The area of a irle is 8 sq.m. Find the length of the ar subtending an angle of 00 at the entre and the area of orresponding setor. Area of irle = r But area is given to be 8 sq.m r = 8 r = 8 r = m = 00 = = Sine, S = r = = m. Area of setor = r S = = sq.m. The measures of angles of a quadrilateral are in the ratio : : 6 : 7. Find their measures in degrees and in radians. Let the measures of angles of the quadrilateral be k, k, 6k and 7k in degrees. k + k + 6k + 7k = 60.(sum of measures of angles of a quadrilateral is 60) 8k = 60 k = 0 the measures of angles are k = 0 = 0 k = 0 = 60 6k = 6 0 = 0 7k = 7 0 = 0 These angles in radians are 0 = 0 80 =

17 Target Publiations Pvt. Ltd. 60 = = 0 = 0 80 = 0 = 0 80 = 7. The angles of a triangle are in A.P. and the greatest angle is 8. Find all the three angles in radians. Let the measures of angles of a triangle be a d, a, a + d in degrees. (a d) + a + (a + d) = 80.(sum of measures of angles of a triangle is 80) a = 80 a = 60 But a + d = 8..[greatest angle is 8] 60 + d = 8 d = the measures of angles are a d = 60 = 6 a = 60 and a + d = 60 + = 8 These angles in radians are 6 = 6 80 = 60 = = 8 = 8 80 = 7. Show that the minute-hand of a lok gains 0 on the hour-hand in one minute. Angle made by hour-hand in one minute 60 = = 60 Angle made by minute-hand in one minute 60 = = 6 60 Gain by minute-hand on the hour-hand in one minute = 6 = + = 0 Chapter 0: Angle and it s Measurement 6. Find the angle between the hour-hand and minute-hand of a lok at i. twenty minutes past two quarter past six i ten past eleven. i. At : 0, the minute-hand is at mark and rd hour hand has rossed of angle between and Angle between two onseutive marks 60 = = 0 Angle traed by hour-hand in 0 minutes = (0) = 0 Angle between marks and = 0 = 60 Angle between two hands of the lok at twenty minutes past two = 60 0 = 0 At 6:, the minute-hand is at mark and hour th hand has rossed of the angle between 6 and Angle between two onseutive marks 60 = = 0 Angle traed by hour-hand in minutes = (0) = (7.) = 7 Angle between mark and 6 = 0 = 0 Angle between two hands of the lok at quarter past six = = 7

18 Target Publiations Pvt. Ltd. i At :0, the minute-hand is at mark and th hour hand has rossed of the angle 6 between and Angle between two onseutive marks 60 = = 0 Angle traed by hour hand in 0 minutes = (0) = 6 Angle between mark and = 0 = 0 Angle between two hands of the lok at ten past eleven = 0 = 8 7. A train is running on a irular trak of radius km at the rate of 6 km per hour. Find the angle to the nearest minute, through whih it will turn in 0 seonds. r = km = 000m (Ar overed by train in 0 seonds) = m S = 00 m Sine, S = r 00 = 000 = 0 80 = 0 = 7 = = (7.8) = = 7 + (0.8 60) = 7 + (0.8) = 7 (approx) 6 Std. XI Si.: Perfet Maths - I 8. The angles of a triangle are in A.P. and the ratio of the number of degrees in the least to the number of radians in the greatest is 60 :. Find the angles of the triangle in degrees and radians. Let the measures of angles of a triangle be a d, a, a + d in degrees. (a d) + a + (a + d) = 80.(sum of measures of angles of a triangle is 80) a = 80 a = 60 Also, greatest angle in radians = (a + d) 80 Aording to the given ondition, a d = 60 (a d) d d 60 d 60 d =. [Dividing throughout by 60] 80 d = 60 + d 0 = d d = 0 the measures of angles are a d = 60 0 = 0 a = 60 and a + d = = 0 These angles in radians are 0 = 0 80 = 6 60 = = 0 = 0 80 =. In a irle of diameter 0 m, the length of a hord is 0 m. Find the length of minor ar of the hord. Let be the entre of the irle and AB be the hord of the irle. Here, d = 0 m 0 0 r = 0 m A 0 B

19 Target Publiations Pvt. Ltd. The angle subtended at the entre by the minor ar AB is = 60 = 60 = 80 (minor ar of hord AB) = r = 0 = 0 m. 0. The angles of a quadrilateral are in A.P. and the greatest angle is double the least. Express the least angle in radians. Let measures of angles of quadrilateral be a d, a d, a + d, a + d in degrees. (a d) + (a d) + (a + d) + (a + d) = 60.(sum of measures of angles of a quadrilateral is 60) a = 60 a = 0 Also, a + d =.(a d) 0 + d =.(0 d) 0 + d = 80 6d d = 0 d = 0 Measure of least angle = a d = 0 (0) = 0 0 = 60 = = Additional Problems for Pratie Based on Exerise.. Determine whih of the following pairs of angles are oterminal: i. 0, 00 0,. Express the following angles in degrees: i. 8 6 i 6 iv. v. (.) Chapter 0: Angle and it s Measurement. Express the following angles in radians: i. 0 0 i iv.. Express the following angles in degrees, minutes and seonds form: i. (.) (0.) i 6. If = and = 0, find and., find 6. In PQR, mp = 0 and mq = the radian measure and the degree measure of R. 7. The differene between two aute angles of a right angled triangle is. Find the angles in degrees. 8. The sum of two angles is and their differene is 0. Find the angles in degrees.. i. The measures of angles of a triangle are in the ratio : 6 : 7. Find their measures in degrees. The measures of angles of a quadrilateral are in the ratio : : : 8. Find their measures in radians. 0. ne angle of a quadrilateral has measure and the measures of other three angles are in the ratio : : 6. Find their measures in degrees and in radians. Based on Exerise.. Find the length of the ar of irle of diameter 6 m, if the ar is subtending an angle of 0 at the entre.. Find the length of an ar of a irle whih subtends an angle of at the entre, if the radius of the irle is m.. The radius of a irle is 7 m. Find the length of an ar of this irle whih uts off a hord of length equal to radius. 7

20 Target Publiations Pvt. Ltd.. A pendulum 8 m long osillates through an angle of. Find the length of the path desribed by its extremity.. Two ars of the same length subtend angles of 60 and 80 at the entre of the irles. What is the ratio of radii of two irles? 6. Find the area of the setor of irle whih subtends an angle of 60 at the entre, if the radius of the irle is m. 7. The area of the irle is 6 sq. m. Find the length of its ar subtending an angle of 0 at the entre. Also, find the area of the orresponding setor. 8. If the perimeter of a setor of a irle is four times the radius of the irle, find the entral angle of orresponding setor in radians.. PQ is a setor of a irle with entre and radius m. If m PQ = 60, find the area enlosed by ar PQ and hord PQ. 0. The perimeter of a setor of a irle, of area sq. m, is m. Find the area of setor. Based on Misellaneous Exerise. Express the following angles into radians: i Express the following angles in degrees, minutes and seonds: i. (8.6) (7.07). A wheel makes 60 revolutions in minute. Through how many radians does it turn in seond?. In XYZ, mx = and mz = 60. Find the measure of Y both in degrees and radians.. Find the radian measure of the interior angle of a regular heptagon. 6. Find the number of sides of a regular polygon, if eah of its interior angles is. 7. Two irles eah of radius m interset eah other. The distane between their entres is m. Find the area ommon to both the irles. 8 Std. XI Si.: Perfet Maths - I 8. ABC is an equilateral triangle with side 6 m. A irle is drawn on segment BC as diameter. Find the length of the ar of this irle interepted within ABC.. Find the radius of the irle in whih a entral angle of 60 interepts an ar of length 8.6 m. Use 7 0. A wire of length 6 m is bent so as to form an ar of a irle of radius 80 m. What is the angle subtended at the entre in degrees?. If the ars of the same lengths in two irles subtend angles 7 and 0 at the entre, then find the ratio of their rad. Find the area of a setor whose ar length is m and angle of the setor is 60.. The measures of angles of a quadrilateral ar in the ratio : : 8 :. Find their measures in degrees and in radians.. The angles of a triangle are in A.P. and the greatest angle is 00. Find all the three angles in radians.. Find the angle between the hour-hand and minute-hand of a lok at i. thirty minutes past eight quarter past one 6. Find the degree and radian measure of the angle between the hour-hand and the minute-hand of a lok at thirty minutes past three. 7. A horse is tied to a post by a rope. If the horse moves along a irular path always keeping the rope tight and desribes 88 metres when it traes an angle of 7 at the entre, find the length of the rope. Use 7 8. In a irle of diameter 66 m, the length of a hord is m. Find the length of minor ar of the hord.. The angles of a quadrilateral are in A.P. and the greatest angle is five times the least. Express the least angle in radians.

21 Target Publiations Pvt. Ltd. Multiple Choie Questions. The angle subtended at the entre of a irle of radius metres by an ar of length metre is equal to (A) 0 (B) 60 (C) radian (D) radians. A irular wire of radius 7 m is ut and bend again into an ar of a irle of radius m. The angle subtended by the ar at the entre is (A) 0 (B) 0 (C) 00 (D) 60. The radius of the irle whose ar of length m makes an angle of radian at the entre is (A) 0 m (B) 0 m (C) m (D) m. Convert into degrees (A) (B) 60 (C) 0 (D). 8 Convert into degrees (A) (B) 80 (C) 80 (D) Convert 6 into radians (A) 6 (B) (C) (D) 7. Convert 0 into radians (A) π (B) (C) π (D) π 6 8. The angles of a triangle are in A. P. suh that greatest is times the least. The angles in degrees are (A) 0, 60, 00 (B) 0,, 0 (C) 0,, 80 (D) 0, 60, 00 π Chapter 0: Angle and it s Measurement. The angles of a quadrilateral are in the ratio : : :. Then the least angle in degrees is (A) 0 (B) (C) 0 (D) The angles of a triangle are in the ratio : 7 : 8. Then the greatest angle in radians is (A) (B) (C) 7 (D) 8 6. The differene between two aute angles of a right angled triangle is. Then the angles in degrees are (A) 0, (B), (C), (D) 60, 7. Angle between the hour hand and minute hand of a lok at quarter past eleven in degrees is (A) (B), 0 (C) 07, 7 (D). The interior angles of a regular polygon of sides in radians is (A) (B) 0 (C) 6 (D). The ar length of a irle is (A) s = rθ (B) S = (C) r θ (D) πθ. The length of ar of a irle of radius m; subtending an angle of 0 at the entre is (A) π m (B) π m (C) m (D) m 6. A and B are two radii of a irle of radius 0 suh that mab =. Then area of the setor AB is (A) 8π sq.m. (B) 0π sq.m. (C) 0π sq.m. (D) 0π sq.m.

22 Target Publiations Pvt. Ltd. 7. The perimeter of a setor of a irle of area 6 sq. m is m. Then the area of setor is (A) 0 sq.m. (B) 6 sq.m. (C) 6 sq.m. (D) 6 sq.m. 8. A semiirle is divided into two setors, whose angles are in the ratio:. Then the ratio of their area is (A) : (B) : (C) : (D) :. If θ is the angle between two radii of a irle of radius r, then the area of orresponding setor is (A) r θ (B) r θ (C) rθ (D) π r 0. A wire m. long is bent so as to lie along the ar of a irle of 80 m radius. The angle subtended at the entre of the ar in degrees is (A), 7 (B) 6, 0 (C) 7, 0 (D) 8, 0 Answers to Additional Pratie Problems Based on Exerise.. i. oterminal not oterminal. i. (.) 0 i 080 iv. v. 6. i. i 6 iv i. 8 0 i 0. = 0, = Std. XI Si.: Perfet Maths - I. i., 7, 8,,, , 00, 0;,,. Based on Exerise.. m. m. 7 6 m. m. : m, sq.m 8.. sq.m 0. 0 sq. m. sq. m Based on Misellaneous Exerise 7. i i (approx.) 7 6 (approx.).. my = = m.. 7. m ( ) sq. m. 6. mr = 60 = 7., 8. 0, :. 7. sq. m.

23 Target Publiations Pvt. Ltd. Chapter 0: Angle and it s Measurement. 0, 7, 0, ; 6,,,.,,. i , m 8. m. 6 Answers to Multiple Choie Questions. (C). (B). (B). (A). (C) 6. (B) 7. (D) 8. (D). (D) 0. (A). (C). (B). (A). (A). (A) 6. (D) 7. (B) 8. (D). (B) 0. (D)