# 15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the circumrdius of the tringle ABC, then ( r R ) equls b c b b c c [ AIEEE 00 ] ( ) If cos - - cos - y = α, then - y cos α y is equl to sin α sin α - sin α [ AIEEE 00 ] ( ) If in tringle ABC, the ltitudes from the vertices A, B, C on opposite sides re in H.P., then sin A, sin B, sin C re in G.P. A.P. Arithmetic-Geometric Progression H.P. [ AIEEE 00 ] ( ) Let α, β be such tht < α - β <. If sin α sin β = -, then the vlue of cos α - β is [ AIEEE 00 ] ( ) If u = sin cos θ b sin θ sin θ b sin θ, then difference between the mimum nd minimum vlues of u is given by ( b ) b ( b ) ( - b ) [ AIEEE 00 ] ( 7 ) The sides of tringle re sin α, cos α nd sin α cos α for some 0 < α < Then the gretest ngle of the tringle is [ AIEEE 00 ]

2 - TRIGONOMETRY Pge ( 8 ) A person stnding on the bnk of river observes tht the ngle of elevtion of the top of tree on the opposite bnk of river is 0 nd when he retires 0 m wy from the tree, the ngle of elevtion becomes 0. The bredth of the river is 0 m 0 m 0 m 0 m [ AIEEE 00 ] ( 9 ) If in tringle cos C c cos A = b, then the sides, b nd c re in A. P. in G. P. in H. P. stisfy b = c [ AIEEE 00 ] ( 0 ) The sum of the rdii of inscribed nd circumscribed circles, for n n sided regulr polygon of side, is cot n b cot n cot n cot n [ AIEEE 00 ] ( ) The upper th portion of verticl pole subtends n ngle tn - t point in the horizontl plne through its foot nd t distnce 0 m from the foot. The height of the verticl pole is 0 m 0 m 0 m 80 m [ AIEEE 00 ] ( ) The vlue of cos α cos ( α 0 ) cos ( α - 0 ) is 0 [ AIEEE 00 ] ( ) The trigonometric eqution sin - = sin - hs solution for l l < l l < l l < ll rel vlues of [ AIEEE 00 ] θ - φ ( ) If sin θ sin φ = nd cos θ cos φ = b, then the vlue of tn is - b - - b b b - b b b [ AIEEE 00 ] b

3 - TRIGONOMETRY Pge ( ) If tn - ( ) cot - ( ) =, then the vlue of is - [ AIEEE 00 ] ( ) The vlue of tn - tn - 7 tn - tn - n n is 0 [ AIEEE 00 ] ( 7 ) The ngles of elevtion of the top of tower ( A ) from the top ( B ) nd bottom ( D ) t building of height re 0 nd respectively. If the tower nd the building stnd t the sme level, then the height of the tower is - ( ) ( - ) [ AIEEE 00 ] ( 8 ) If cos ( α - β ) = nd cos ( α β ) = e, - α, β, then the number of ordered pirs ( α, β ) = 0 [ IIT 00 ] ( 9 ) Which of the following is correct for tringle ABC hving sides, b, c opposite to the ngles A, B, C respectively B - C sin B C ( b c ) sin = ( b - c ) cos A B C sin A B - C = cos sin = ( b c ) cos A = cos A [ IIT 00 ] ( 0 ) Three circles of unit rdii re inscribed in n equilterl tringle touching the sides of the tringle s shown in the figure. Then, the re of the tringle is [ IIT 00 ]

4 - TRIGONOMETRY Pge ( ) If θ nd φ re cute ngles such tht sin θ = nd cos θ =, then θ nd φ lies in,,,, [ IIT 00 ] ( ) For which vlue of, sin [ cot - ( ) ] = cos ( tn - )? 0 - [ IIT 00 ] ( ) If, b, c re the sides of tringle such tht : b : c = : :, then A : B : C is : : : : : : : : [ IIT 00 ] ( ) Vlue of equl to tn α, > 0, α 0, is lwys greter thn or tn α sec α [ IIT 00 ] ( ) If the ngles of tringle re in the rtio : :, then the rtio of the lrgest side to the perimeter is equl to : : : : [ IIT 00 ] ( ) The nturl domin of - sin ( ) for ll R, is -, -, -, -, [ IIT 00 ] ( 7 ) The length of longest intervl in which the function sin - sin is incresing is [ IIT 00 ] ( 8 ) Which of the following pieces of dt does NOT uniquely determine n cute-ngled tringle ABC ( R being the rdius of the circumcircle )? sin A, sin B, b, c, sin B, R, sin A, R [ IIT 00 ]

5 - TRIGONOMETRY Pge ( 9 ) The number of integrl vlues of k for which the eqution 7 cos sin = k hs solution is 8 0 [ IIT 00 ] ( 0 ) Let 0 < α < be fied ngle. If P = ( cos θ, sin θ ) nd Q = [ cos ( α - θ ), sin (α - θ ) ], then Q is obtined from P by clockwise rottion round origin through n ngle α nticlockwise rottion round origin through n ngle α reflection in the line through origin with slope tn α reflection in the line through origin with slope tn α [ IIT 00 ] ( ) Let PQ nd RS be tngents t the etremities of the dimeter PR of circle of rdius r. If PS nd RQ intersect t point X on the circumference of the circle, then r equls PQ RS PQ RS PQ RS PQ RS PQ RS [ IIT 00 ] ( ) A mn from the top of 00 metres high tower sees cr moving towrds the tower t n ngle of depression of 0. After some time, the ngle of depression becomes 0. The distnce in ( metres ) trveled by the cr during this time is [ IIT 00 ] ( ) If α β = nd β γ = α, then tn α equls ( tn β tn γ ) tn β tn γ tn β tn γ tn β tn γ [ IIT 00 ] ( ) If sin -... cos = for 0 < l l <, then equls - - [ IIT 00 ]

6 - TRIGONOMETRY Pge ( ) The mimum vlue of ( cos α ) ( cos α ).. ( cos α n ), under the restrictions 0 α, α,.. α n nd ( cos α ) ( cos α ).. ( cos α n ) = is n n n [ IIT 00 ] ( ) The number of distinct rel roots of sin cos cos cos sin cos cos cos sin = 0 in the intervl - is 0 [ IIT 00 ] ( 7 ) If f ( θ ) = sin θ ( sin θ sin θ ), then f ( θ ) 0 only when θ 0 0 for ll rel θ 0 for ll rel θ 0 only when θ 0 [ IIT 000 ] ( 8 ) In tringle ABC, c sin ( A - B C ) = b - c c - b b - c - c - - b [ IIT 000 ] ( 9 ) In tringle ABC, if C =, r = inrdius nd R = circum-rdius, then ( r R ) = b b c c b c [ IIT 000 ] ( 0 ) A pole stnds verticlly inside tringulr prk Δ ABC. If the ngle of elevtion of the top of the pole from ech corner of the prk is sme, then in Δ ABC, the foot of the pole is t the centroid circumcentre incentre orthocentre [ IIT 000 ]

7 - TRIGONOMETRY Pge 7 P ( ) In tringle PQR, R =. If tn eqution b c = 0 ( 0 ), then Q nd tn re the roots of the b = c b c = c = b b = c [ IIT 999 ] ( ) The number of rel solutions of tn - ( ) sin - = is zero one two infinite [ IIT 999 ] ( ) The number of vlues of where the function f ( ) = cos cos ( ) ttins its mimum is 0 infinite [ IIT 998 ] ( ) If, for positive integer n, θ f ( θ tn ( sec θ ) ( sec θ )... ( sec n n ) = θ ), then f = f = f = f 8 = [ IIT 999 ] ( ) If in tringle PQR, sin P, sin Q, sin R re in A. P., then the ltitudes re in A. P. the ltitudes re in H. P. the medins re in G. P. the medins re in A. P. [ IIT 998 ] ( ) The number of vlues of in the intervl [ 0, ] stisfying the eqution sin - 7 sin = 0 is 0 (b ) 0 [ IIT 998 ] ( 7 ) Which of the following number( s ) is / re rtionl? sin cos sin cos sin cos 7 [ IIT 998 ]

8 - TRIGONOMETRY Pge 8 n ( 8 ) Let n be n odd integer. If sin nθ = b r r sin r = 0 b respectively re θ, for every vlue of θ, then b 0 nd, 0, n -, n 0, n - n [ IIT 998 ] ( 9 ) The prmeter, on which the vlue of the determinnt cos ( p sin ( p - d ) cos p cos ( p d ) does not depend upon is - d ) sin p sin ( p d ) p d [ IIT 997 ] ( 0 ) The grph of the function cos cos ( ) - cos ( ) is stright line pssing through the point, - sin nd prllel to the X-is stright line pssing through ( 0, - sin ) with slope stright line pssing through ( 0, 0 ) prbol with verte (, - sin ) [ IIT 997 ] ( ) If A 0 A A A A A be regulr hegon inscribed in circle of unit rdius, then the product of the lengths of the line segments A 0 A, A 0 A nd A 0 A is [ IIT 998 ] ( ) sec θ = ( y y ) is true if nd only if y 0 = y, 0 = y 0, y 0 [ IIT 99 ] ( ) The minimum vlue of the epression sin α sin β sin γ, where α, β, γ re the rel numbers stisfying α β γ = is positive zero negtive ( D ) - [ IIT 99 ]

9 - TRIGONOMETRY Pge 9 ( ) In tringle ABC, B = nd C = sin BAD :, then equls sin CAD. If D divides BC internlly in the rtio [ IIT 99 ] ( ) Number of solutions of the eqution tn sec = cos, lying in the intervl [ 0, ], is 0 [ IIT 99 ] ( ) If = cos n φ, y = sin n φ, z = cos n φ sin n φ, for 0 < φ <, n = 0 n = 0 n = 0 then yz = z y yz = y z yz = y z yz = yz [ IIT 99 ] ( 7 ) If f ( ) = cos [ ] cos [ - ], where [ ] stnds for the gretest integer function, then f = - f ( ) = f ( - ) = 0 f = [ IIT 99 ] ( 8 ) The eqution ( cos p - ) ( cos p ) sin p = 0 in the vrible hs rel roots. Then p cn tke ny vlue in the intervl ( 0, ) ( -, 0 ), - ( 0, ) [ IIT 990 ] ( 9 ) In tringle ABC, ngle A is greter thn ngle B. If the mesures of ngles A nd B stisfy the eqution sin - sin - k = 0, 0 < k <, then the mesure of ngle C is [ IIT 990 ]

10 - TRIGONOMETRY Pge 0 ( 0 ) The number of rel solutions of the eqution sin ( e ) = is 0 infinitely mny [ IIT 990 ] ( ) The generl solution of sin - sin sin = cos - cos cos is n 8 ( - ) n n 8 n 8 n cos - [ IIT 989 ] ( ) The vlue of the epression cosec 0 - sec 0 is equl to sin 0 sin 0 o o sin 0 sin 0 o o [ IIT 988 ] ( ) The vlues of θ lying between θ = 0 nd θ = sin θ sin θ sin θ cos θ cos θ cos θ sin θ sin θ sin θ = 0 re nd stisfying the eqution 7 [ IIT 988 ] ( ) In tringle, the lengths of the two lrger sides re 0 nd 9 respectively. If the ngles re in A. P., then the lengths of the third side cn be - [ IIT 987 ] ( ) The smllest positive root of the eqution tn = lies in 0,,,, [ IIT 987 ] ( ) The number of ll triplets (,, ) such tht cos sin = 0 for ll is 0 infinite ( e ) none of these [ IIT 987 ]

11 - TRIGONOMETRY Pge ( 7 ) The principl vlue of sin sin is - ( e ) none of these [ IIT 98 ] ( 8 ) The epression sin - α sin ( α ) - sin α sin ( - α ) is equl to 0 sin α cos α ( e ) none of these [ IIT 98 ] ( 9 ) There eists tringle ABC stisfying the conditions b sin A =, A < b sin A >, A < ( e ) b sin A <, A > b sin A >, A > b sin A <, A <, b >, b = [ IIT 98 ] ( 70 ) 7 cos cos cos cos is equl to cos 8 8 [ IIT 98 ] ( 7 ) From the top of light-house 0 m high with its bse t the se-level, the ngle of depression of bot is. The distnce of the bot from the foot of the lighthouse is - 0 metres 0 metres - metres - None of these [ IIT 98 ] ( 7 ) The vlue of tn cos - tn - is None of these [ IIT 98 ]

12 - TRIGONOMETRY Pge ( 7 ) If f ( ) = cos ( ln ), then f ( ) f ( y ) - f f ( y ) hs the vlue y - - none of these [ IIT 98 ] ( 7 ) The generl solution of the trigonometric eqution sin cos = is given by = n, n = 0, ±, ±, = n, n = 0, ±, ±, n ( - ) n -, n = 0, ±, ±, none of these [ IIT 98 ] ( 7 ) If A = sin θ cos θ, then for ll rel vlues of θ A A A A [ IIT 980 ] ( 7 ) The eqution cos sin = -, 0 < hs no rel solution one rel solution more thn one rel solution [ IIT 980 ] ( 77 ) If tn θ = -, then sin θ is - but not but not - - or none of these [ IIT 979 ] ( 78 ) If α β γ =, then tn γ tn β tn α = tn α tn β tn γ tn α tn β tn β tn γ tn γ tn α = γ β α α β γ tn tn tn = - tn tn tn none of these [ IIT 979 ]

13 - TRIGONOMETRY Pge Answers b b c b d c c b b c b c d b d d c c d b d b c b c c b b c,b,c,d d c c b b c b c d b,c b c b b,c,c d e b,d c c d d c b b

### SOLUTION OF TRIANGLES

SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :

### CET MATHEMATICS 2013

CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The

### DEEPAWALI ASSIGNMENT

DEEPWLI SSIGNMENT CLSS & DOPPE FO TGET IIT JEE Get Solution & Video Tutorils online www.mthsbysuhg.com Downlod FEE Study Pckges, Test Series from w ww.tekoclsses.com Bhopl : Phone : (0755) 00 000 Wishing

### MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC

FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot

### Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A

Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new

### Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

### Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38

Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score

### 1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D

. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )

### Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).

### Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

### Form 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6

Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms

### R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

### 1. If y 2 2x 2y + 5 = 0 is (A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2) 9 (A) 0, (B) 4, (C) (4, 4) (D) a (C) c = am m.

SET I. If y x y + 5 = 0 is (A) circle with centre (, ) (B) prbol with vertex (, ) (C) prbol with directrix x = 3. The focus of the prbol x 8x + y + 7 = 0 is (D) prbol with directrix x = 9 9 (A) 0, (B)

### Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,

R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice

### KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

### Mathematics Extension 1

04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

### 2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B

Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,

### , MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

### JEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS

JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle

### Trigonometric Functions

Trget Publictions Pvt. Ltd. Chpter 0: Trigonometric Functions 0 Trigonometric Functions. ( ) cos cos cos cos (cos + cos ) Given, cos cos + 0 cos (cos + cos ) + ( ) 0 cos cos cos + 0 + cos + (cos cos +

### / 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

### IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB

` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd - 5 7 Ph: -66 Regd

### Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

MT TRIGONOMETRIC FUNCTIONS AND TRIGONOMETRIC EQUATIONS C Trigonometric Functions : Bsic Trigonometric Identities : + cos = ; ; cos R sec tn = ; sec R (n ),n cosec cot = ; cosec R {n, n I} Circulr Definition

### + R 2 where R 1. MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark)

2. C h p t e r t G l n c e is the set of ll points in plne which re t constnt distnce from fixed point clled centre nd constnt distnce is known s rdius of circle. A tngent t ny point of circle is perpendiculr

### Mathematics Extension 2

00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

### CONIC SECTIONS. Chapter 11

CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

### A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

### PART - III : MATHEMATICS

JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out

### GEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS:

GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0

### Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

### PARABOLA EXERCISE 3(B)

PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution

### THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

### Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

### Mathematics. Area under Curve.

Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

### P 1 (x 1, y 1 ) is given by,.

MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

### US01CMTH02 UNIT Curvature

Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently

### x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

### Set 1 Paper 2. 1 Pearson Education Asia Limited 2017

. A. A. C. B. C 6. A 7. A 8. B 9. C. D. A. B. A. B. C 6. D 7. C 8. B 9. C. D. C. A. B. A. A 6. A 7. A 8. D 9. B. C. B. D. D. D. D 6. D 7. B 8. C 9. C. D. B. B. A. D. C Section A. A (68 ) [ ( ) n ( n 6n

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### ICSE Board Class IX Mathematics Paper 4 Solution

ICSE Bord Clss IX Mthemtics Pper Solution SECTION A (0 Mrks) Q.. () Consider x y 6 5 5 x y 6 5 5 0 6 0 6 x y 6 50 8 5 6 7 6 x y 6 7 6 x y 6 x 7,y (b) Dimensions of the brick: Length (l) = 0 cm, bredth

### EXERCISE I. 1 at the point x = 2 and is bisected by that point. Find 'a'. Q.13 If the tangent at the point (x ax 4 touches the curve y =

TANGENT & NORMAL EXERCISE I Q. Find the equtions of the tngents drwn to the curve y 4y + 8 = 0 from the point (, ). Q. Find the point of intersection of the tngents drwn to the curve y = y t the points

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

### Answers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2

Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge

### Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

### Year 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks

THGS Mthemtics etension Tril 00 Yer Tril Emintion Mthemtics Etension Question One mrks (Strt on new pge) Mrks ) If P is the point (-, 5) nd Q is the point (, -), find the co-ordintes of the point R which

### SOLUTION OF TRIANGLE GENERAL NOTATION : 1. In a triangle ABC angles at vertices are usually denoted by A, B, C

GENERL NOTTION : SOLUTION OF TRINGLE In tringle BC ngles t vertices re usully denoted by, B, C PGE # sides opposite to these vertices re denoted by, b, c respectively Circumrdius is denoted by R 3 Inrdius

### First Semester Review Calculus BC

First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

### MTH 4-16a Trigonometry

MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

### Ch AP Problems

Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

### Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES

Leon-5 PROPERTIES ND SOLUTIONS OF TRINGLES Reltion etween the ide nd trigonometri rtio of the ngle of tringle In ny tringle, the ide, oppoite to the ngle, i denoted y ; the ide nd, oppoite to the ngle

### PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### Thomas Whitham Sixth Form

Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

### Section 13.1 Right Triangles

Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl

### UNIT-4. File downloaded from For the things of this world cannot be made known without a knowledge of mathematics.

File downloded from http://jsuniltutoril.weely.com UNIT- For the things of this world cnnot e mde known without knowledge of mthemtics.. Solve y fctoriztion. - + ( ) = 0 Ans: + ( - ) = 0. [( + ) + ( -

### APPM 1360 Exam 2 Spring 2016

APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### Chapter 1: Fundamentals

Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

### Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

### SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

### APPLICATIONS OF THE DEFINITE INTEGRAL

APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through

### Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

### FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

### 03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

### 03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

### Use the diagram to identify each angle pair as a linear pair, vertical angles, or neither.

inl xm Review hpter 1 6 & hpter 9 Nme Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points

### Cat Solved Paper. Fresh Paper. No. of Questions : 50 Time : 40 min

t Solved Pper Fresh Pper No. of Questions : 50 Time : 0 min Note Ech wrong nswer crry rd negtive mrk. irections for question number to 5 : nswer the questions independently of ech other. 9 6 5. The infinite

### Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

### Pre-Calculus TMTA Test 2018

. For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to

### NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then

3. Stellr Prllx y terrestril stndrds, the strs re extremely distnt: the nerest, Proxim Centuri, is 4.24 light yers (~ 10 13 km) wy. This mens tht their prllx is extremely smll. Prllx is the pprent shifting

### Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd

### Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

### Coimisiún na Scrúduithe Stáit State Examinations Commission

M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions

### Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

### 3.1 Review of Sine, Cosine and Tangent for Right Angles

Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

### Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

### ( β ) touches the x-axis if = 1

Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without

### 1 (=0.5) I3 a 7 I4 a 15 I5 a (=0.5) c 4 N 10 1 (=0.5) N 6 A 52 S 2

Answers: (98-84 HKMO Finl Events) Creted by Mr. Frncis Hung Lst updted: December 05 Individul Events SI 900 I 0 I (=0.5) I 7 I4 5 I5 80 b 7 b b 5 b 6 b 8 b 4 c c 4 c 0 x (=0.5) c 4 N 0 d 9 d 5 d 5 y d

### 10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm

HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then

### 1 cos. cos cos cos cos MAT 126H Solutions Take-Home Exam 4. Problem 1

MAT 16H Solutions Tke-Home Exm 4 Problem 1 ) & b) Using the hlf-ngle formul for cosine, we get: 1 cos 1 4 4 cos cos 8 4 nd 1 8 cos cos 16 4 c) Using the hlf-ngle formul for tngent, we get: cot ( 3π 1 )

### SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 2012

MOCK TEST 0 SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 0 ANSWERS. () π π Tke cos - (- ) then sin [ cos - (- )]sin [ ]/. () Since sin - + sin - y + sin - z π, -; y -, z - 50 + y 50 + z 50 - + +

### Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

### Ellipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION

FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied

### Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### Answers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017

Answers: (0- HKMO Het Events) reted y: Mr. Frncis Hung Lst updted: 5 Decemer 07 - Individul - Group Individul Events 6 80 0 4 5 5 0 6 4 7 8 5 9 9 0 9 609 4 808 5 0 6 6 7 6 8 0 9 67 0 0 I Simplify 94 0.

### Answers for Lesson 3-1, pp Exercises

Answers for Lesson -, pp. Eercises * ) PQ * ) PS * ) PS * ) PS * ) SR * ) QR * ) QR * ) QR. nd with trnsversl ; lt. int. '. nd with trnsversl ; lt. int. '. nd with trnsversl ; sme-side int. '. nd with

### TO: Next Year s AP Calculus Students

TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

### HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

### Definite integral. Mathematics FRDIS MENDELU

Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

### I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles

UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet STRAND I: Geometry nd Trigonometry I1 Pythgors' Theorem nd Trigonometric Rtios Tet Contents Section I1.1 Pythgors' Theorem I1. Further Work With Pythgors'

### Mathematics Extension 2

S Y D N E Y B O Y S H I G H S C H O O L M O O R E P A R K, S U R R Y H I L L S 005 HIGHER SCHOOL CERTIFICATE TRIAL PAPER Mthemtics Extension Generl Instructions Totl Mrks 0 Reding Time 5 Minutes Attempt

### sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to