|
|
- Shavonne Lane
- 6 years ago
- Views:
Transcription
1 TRIGONOMETRY UNIT-6 "Te matematician is fascinated wit te marvelous beauty of te forms e constructs, and in teir beauty e finds everlasting trut.". If xcosθ ysinθ a, xsinθ + ycos θ b, prove tat x +y a +b. xcosθ - y sinθ a xsinθ + y cosθ b Squaring and adding x +y a +b.. Prove tat sec θ+cosec θ can never be less tan. S.T Sec θ + Cosec θ can never be less tan. If possible let it be less tan. + Tan θ + + Cot θ <. + Tan θ + Cot θ (Tanθ + Cotθ) <. Wic is not possible. 3. If sinϕ, sow tat 3cosϕ-4cos 3 ϕ 0. Sin ϕ ½ ϕ 30 o Substituting in place of ϕ 30 o. We get If 7sin ϕ+3cos ϕ 4, sow tat tanϕ. If 7 Sin ϕ + 3 Cos ϕ 4 S.T. Tanϕ 3 7 Sin ϕ + 3 Cos ϕ 4 (Sin ϕ + Cos ϕ) 3 Sin ϕ Cos ϕ Sin ϕ Cos ϕ 3 43
2 Tan ϕ 3 Tanϕ 3 5. If cosϕ+sinϕ cosϕ, prove tat cosϕ - sinϕ sin ϕ. Cosϕ + Sinϕ Cosϕ ( Cosϕ + Sinϕ) Cos ϕ Cos ϕ + Sin ϕ+cosϕ Sinϕ Cos ϕ Cos ϕ - Cosϕ Sinϕ+ Sin ϕ Sin ϕ Sin ϕ - Cos ϕ (Cosϕ - Sinϕ) Sin ϕ - Cos ϕ Sin ϕ & - Sin ϕ Cos ϕ or Cosϕ - Sinϕ Sinϕ. 6. If tana+sinam and tana-sinan, sow tat m -n 4 TanA + SinA m TanA SinA n. m -n 4 mn. m -n (TanA + SinA) -(TanA - SinA) 4 TanA SinA TanA + SinA ( TanA SinA RHS 4 mn 4 ( ) ) 4 Tan A Sin A 4 Sin A Sin ACos A Cos A 4 Sin A 4 Cos A Sin A 4 4 TanA SinA Cos A m n 4 mn 7. If seca, prove tat seca+tanax or. 44
3 Secϕ x + 4x Sec ϕ ( x + ) 4x Tan ϕ ( x + ) - 4x Tan ϕ ( x - ) 4x (Sec ϕ + Tan ϕ) Tanϕ + x - 4x Secϕ + Tanϕ x + x or x + x - 4x 4x 8. If A, B are acute angles and sina cosb, ten find te value of A+B. A + B 90 o 9. a)solve for ϕ, if tan5ϕ. Tan 5ϕ 45 ϕ ϕ9 o. 5 Sinϕ + Cosϕ b)solve for ϕ if Cosϕ Sinϕ Sinϕ + Cosϕ Cosϕ Sinϕ Sin ϕ + (Cosφ) Sinϕ( + Cosϕ) 4 Sin ϕ + + Cos ϕ + Cosϕ 4 Sinϕ + SinϕCosϕ + Cosϕ Sinϕ( + Cosϕ) 4 45
4 + ( + Cosϕ) 4 Sinϕ( + Cosϕ) 4 Sinϕ Sinϕ Sinϕ Sin30 ϕ 30 o 0. If Cosα m Cosβ m Cos α Cos β LHS (m +n ) Cos β Cosα n Sinβ n Cos α Cos α + Cos β Cos β Sin β Cos α Cos βsin Cos α n Sin β (m +n ) Cos α Sin Cos β Cos β n β β. If 7 cosecϕ-3cotϕ 7, prove tat 7cotϕ - 3cosecϕ 3. 7 Cosecϕ-Cotϕ7 P.T 7Cotϕ - 3 Cosecϕ3 7 Cosecϕ-3Cotϕ7 7Cosecϕ-73Cotϕ 7(Cosecϕ-)3Cotϕ 46
5 7(Cosecϕ-) (Cosecϕ+)3Cotϕ(Cosecϕ+) 7(Cosec ϕ-)3cotϕ(cosecϕ+) 7Cot ϕ3 Cotϕ (Cosecϕ+) 7Cotϕ 3(Cosecϕ+) 7Cotϕ-3 Cosecϕ3. (sin 6 ϕ+cos 6 ϕ) 3(sin 4 ϕ+cos 4 ϕ)+ 0 (Sin ϕ) 3 + (Cos ϕ) 3-3 (Sin 4 ϕ+(cos 4 ϕ)+0 Consider (Sin ϕ) 3 +(Cos ϕ) 3 (Sin ϕ+cos ϕ) 3-3 Sin ϕcos ϕ( Sin ϕ+cos ϕ) - 3Sin ϕ Cos ϕ Sin 4 ϕ+cos 4 ϕ(sin ϕ) +(Cos ϕ) (Sin ϕ+cos ϕ) - Sin ϕ Cos ϕ - Sin ϕ Cos ϕ (Sin 6 ϕ+cos 6 ϕ)-3(sin 4 ϕ+cos 4 ϕ) + (-3 Sin ϕ Cos ϕ)-3 (- Sin ϕ+cos ϕ)+ 3. 5(sin 8 A- cos 8 A) (sin A ) (- sin A cos A) Proceed as in Question No. 4. If tanθ 6 5 & θ +φ 90 o wat is te value of cotφ. Tanθ 5 i.e. Cotφ 6 5. Wat is te value of tanϕ in terms of sinϕ. Sinϕ Tan ϕ Cosϕ Tan ϕ Sinϕ Sin ϕ 6. If Secϕ+Tanϕ4 find sin ϕ, cosϕ Sec ϕ + Tan ϕ 4 5 Since ϕ + θ 90 6 o. + Cosϕ Sinϕ 4 Cosϕ + Sinϕ 4 Cosϕ 47
6 ( + Sinϕ) 6 Cos ϕ apply (C & D) ( + Sinφ) ( + Sinφ) + Cos φ 6 + Cos φ 6 ( + Sinφ) Sinφ( + Sinφ) 7 Sinφ 5 5 Sinϕ 7 Cosϕ Sin ϕ p 7. Secϕ+Tanϕp, prove tat sinϕ p P Secϕ + Tanϕ P. P.T Sinϕ P Proceed as in Question No Prove geometrically te value of Sin 60 o Exercise for practice. 9. tan θ 3 sin θ If,sow tat + tan θ 3 + cos θ Exercise for practice. 0. If xsecθ and tanθ,ten find te value of x x. x () Exercise for practice. 48
7 HEIGHTS AND DISTANCES. If te angle of elevation of cloud from a point meters above a lake is and te angle of depression of its reflection in te lake is cloud is., prove tat te eigt of te Ans : If te angle of elevation of cloud from a point n meters above a lake is and te angle of depression of its reflection in te lake is β, prove tat te eigt of te tan β + tanα cloud is tan β tanα Let AB be te surface of te lake and Let p be an point of observation suc tat AP meters. Let c be te position of te cloud and c be its reflection in te lake. Ten CPM and MPC β. Let CM x. Ten, CB CM + MB CM + PA x + CM In CPM, we ave tan PM x tan AB [ PM AB] AB x cot In PMC, we ave C' M tanβ PM.. x + tanβ [Θ C MC B+BM x + + n] AB AB (x + ) cot β From & x cot (x + ) cot β 49
8 x (cot - cot β) cot β (on equating te values of AB) tan β tanα x x tanα tan β tan β tanα + tan β tan β tanα x tan β tanα Hence, te eigt CB of te cloud is given by CB is given by CB x + tanα CB + tan β tanα tanα + tan β tanα (tanα + tan β ) CB- tan β tanα tan β tanα. From an aero plane vertically above a straigt orizontal road, te angles of depression of two consecutive milestones on opposite sides of te aero plane are observed to be α and β. Sow tat te eigt of te aero plane above te road is Let P Q be QB be x Given : AB mile QB x AQ (- x) mile in PAQ PQ Tan α AQ Tan α x. x Tanα. In PQB Tan β x x Tanβ Substitute for x in equation () + Tanβ Tanα 50
9 + Tanβ Tanα Tan β + Tanα TanβTanα 3. Two stations due sout of a tower, wic leans towards nort are at distances a and b from its foot. If α and β be te elevations of te top of te tower from te situation, prove tat its inclination θ to te orizontal given by Let AB be te leaning tower and C and D be te given stations. Draw BL DA produced. Ten, BAL 0, BCA α, BDC a and DA b. Let AL x and BL In rigt ALB, we ave : AL x Cot θ Cot θ BL x Cot θ x cot θ In rigt BCL, we ave : CL Cot α a + x cot α BL a (cot α - cot θ) a (cotα cotθ )..(i)...(ii) In rigt BDL, we ave : DL DA + cot β cot β BL BLAL b + x cot β b + x b cot β b ((cot β - cot θ) b (cot β cotθ ) [using (i)]..(iii) equating te value of in (ii) and (iii), we get: a b (cotα cotθ ) (cot β cotθ ) 5
10 a cot β - a cot θ b cot α - b cot θ (b a) cot θ b cot α - a cotβ b cotα a cot βθ cot θ ( b a) 4. Te angle of elevation of te top of a tower from a point on te same level as te foot of te tower is α. On advancing p meters towards te foot of te tower, te angle of elevation becomes β. sow tat te eigt of te tower is given by 5. A boy standing on a orizontal plane finds a bird flying at a distance of 00m from im at an elevation of A girl standing on te roof of 0 meter ig building finds te angle of elevation of te same bird to be Bot te boy and te girl are on opposite sides of te bird. Find te distance of te bird from te girl. ( 4.4m) In rigt ACB AC Sin 30 AB AC 00 AC 00 AC 50m AF (50 0) 30m In rigt AFE AF Sin 45 AE 30 AE AE x m 6. From a window x meters ig above te ground in a street, te angles of elevation and depression of te top and te foot of te oter ouse on te opposite side of te street are α and β respectively. Sow tat te eigt of te opposite ouse is Meters. Let AB be te ouse and P be te window Let BQ x PC x Let AC a 5
11 PQ In PQB, tan θ or tan θ QB x x cot θ tanθ AC a In PAC, tan θ or tan θ PC x a x tan θ > ( cot θ) tan θ tan θ cot θ. te eigt of te tower AB AC + BC a + tan θ cot θ + (tanθ cot θ + ) 7. Two sips are sailing in te sea on eiter side of a ligtouse; te angles of depression of two sips as observed from te top of te ligtouse are 60 0 and 45 0 respectively. If te distance between te sips is eigt of te ligtouse. (00m) In rigt ABC Tan 60 BC 3 BC H 3 BC In rigt ABD Tan 45 BD BD BC + BD 00 BC + 3 BC 00 BC 00( ( + 3) BC m eigt of ligt ouse 00m 3 meters, find te 53
12 8. A round balloon of radius a subtends an angle θ at te eye of te observer wile te angle of elevation of its centre is Φ. Prove tat te eigt of te center of te balloon is a sin θ cosec Φ /. Let θ be te centre of te ballon of radius r and p te eye of te observer. Let PA, PB be tangents from P to ballong. Ten APB θ. APO BPO θ Let OL be perpendicular from O on te orizontal PX. We are given tat te angle of te elevation of te centre of te ballon is φ i.e., OPL φ θ OA In OAP, we ave sin OP θ a sin OP θ OP a cosec OL In OP L, we ave sinφ OP OL OP sin φ a cosec φ sin θ. Hence, te eigt of te center of te balloon is a sin θ cosec Φ /. 9. Te angle of elevation of a jet figter from a point A on te ground is After a fligt of 5 seconds, te angle of elevation canges to If te jet is flying at a speed of 70 km/r, find te constant eigt at wic te jet is flying.(use 3.73 ( 598m) 36 km / r 0m / sec 70 km / 0 x Speed 00 m/s Distance of jet from AE speed x time 00 x m tan 60 o AC oppositeside BC adjacentside AC 3 BC BC 3 AC 54
13 AC ED (constant eigt) BC 3 ED. tan 30 o ED oppositeside BC + CD adjacentside ED 3 BC BC ED 3 BC BC 3 (from equation ) 3 BC BC 3BC BC 3000 BC 3000 BC 3000 BC 500 m ED BC 3 (from equation ) x.73 ED 598m Te eigt of te jet figter is 598m. 0. A vertical post stands on a orizontal plane. Te angle of elevation of te top is 60 o and tat of a point x metre be te eigt of te post, ten prove tat Self Practice x. 3. A fire in a building B is reported on telepone to two fire stations P and Q, 0km apart from eac oter on a straigt road. P observes tat te fire is at an angle of 60 o to te road and Q observes tat it is an angle of 45 o to te road. Wic station sould send its team and ow muc will tis team ave to travel? (7.3km) Self Practice. A ladder sets against a wall at an angle α to te orizontal. If te foot is pulled away from te wall troug a distance of a, so tat is slides a distance b down te wall cosα cos β a making an angle β wit te orizontal. Sow tat. sin β sinα b Let CB x m. Lengt of ladder remains same 55
14 CB Cos α CA x Cos α x cos α ED AC Let Ed be ED AC () DC + CB cos β ED a + x cos β a + x cos β x cos β a () from () & () cos α cos β - a cos α - cos β - a -a (cosα - cosβ)...(3) AE + EB Sin α AC b + EB Sin α Sin α b EB EB Sin α b...(4) EB Sin β DE EB Sin β EB Sin β From (4) & (5) (5) Sin β Sin α b b sin α Sin β -b (Sin β - Sin α)...(6) Divide equation (3) wit equation (6) a (cosα cos β ) b (sin β Sinα) a b Cosα Cosβ Sinβ Sinα 56
15 3. Two stations due sout of a leaning tower wic leans towards te nort are at distances a and b from its foot. If α, β be te elevations of te top of te tower bcotα acot β from tese stations, prove tat its inclination ϕ is given by cotϕ. b a Let AE x, BE BE Tan φ AE x x x tanφ x cot φ BE tan α CE a + x a + x cot α x cot α - a BE tan β DE b + x b+x cot β x cot β - b from and cot φ cot α - a ( cot φ + cot α ) a a cot φ + cotα from and 3 cot φ cot β - b ( cot φ - cot β) b b cot φ + cot β from 4 and 5 a b cot φ + cotα cot φ + cot β a (cot β - cot φ ) b ( cot α - cot φ ) - a cot φ + b cot φ b cot α - a cot β (b a) cot φ b cot α - a cot β 57
16 b cot α - a cot β cot φ b a 4. In Figure, wat are te angles of depression from te observing positions O and O of te object at A? Self Practice ( 30 o,45 o ) 5. Te angle of elevation of te top of a tower standing on a orizontal plane from a point A is α. After walking a distance d towards te foot of te tower te angle d of elevation is found to be β. Find te eigt of te tower. ( ) cotα cot β Let BC x AB tan β CB tan β x x tan β x cot β () AB tan α DC + CB tan α d + x d + x cotα tan α x cot α - d () from and cot β cot α - d (cot α - cot β ) d d cot α cot β 58
17 6. A man on a top of a tower observes a truck at an angle of depression α were tanα and sees tat it is moving towards te base of te tower. Ten minutes 5 later, te angle of depression of te truck is found to be β were tan β 5, if te truck is moving at a uniform speed, determine ow muc more time it will take to reac te base of te tower... 0 minutes600sec A Let te speed of te truck be x m/sec CDBC-BD In rigt triangle ABC tanα BC BC 5. In rigt triangle ABD tanβ BD tanα 5 5BD ( tan β 5 ) CDBC-BD ( CD600x ) 600x 5BD-BD BD50x 50x Time taken x 50 seconds Time taken by te truck to reac te tower is 50 sec. C α D β B 59
Downloaded from
HEIGHTS AND DISTANCES 1. If te angle of elevation of cloud from a point meters aove a lake is and te angle of depression of its reflection in te lake is cloud is., prove tat te eigt of te Ans : If te angle
More informationSTUDY MATERIAL CLASS X MATHEMATICS
STUDY MATERIAL CLASS X MATHEMATICS 008-09 STUDY MATERIAL HOTS QUESTIONS & SOLUTIONS CLASS X MATHEMATICS 008-09 09 Mathematics is an independent world Created out of pure intelligence. STUDY MATERIAL PREPRATION
More informationDownloaded from APPLICATION OF TRIGONOMETRY
MULTIPLE CHOICE QUESTIONS APPLICATION OF TRIGONOMETRY Write the correct answer for each of the following : 1. Write the altitude of the sun is at 60 o, then the height of the vertical tower that will cost
More informationMATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart
MATH 121 - Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line
More informationTHE COMPOUND ANGLE IDENTITIES
TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x + + 3 sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos
More informationSome Applications of trigonometry
Some Applications of trigonometry 1. A flag of 3m fixed on the top of a building. The angle of elevation of the top of the flag observed from a point on the ground is 60º and the angle of depression of
More informationMathematics. Sample Question Paper. Class 10th. (Detailed Solutions) Mathematics Class X. 2. Given, equa tion is 4 5 x 5x
Sample Question Paper (Detailed Solutions Matematics lass 0t 4 Matematics lass X. Let p( a 6 a be divisible by ( a, if p( a 0. Ten, p( a a a( a 6 a a a 6 a 6 a 0 Hence, remainder is (6 a.. Given, equa
More informationHigher Order Thinking Skill questions
Higher Order Thinking Skill questions TOPIC- Constructions (Class- X) 1. Draw a triangle ABC with sides BC = 6.3cm, AB = 5.2cm and ÐABC = 60. Then construct a triangle whose sides are times the corresponding
More informationClass 10 Application of Trigonometry [Height and Distance] Solved Problems
Class 10 Application of Trigonometry [Height and Distance] Solved Problems Question 01: The angle of elevation of an areoplane from a point on the ground is 45 o. After a flight of 15 seconds, the elevation
More informationAptitude Height and Distance Practice QA - Difficult
Aptitude Height and Distance Practice QA - Difficult 1. A flagstaff is placed on top of a building. The flagstaff and building subtend equal angles at a point on level ground which is 200 m away from the
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationMath 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
More information( ( ) cos Chapter 21 Exercise 21.1 Q = 13 (ii) x = Q. 1. (i) x = = 37 (iii) x = = 99 (iv) x =
Capter 1 Eercise 1.1 Q. 1. (i) = 1 + 5 = 1 (ii) = 1 + 5 = 7 (iii) = 1 0 = 99 (iv) = 41 40 = 9 (v) = 61 11 = 60 (vi) = 65 6 = 16 Q.. (i) = 8 sin 1 = 4.1 5.5 (ii) = cos 68 = 14.7 1 (iii) = sin 49 = 15.9
More information(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?
Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is
More informationExcerpt from "Calculus" 2013 AoPS Inc.
Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities
More informationPhy 231 Sp 02 Homework #6 Page 1 of 4
Py 231 Sp 02 Homework #6 Page 1 of 4 6-1A. Te force sown in te force-time diagram at te rigt versus time acts on a 2 kg mass. Wat is te impulse of te force on te mass from 0 to 5 sec? (a) 9 N-s (b) 6 N-s
More informationMVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
More information6.2 TRIGONOMETRY OF RIGHT TRIANGLES
8 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac 6. TRIGONOMETRY OF RIGHT TRIANGLES Trigonometric Ratios Special Triangles; Calculators Applications of Trigonometry of Rigt Triangles In tis section
More informationCalculus I, Fall Solutions to Review Problems II
Calculus I, Fall 202 - Solutions to Review Problems II. Find te following limits. tan a. lim 0. sin 2 b. lim 0 sin 3. tan( + π/4) c. lim 0. cos 2 d. lim 0. a. From tan = sin, we ave cos tan = sin cos =
More informationCh6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2
Ch6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2 2. Find the degree measure of the angle with the given radian measure.
More informationExam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationINTRODUCTION AND MATHEMATICAL CONCEPTS
INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,
More informationD) sin A = D) tan A = D) cos B =
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the function requested. Write your answer as a fraction in lowest terms. 1) 1) Find sin A.
More information( )( ) PR PQ = QR. Mathematics Class X TOPPER SAMPLE PAPER-1 SOLUTIONS. HCF x LCM = Product of the 2 numbers 126 x LCM = 252 x 378
Mathematics Class X TOPPER SAMPLE PAPER- SOLUTIONS Ans HCF x LCM Product of the numbers 6 x LCM 5 x 378 LCM 756 ( Mark) Ans The zeroes are, 4 p( x) x + x 4 x 3x 4 ( Mark) Ans3 For intersecting lines: a
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More information= 0 and states ''hence there is a stationary point'' All aspects of the proof dx must be correct (c)
Paper 1: Pure Matematics 1 Mark Sceme 1(a) (i) (ii) d d y 3 1x 4x x M1 A1 d y dx 1.1b 1.1b 36x 48x A1ft 1.1b Substitutes x = into teir dx (3) 3 1 4 Sows d y 0 and states ''ence tere is a stationary point''
More informationINTRODUCTION AND MATHEMATICAL CONCEPTS
Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips
More informationIntroduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
More information1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks : 100 QUESTION PAPER CODE 30/1/1 SECTION - A
MATHEMATICS Time allowed : 3 hours Maximum Marks : 100 GENERAL INSTRUCTIONS : 1. All questions are compulsory 2. The question paper consists of 30 questions divided into four sections - A, B, C and D.
More informationChapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.
Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+
More informationAB AB 10 2 Therefore, the height of the pole is 10 m.
Class X - NCERT Maths EXERCISE NO: 9.1 Question 1: A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the
More information= h. Geometrically this quantity represents the slope of the secant line connecting the points
Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (
More informationKARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE G È.G È.G È.. Æ fioê, d È 2018 S. S. L. C. EXAMINATION, JUNE, 2018
CCE RR REVISED & UN-REVISED O %lo ÆË v ÃO y Æ fio» flms ÿ,» fl Ê«fiÀ M, ÊMV fl 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 G È.G È.G È.. Æ fioê, d È 08 S. S. L.
More information1 Limits and Continuity
1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function
More informationBOARD QUESTION PAPER : MARCH 2016 GEOMETRY
BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential
More informationThe distance between City C and City A is just the magnitude of the vector, namely,
Pysics 11 Homework III Solutions C. 3 - Problems 2, 15, 18, 23, 24, 30, 39, 58. Problem 2 So, we fly 200km due west from City A to City B, ten 300km 30 nort of west from City B to City C. (a) We want te
More information2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
More informationCHAPTER 3: DERIVATIVES
(Answers to Exercises for Capter 3: Derivatives) A.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE 1) a) f ( 3) f 3.1 3.1 3 f 3.01 f ( 3) 3.01 3 f 3.001 f ( 3) 3.001
More informationVIVEKANANDA VIDYALAYA MATRIC HR SEC SCHOOL, RAMESWARAM. Lesson 1 & 7 & Exercise 9.1 ( Unit Test -1 ) 10th Standard
VIVEKANANDA VIDYALAYA MATRIC HR SEC SCHOOL, RAMESWARAM Lesson 1 & 7 & Exercise 9.1 ( Unit Test -1 ) 10 Standard Date : 6-Oct-18 MATHEMATICS Reg.No. : Time : 01:30:00 Hrs Total Marks : 50 I. CHOOSE THE
More informationDownloaded from
Exercise 9.1 Question 1: A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made
More information2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
More informationChapter 9 Some Applications of Trigonometry Exercise 9.1 Question 1: A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find
More information1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6
A Answers Important Note about Precision of Answers: In many of te problems in tis book you are required to read information from a grap and to calculate wit tat information. You sould take reasonable
More informationChapter 2. Limits and Continuity 16( ) 16( 9) = = 001. Section 2.1 Rates of Change and Limits (pp ) Quick Review 2.1
Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 969) Quick Review..... f ( ) ( ) ( ) 0 ( ) f ( ) f ( ) sin π sin π 0 f ( ). < < < 6. < c c < < c 7. < < < < < 8. 9. 0. c < d d < c
More informationTime: 3 Hrs. M.M. 90
Class: X Subject: Mathematics Topic: SA1 No. of Questions: 34 Time: 3 Hrs. M.M. 90 General Instructions: 1. All questions are compulsory. 2. The questions paper consists of 34 questions divided into four
More information1 Calculus. 1.1 Gradients and the Derivative. Q f(x+h) f(x)
Calculus. Gradients and te Derivative Q f(x+) δy P T δx R f(x) 0 x x+ Let P (x, f(x)) and Q(x+, f(x+)) denote two points on te curve of te function y = f(x) and let R denote te point of intersection of
More informationTRIGONOMETRY - Angle Of Elevation And Angle Of Depression Based Questions.
TRIGONOMETRY - Angle Of Elevation And Angle Of Depression Based Questions. 1. A man 1.7 m tall standing 10 m away from a tree sees the top of the tree at an angle of elevation 50 0. What is the height
More informationSIMG Solution Set #5
SIMG-303-0033 Solution Set #5. Describe completely te state of polarization of eac of te following waves: (a) E [z,t] =ˆxE 0 cos [k 0 z ω 0 t] ŷe 0 cos [k 0 z ω 0 t] Bot components are traveling down te
More informationCCE PR Revised & Un-Revised
D CCE PR Revised & Un-Revised 560 00 KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE 560 00 08 S.S.L.C. EXAMINATION, JUNE, 08 :. 06. 08 ] MODEL ANSWERS : 8-K Date :. 06. 08 ] CODE
More informationSection 3: The Derivative Definition of the Derivative
Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope
More informationCBSE Board Class X Mathematics
CBSE Board Class X Mathematics Time: 3 hrs Total Marks: 80 General Instructions: 1. All questions are compulsory.. The question paper consists of 30 questions divided into four sections A, B, C, and D.
More informationUdaan School Of Mathematics Class X Chapter 10 Circles Maths
Exercise 10.1 1. Fill in the blanks (i) The common point of tangent and the circle is called point of contact. (ii) A circle may have two parallel tangents. (iii) A tangent to a circle intersects it in
More informationContinuity and Differentiability of the Trigonometric Functions
[Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te
More information; approximate b to the nearest tenth and B or β to the nearest minute. Hint: Draw a triangle. B = = B. b cos 49.7 = 215.
M 1500 am Summer 009 1) Given with 90, c 15.1, and α 9 ; approimate b to the nearest tenth and or β to the nearest minute. Hint: raw a triangle. b 18., 0 18 90 9 0 18 b 19.9, 0 58 b b 1.0, 0 18 cos 9.7
More information1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine?
1 1 Power is transferred troug a macine as sown. power input P I macine power output P O power loss P L Wat is te efficiency of te macine? P I P L P P P O + P L I O P L P O P I 2 ir in a bicycle pump is
More information= ( +1) BP AC = AP + (1+ )BP Proved UNIT-9 CIRCLES 1. Prove that the parallelogram circumscribing a circle is rhombus. Ans Given : ABCD is a parallelogram circumscribing a circle. To prove : - ABCD is
More information1 / 23
CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question
More informationQuestion 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain =
Question 1 ( 1.0 marks) The decimal expansion of the rational number places of decimals? will terminate after how many The given expression i.e., can be rewritten as Now, on dividing 0.043 by 2, we obtain
More informationHonors Calculus Midterm Review Packet
Name Date Period Honors Calculus Midterm Review Packet TOPICS THAT WILL APPEAR ON THE EXAM Capter Capter Capter (Sections. to.6) STRUCTURE OF THE EXAM Part No Calculators Miture o multiple-coice, matcing,
More informationMAT 1800 FINAL EXAM HOMEWORK
MAT 800 FINAL EXAM HOMEWORK Read te directions to eac problem careully ALL WORK MUST BE SHOWN DO NOT USE A CALCULATOR Problems come rom old inal eams (SS4, W4, F, SS, W) Solving Equations: Let 5 Find all
More informationSFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00
SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only
More informationTangent Lines-1. Tangent Lines
Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property
More informationCBSE CLASS-10 MARCH 2018
CBSE CLASS-10 MARCH 2018 MATHEMATICS Time : 2.30 hrs QUESTION & ANSWER Marks : 80 General Instructions : i. All questions are compulsory ii. This question paper consists of 30 questions divided into four
More informationReview for Exam IV MATH 1113 sections 51 & 52 Fall 2018
Review for Exam IV MATH 111 sections 51 & 52 Fall 2018 Sections Covered: 6., 6., 6.5, 6.6, 7., 7.1, 7.2, 7., 7.5 Calculator Policy: Calculator use may be allowed on part of te exam. Wen instructions call
More informationMT - GEOMETRY - SEMI PRELIM - I : PAPER - 4
07 00 MT A.. Attempt ANY FIVE of the following : (i) Slope of the line (m) 4 y intercept of the line (c) 0 By slope intercept form, The equation of the line is y m + c y (4) + (0) y 4 MT - GEOMETRY - SEMI
More informationPractice Assessment Task SET 3
PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,
More informationSection 2: The Derivative Definition of the Derivative
Capter 2 Te Derivative Applied Calculus 80 Section 2: Te Derivative Definition of te Derivative Suppose we drop a tomato from te top of a 00 foot building and time its fall. Time (sec) Heigt (ft) 0.0 00
More information2013 HSC Mathematics Extension 2 Marking Guidelines
3 HSC Mathematics Extension Marking Guidelines Section I Multiple-choice Answer Key Question Answer B A 3 D 4 A 5 B 6 D 7 C 8 C 9 B A 3 HSC Mathematics Extension Marking Guidelines Section II Question
More informationSAMPLE QUESTION PAPER 11 Class-X ( ) Mathematics
SAMPLE QUESTION PAPER 11 Class-X (2017 18) Mathematics GENERAL INSTRUCTIONS (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B,C and D. (iii)
More information1. For Cosine Rule of any triangle ABC, b² is equal to A. a² - c² 4bc cos A B. a² + c² - 2ac cos B C. a² - c² + 2ab cos A D. a³ + c³ - 3ab cos A
1. For Cosine Rule of any triangle ABC, b² is equal to A. a² - c² 4bc cos A B. a² + c² - 2ac cos B C. a² - c² + 2ab cos A D. a³ + c³ - 3ab cos A 2. For Cosine Rule of any triangle ABC, c² is equal to A.
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More information1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
More information. If lim. x 2 x 1. f(x+h) f(x)
Review of Differential Calculus Wen te value of one variable y is uniquely determined by te value of anoter variable x, ten te relationsip between x and y is described by a function f tat assigns a value
More informationCBSE QUESTION PAPER CLASS-X MATHS
CBSE QUESTION PAPER CLASS-X MATHS SECTION - A Question 1: In figure, AB = 5 3 cm, DC = 4cm, BD = 3cm, then tan θ is (a) (b) (c) (d) 1 3 2 3 4 3 5 3 Question 2: In figure, what values of x will make DE
More informationClass X Mathematics Sample Question Paper Time allowed: 3 Hours Max. Marks: 80. Section-A
Class X Mathematics Sample Question Paper 08-9 Time allowed: Hours Max. Marks: 80 General Instructions:. All the questions are compulsory.. The questions paper consists of 0 questions divided into sections
More informationDerivatives of Exponentials
mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function
More informationTrigonometry Final Exam Review
Name Period Trigonometry Final Exam Review 2014-2015 CHAPTER 2 RIGHT TRIANGLES 8 1. Given sin θ = and θ terminates in quadrant III, find the following: 17 a) cos θ b) tan θ c) sec θ d) csc θ 2. Use a calculator
More informationSection 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is
Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit
More informationChapter 1: Analytic Trigonometry
Chapter 1: Analytic Trigonometry Chapter 1 Overview Trigonometry is, literally, the study of triangle measures. Geometry investigated the special significance of the relationships between the angles and
More information3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
More informationRecall from Geometry the following facts about trigonometry: SOHCAHTOA: adjacent hypotenuse. cosa =
Chapter 1 Overview Trigonometry is, literally, the study of triangle measures. Geometry investigated the special significance of the relationships between the angles and sides of a triangle, especially
More informationWYSE Academic Challenge 2004 Sectional Mathematics Solution Set
WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means
More informationProblem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems
Proble Set 7: Potential Energy and Conservation of Energy AP Pysics C Suppleentary Probles 1. Approxiately 5.5 x 10 6 kg of water drops 50 over Niagara Falls every second. (a) Calculate te aount of potential
More informationLesson 4 - Limits & Instantaneous Rates of Change
Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous
More informationPre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
More informationCHAPTER 10 TRIGONOMETRY
CHAPTER 10 TRIGONOMETRY EXERCISE 39, Page 87 1. Find the length of side x in the diagram below. By Pythagoras, from which, 2 25 x 7 2 x 25 7 and x = 25 7 = 24 m 2. Find the length of side x in the diagram
More information11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
More informationTOPPER SAMPLE PAPER 3 Summative Assessment-II MATHEMATICS CLASS X
TOPPER SAMPLE PAPER 3 Summative Assessment-II MATHEMATICS CLASS X M.M: 80 TIME : 3-3 2 Hrs. GENERAL INSTRUCTIONS :. All questions are compulsory. 2. The question paper consists of 34 questions divided
More informationC3 Exam Workshop 2. Workbook. 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2
C3 Exam Workshop 2 Workbook 1. (a) Express 7 cos x 24 sin x in the form R cos (x + α) where R > 0 and 0 < α < 2 π. Give the value of α to 3 decimal places. (b) Hence write down the minimum value of 7 cos
More informationWYSE ACADEMIC CHALLENGE State Math Exam 2009 Solution Set. 2. Ans E: Function f(x) is an infinite geometric series with the ratio r = :
WYSE ACADEMIC CHALLENGE State Math Eam 009 Solution Set 40. Ans A: ( C( 40,8 ) * C( 3,8 ) * C( 4,8 ) * C( 6,8 ) * C( 8,8 )) / 5 = 0.00084. Ans E: Function f() is an infinite geometric series with the ratio
More information1 / 22
CBSE-XII-017 EXAMINATION MATHEMATICS Paper & Solution Time: 3 Hrs. Max. Marks: 90 General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into
More informationMathematics 123.3: Solutions to Lab Assignment #5
Matematics 3.3: Solutions to Lab Assignment #5 Find te derivative of te given function using te definition of derivative. State te domain of te function and te domain of its derivative..: f(x) 6 x Solution:
More informationMath 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
More informationAPPLICATIONS OF DERIVATIVES
ALICATIONS OF DERIVATIVES 6 INTRODUCTION Derivatives have a wide range of applications in engineering, sciences, social sciences, economics and in many other disciplines In this chapter, we shall learn
More information1watt=1W=1kg m 2 /s 3
Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory
More informationSlopes of Secant and!angent (ines - 2omework
Slopes o Secant and!angent (ines - omework. For te unction ( x) x +, ind te ollowing. Conirm c) on your calculator. between x and x" at x. at x. ( )! ( ) 4! + +!. For te unction ( x) x!, ind te ollowing.
More informationSolutions to RSPL/1. Mathematics 10
Solutions to RSPL/. It is given that 3 is a zero of f(x) x 3x + p. \ (x 3) is a factor of f(x). So, (3) 3(3) + p 0 8 9 + p 0 p 9 Thus, the polynomial is x 3x 9. Now, x 3x 9 x 6x + 3x 9 x(x 3) + 3(x 3)
More information10 th MATHS SPECIAL TEST I. Geometry, Graph and One Mark (Unit: 2,3,5,6,7) , then the 13th term of the A.P is A) = 3 2 C) 0 D) 1
Time: Hour ] 0 th MATHS SPECIAL TEST I Geometry, Graph and One Mark (Unit:,3,5,6,7) [ Marks: 50 I. Answer all the questions: ( 30 x = 30). If a, b, c, l, m are in A.P. then the value of a b + 6c l + m
More information