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

Save this PDF as:
Size: px
Start display at page:

Download "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"

Transcription

1 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 the function f? Which digrm shows sketch of f ( )? omin Rnge f() - - f() 8 f ( ) f() f ( ) 8 f ( ) The digrm shows prt of the grph of f ( ). Q(, ) f ( ) P(, 8) The curve psses through the points P(, 8) nd Q(, ). Which of the following represents the eqution of the curve? 8 f ( ) 7 log 7 Pge

2 The digrm elow shows sketch of f ( ). f ( ) Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows sketch of f ( ). f ( ) Which eqution elow represents f ( )? sin sin( ) Which digrm elow shows sketch of f ( )? sin sin The digrm elow shows sketch of g( ). g( ) Which eqution elow represents g( )? cos cos cos cos Pge

3 Higher Mthemtics Ojective Test Prctice ook 7 The digrm shows sketch of f ( ). f ( ) 8 Functions f nd g re defined on suitle domins f ( ) nd g( ) Find n epression for f ( g( )). ( ) Which digrm elow shows sketch of f ( )? 9 When 8 is written in the form ( p) q, wht is the vlue of q? 9 7 Here re two sttements out the grph of the curve with eqution ( ) ( ) () There is turning point on the -is. () The minimum vlue of is t. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Wht is the minimum vlue of f( ) ( )( )? 7 Pge

4 Higher Mthemtics Ojective Test Prctice ook qudrtic function f is given f ( ) c, where nd c Which grph elow is possile sketch of f ( )? The digrm shows prt of the grph of f ( ). f ( ) The curve cuts the -is t (, ) nd the -is t (-,), (, ) nd (, ). Which of the following represents the eqution of the curve? cos cos Here re two sttements out the circle with eqution ( ) ( ) nd the line with eqution. () The line intersects the circle t (6, 6) () Prt of the line is dimeter of the circle. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. sequence is defined the recurrence reltion u u 9, u. n n Wht is the vlue of u? Pge

5 Higher Mthemtics Ojective Test Prctice ook 6 sequence is defined the recurrence reltion u u, u n n Wht is the vlue of u? 7 sequence is defined the recurrence reltion n n u u, u Wht is the vlue of u? Here re two sttements out the limits of sequences. () The sequence generted un 7u n 6 hs limit 8 s n. () The sequence generted u n hs no limit. Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 9 sequence is generted the recurrence reltion un u n 6. Wht is the limit of this sequence s n? u n Wht re the roots of the qudrtic eqution ( )( )? nd nd nd nd Solve the inequlit Solve ( )( ). or or Solve ( 8)( ). 8 or 8 or 8 8 Wht is the nture of the roots of the qudrtic eqution Two rel equl roots Two rel distinct roots No rel roots 6 7 6? Three rel distinct roots Pge

6 Higher Mthemtics Ojective Test Prctice ook For wht vlues of p does p (p ) p hve equl roots? 6 Wht is the eqution of the prol with roots nd, pssing through the point (, ) qudrtic eqution is given 8( )( ) It hs roots t nd. Wht is the -intercept? 6 9 Wht is the reminder when is divided ( )? 7 7 The digrm shows sketch of the curve with eqution k( )( )( ) Evlute log 9 log 6 log 8. Wht re the vlues of nd k? (, ) k Epress log log 8 log in form without using logrithms. 8 8 Pge 6

7 Higher Mthemtics Ojective Test Prctice ook The epression log ( e) log ( e ) cn e written in the form e P loge Q. Wht re the vlues of P nd Q? P Q e Which digrm elow shows the grph of log? e The digrm shows the grph with eqution log ( ). (, ) log ( ) Wht re the vlues of nd? 6 Wht is the solution of log? If log 6, wht is the vlue of? Which of the following equtions is equivlent to log log log? 8 Pge 7

8 Higher Mthemtics Ojective Test Prctice ook 8 Solve log log. 6 9 The digrm shows the grph of log plotted ginst. log The curve with eqution log ( ) cuts the -is t P. Wht is the -coordinte of P? 9 Given tht f ( ) ( ), find f( ). 8 6 The grph is stright line through the origin with grdient. Find n epression for in terms of. d Given tht, find. d 6 7 n The grph illustrtes the lw p kv. 8 7 log p Wht is the derivtive of? log v Wht is the vlue of k? Pge 8

9 Higher Mthemtics Ojective Test Prctice ook tngent to the curve with eqution hs grdient. Wht is the -coordinte of the point of contct of the tngent nd curve? Given tht, for f ( ), for, for Which digrm shows the curve with eqution f ( )? 6 Given tht s t, find the rte of chnge of s with respect to t, when t For wht vlues of is the function with derivtive f ( ) strictl decresing? or 8 For wht vlues of is the function with derivtive f ( ) neither incresing or decresing? nd nd nd nd 9 function is given f ( ). Find the mimum vlue of f in the intervl. 7 Pge 9

10 function f( ) is such tht f ( ) ( ). Which grph could e sketch of f ( )? (, ) Higher Mthemtics Ojective Test Prctice ook Find 6 d 6 c c c 9 c Wht is the integrl of respect to? c Find c c c d, c c c c ( ) with Find d 7 c 7 c 7 c 7 c 7 6 Find 6 d, c c 7 c c Pge

11 7 Find d 6 c 6 c c c Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows the curve with eqution f ( ). c 8 Evlute 8 8 d 9 Evlute 8 d Which of the following gives the re etween the curve nd -is? c f ( ) d f ( ) d c f ( ) d f ( ) d c f ( ) d f ( ) d Pge

12 Higher Mthemtics Ojective Test Prctice ook 6 The digrm shows the curve with eqution. 6 The digrm shows the curves with equtions. nd. Which of the following gives the vlue of the shded re? d d d d d Which of the following gives the vlue of the shded re? d d d d Pge

13 Higher Mthemtics Ojective Test Prctice ook 6 6 f ( ) ( cd, ) f ( ) g( ) (, ) Wht is n epression for the shded re in the digrm ove? c ( ) ( ) c f g d ( ) ( ) d f g d ( ) ( ) c f g d ( ) ( ) g f d Find n epression for the totl shded re in the digrm ove. f ( ) d ( ) ( ) f d f d ( ) ( ) f d f d ( ) ( ) f d f d 6 The grdient of the tngent to curve is d given. d If the curve psses through the point (, ) find its eqution. 6 Pge

14 Higher Mthemtics Ojective Test Prctice ook d 66 Given tht d nd when, find n epression for in terms of. = + 67 Given tht p. 8 8 p 6 d, find the vlue of d 68 If 6sin( ), find. d 6 cos( + ) -6 cos( + ) 8 cos( + ) cos( + ) d 69 If cos( ), find. d sin( ) sin( ) sin( ) 6sin( ) 7 Wht is the derivtive of respect to? cos cos sin sin cos 7 Find d ( ) c () c ( ) ( ) c 7 Find d c c ( ) ( ) () c c c 7 Find 6cos d sin() c 6sin() c 8sin() c 8sin( ) c sin with 7 If f find f ( ). ( ) ( ), ( ) 6 ( ) ( ) 7 Find sin( ) d cos( ) c cos( ) c cos() c cos() c Pge

15 Higher Mthemtics Ojective Test Prctice ook 76 Wht is the derivtive of respect to? cos sin sin cos sin with 8 Wht is the distnce etween the points (,, ) nd (,, )? G hs coordintes (,, ) nd H(,, ). Wht is the distnce etween G nd H? 6 78 P is the point (, ), Q (, ), R(,) nd S(, ). Here re two sttements out PQ nd RS. () PQ 9 units () PQ RS Neither sttements is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 79 The point (, ) lies on the circle with centre (, ). Find the length of the dimeter of the circle. 8 lculte the length of the line joining P(,) to Q(,) Find the grdient of the line in the digrm ove. 8 Find the grdient of the line with eqution. 8 The line with eqution 6 meets the -is t the point. Wht is the grdient of the line joining to the point (, )? 9 9 Pge

16 Higher Mthemtics Ojective Test Prctice ook 8 The line through K(, ) nd L(7, ) hs grdient of. Wht is the vlue of? Wht is the eqution of the line through the points (, ) nd (, )? ( ) ( ) ( ) ( ) 89 Wht is the eqution of the line joining the points (, ) nd (6, )? line, pssing through the point (, ), mkes n ngle of with the positive direction of the -is. Wht is the eqution of the line? Find the eqution of the line through the points P(, ) nd Q(, 9). 88 line psses through the points (, ) nd (, ). Wht is the grdient nd -intercept of this line? m -intercept - 9 Find the eqution of the line through the points (, ) nd (, ) Wht is the grdient of line prllel to the line with eqution 7? 9 Here re two sttements out the lines PQ : nd RS : 6 () PQ nd RS re perpendiculr () PQ nd RS hve the sme -intercept Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Pge 6

17 Higher Mthemtics Ojective Test Prctice ook 9 line L is perpendiculr to the line with eqution 6 Wht is the grdient of the line L? 6 9 line L hs grdient. Wht is the eqution of the line through the point (, ) nd perpendiculr to line L? 98 P is the point (, ) nd Q(, 7). Wht is the grdient nd the midpoint R of the line joining P to Q? m 99 Wht re the coordintes of M, the midpoint of the line joining P(, 7, ) nd Q(,, ). (,, ) (,, ) (,, ) (,, ) R (,) (,) (, ) (, ) 96 line hs eqution. Which of the following lines is perpendiculr to this line? The digrm shows tringle PQR with ltitude PS. Q S R P(, ) 97 line L is perpendiculr to the line with eqution 6 7. Wht is the grdient of line L? P hs coordintes (,) nd the grdient of QR is. Wht is the eqution of ltitude PS? 9 7 Pge 7

18 Higher Mthemtics Ojective Test Prctice ook The digrm shows tringle EFG with medin EH drwn. EH hs grdient. F hs coordintes (,7) nd G is (7,). Wht is the eqution of medin EH? 7 8 The digrm shows tringle PQR with ltitude PS drwn. Q E S R F(, 7) H P(, ) G(7, ) The points P( 9, ), Q(,) nd R(,) re colliner. Wht is the vlue of? 7 circle hs eqution ( ) ( ) 8 Wht re the coordintes of its centre nd length of its rdius? circle hs eqution 6 6 Wht is the rdius of this circle? entre (,) Rdius 8 (, ) 8 (,) 8 (, ) 8 P hs coordintes (,) nd the grdient of QR is. Wht is the eqution of ltitude PS? 7 6 circle hs eqution 7 Wht is the rdius of this circle? 7 Pge 8

19 Higher Mthemtics Ojective Test Prctice ook 7 circle with centre (, ) psses through the point (6, ). Wht is the eqution of the circle? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The point P(,) lies on the circle with eqution ( ) ( ). Wht is the grdient of the tngent t P? 8 circle with centre (, ) psses through the point (, ). Wht is the eqution of the circle? ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The point T(9, ) lies on the circle with eqution 6. Wht is the grdient of the tngent t T? 9 9 circle hs centre (, ) nd rdius 6 units. Wht is the eqution of the circle? The point P(, ) lies on the circle with eqution 6 7. Wht is the grdient of the tngent t P? To find the points of intersection of line nd circle the eqution k is formed. Wht is the vlue of k for which the line is tngent to the circle? To find the points of intersection of line nd circle the eqution p is formed. Wht is the vlue of p for which the line tngent to the circle? Pge 9

20 Higher Mthemtics Ojective Test Prctice ook The circle with eqution 6 intersects the -is t the points (, ) nd (, ) (, ) nd (, ) ( 6, ) nd (, ) (, ) nd (6, ) 6 PQ is dimeter of the circle with eqution ( ) ( ) P hs coordintes (, 7), wht re the coordintes of Q? (,) (, ) (, 8) (-, -9) 7 The vectors nd cn e epressed in the form i jk nd i j k respectivel. Find? Wht is the mgnitude of vector 7 where hs components? The vector GH cn e epressed s 7i 7jk. The coordintes of H re (,, ). Wht re the coordintes of G? (, 9, ) (,, ) (,, ) (, 9, 7) The points K nd L hve coordintes (6,, ) nd (,, ) respectivel. Epress KL in terms of i, j nd k? -8i + j k -8i + 7j k -8i + 7j k 8i - 7j + k 8 The points nd hve coordintes (6,, ) nd (,, ) respectivel. Find If L is the point (7,, ) nd Wht re the coordintes of K? (,, ) (,, ) (9,, ) (9,, ) KL 6 Pge

21 Higher Mthemtics Ojective Test Prctice ook Vector u, where u k, is unit vector. Given tht k, find the vlue of k. Given tht p i j k, find p Which of the following vectors is prllel to vector u, where u? Given tht u i j k nd v i j k, find u v. 7 6 Here re two sttements out 6 u nd v 8 () u is prllel to v () u nd v hve the sme direction Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. 8 Given tht u nd v, Find the components of vector u v Pge

22 Higher Mthemtics Ojective Test Prctice ook 9 Given tht u nd v, Wht re the components of u v? Vectors u nd v re shown in the digrm elow. 6 E Find in terms of u nd v. v u v u v u v u v u v E Here re two sttements out 6 nd 8 6 () () is prllel to Which of the following is true? Neither sttement is correct. Onl sttement () is correct. Onl sttement () is correct. oth sttements re correct. Which of the following is equivlent to c? d e e d c d e Pge

23 Higher Mthemtics Ojective Test Prctice ook Pge is the point (,, ), (8,, ) nd G(8, 7, ). Wht re the components of H? 7 OPQRS is prmid s shown SM is perpendiculr to the plne OPQR. SM Wht re the components of SQ? 8 6 The points (,, ), (, 7, 9) nd (,, 7) re colliner. Wht is the vlue of? The points P(,, ), Q(,, 6) nd R(,, ) re colliner. Find the rtio PQ : QR. : : : : E F G H P Q S R 8 6 M z

24 Higher Mthemtics Ojective Test Prctice ook 7 The points P(,, ), Q(,, ) nd R(9,, ) re colliner. Find the rtio PQ : QR. : : : : Given tht nd, find P is the point (,, ) nd Q is (6,, ). R divides PQ in the rtio :. Wht re the coordintes of R? (,, ) (,, ) (,, ) (,, 9) 9 P is the point (,, -) nd Q is (,, ). R divides PQ in the rtio :. Wht re the coordintes of R? ( 9, 6, ) (, 7, ) (,, ) (,, ) Given tht PQ nd P is (,, ). Wht re the coordintes of R, the midpoint of PQ? (,, ) (,, ) (,, ) (,, ) 6 Given tht p nd q, find p. q q. q Given tht nd, find 7 if is perpendiculr to. - - Given tht nd, find if is perpendiculr to. Pge

25 Higher Mthemtics Ojective Test Prctice ook u nd v hve components nd respectivel. Find if u nd v re perpendiculr u nd v re the vectors with components nd respectivel. Wht is the ect vlue of cos where is the ngle etween the vectors? ½ 6 8 In the digrm, r s nd the ngle is. Wht is the vlue of r.( r s )? r s c 7 6 O(,, ) P(,, ) Q(,, ) In the isosceles tringle shown nd c. Find.( c ). Wht is the ngle etween OP nd OQ? 6 9 Pge

26 Higher Mthemtics Ojective Test Prctice ook p r q In the equilterl tringle shown the length of ech side is units. Find q. r. Pge 6

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

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

More information

CONIC SECTIONS. Chapter 11

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

More information

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

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

More information

Lesson-5 ELLIPSE 2 1 = 0

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).

More information

/ 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

/ 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

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

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

More information

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

, 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

More information

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

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

More information

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

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

More information

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

More information

Thomas Whitham Sixth Form

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

More information

( β ) touches the x-axis if = 1

( 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

More information

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

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

More information

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

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 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

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

Polynomials and Division Theory

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

More information

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

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) (, )

More information

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra

Believethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper

More information

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

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

More information

Mathematics. Area under Curve.

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

More information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

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

More information

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

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..

More information

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.

( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12. Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k

More information

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

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

More information

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)

More information

PARABOLA EXERCISE 3(B)

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

More information

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

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

More information

Mathematics Extension 2

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

More information

IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB

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

More information

CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee

CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Ee ADDITIONAL MATHEMATICS FORM 5 MODULE 4 INTEGRATION CHAPTER : INTEGRATION Content pge Concept Mp 4. Integrtion of Algeric Functions 4 Eercise A 5 4. The Eqution of Curve from Functions of Grdients. 6 Eercise

More information

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

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

More information

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

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

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

Eigen Values and Eigen Vectors of a given matrix

Eigen Values and Eigen Vectors of a given matrix Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme

More information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th  Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd

More information

Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.

Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100. Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte

More information

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

Mathematics Extension 1

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

More information

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS

GEOMETRY OF THE CIRCLE TANGENTS & SECANTS Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the

More information

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

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)

More information

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

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

More information

Ch AP Problems

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,

More information

Loudoun Valley High School Calculus Summertime Fun Packet

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!

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

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

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 ) ) (,

More information

y = f(x) This means that there must be a point, c, where the Figure 1

y = f(x) This means that there must be a point, c, where the Figure 1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

More information

MTH 4-16a Trigonometry

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

More information

Answer: A. Answer: A. k k. Answer: D. 8. Midpt. BC = (3, 6) Answer: C

Answer: A. Answer: A. k k. Answer: D. 8. Midpt. BC = (3, 6) Answer: C THE STRAIGHT LINE. (, p) p p p. ( ) AB. D p p 9. A(, ) B(k, l) I. ( ) 9 II III. AB. tn - () = o. Midpt. A = (, ) Midpt. BD = (, ). p p p AB A k k k k. Midpt. B = (, ).. perp 9 k ( ) k k k k k Pegss Higher

More information

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.

More information

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS

Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A

MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution

More information

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSS-SECTIONS. A() ;, ; (digonl) ˆ Ȉ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8

More information

REVIEW SHEET FOR PRE-CALCULUS MIDTERM

REVIEW SHEET FOR PRE-CALCULUS MIDTERM . If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,

More information

SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING

SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS

More information

CET MATHEMATICS 2013

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

More information

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

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

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

More information

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?

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

More information

APPLICATIONS OF DEFINITE INTEGRALS

APPLICATIONS OF DEFINITE INTEGRALS Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

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

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

More information

CHAPTER 6 Introduction to Vectors

CHAPTER 6 Introduction to Vectors CHAPTER 6 Introduction to Vectors Review of Prerequisite Skills, p. 73 "3 ".. e. "3. "3 d. f.. Find BC using the Pthgoren theorem, AC AB BC. BC AC AB 6 64 BC 8 Net, use the rtio tn A opposite tn A BC djcent.

More information

Answers for Lesson 3-1, pp Exercises

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

More information

First Semester Review Calculus BC

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.

More information

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

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,

More information

Mathematics Extension Two

Mathematics Extension Two Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student

More information

ICSE Board Class IX Mathematics Paper 4 Solution

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

More information

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

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

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

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

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

More information

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

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

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS 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.

More information

Calculus AB. For a function f(x), the derivative would be f '(

Calculus AB. For a function f(x), the derivative would be f '( lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

More information

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up 9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute

More information

Unit 1 Exponentials and Logarithms

Unit 1 Exponentials and Logarithms HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

More information

Review Exercises for Chapter 4

Review Exercises for Chapter 4 _R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

Set 1 Paper 2. 1 Pearson Education Asia Limited 2017

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 information

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

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

More information

SAINT IGNATIUS COLLEGE

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

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

Stage 11 Prompt Sheet

Stage 11 Prompt Sheet Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

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

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

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

HYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..

HYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3.. HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of

More information

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

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

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+ .1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

More information

If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a

If C = 60 and = P, find the value of P. c 2 = a 2 + b 2 2abcos 60 = a 2 + b 2 ab a 2 + b 2 = c 2 + ab. c a Answers: (000-0 HKMO Finl Events) Creted : Mr. Frncis Hung Lst updted: 0 June 08 Individul Events I P I P I P I P 5 7 0 0 S S S S Group Events G G G G 80 00 0 c 8 c c c d d 6 d 5 d 85 Individul Event I.,

More information