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


 Marilynn Phebe Bryant
 4 years ago
 Views:
Transcription
1 DOWNLOAD FREE FROM PH.: , Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs 4 choices (A), (B), (C) nd (D) out of which ONLY ONE is correct. So select the correct choice : Choices re : (A) Sttement is True, Sttement is True; Sttement is correct eplntion for Sttement. (B) Sttement is True, Sttement is True; Sttement is NOT correct eplntion for Sttement. (C) Sttement is True, Sttement is Flse. (D) Sttement is Flse, Sttement is True. PARABOLA 86. Sttement : Slope of tngents drwn from (4, 0) to prol = 9 re, Sttement : Ever prol is smmetric out its directri. 87. Sttement : Though (λ, λ + ) there cn t e more thn one norml to the prol = 4, if λ <. Sttement : The point (λ, λ + ) lies outside the prol for ll λ. 88. Sttement : If + = k is norml to the prol =, then k is 9. Sttement : Eqution of norml to the prol = 4 is m + m + m 3 = Sttement : If, k re the segments of focl chord of the prol = 4, then k is equl to /. Sttement : Ltus rectum of the prol = 4 is H.M. etween the segments of n focl chord of the prol 90. Sttement : Two prols = 4 nd = 4 hve common tngent + + = 0 Sttement : + + = 0 is common tngent to the prols = 4 nd = 4 nd point of contcts lie on their respective end points of ltus rectum. 9. Sttement : In prol = 4, the circle drwn tking focl rdii s dimeter touches is. Sttement : The portion of the tngent intercepted etween point of contct nd directi sutends 90 ngle t focus. 9. Sttement : The joining points (8, 8) & (/, ), which re ling on prol = 4, pss through focus of prol. Sttement : Tngents drwn t (8, 8) & (/, ) on the prol = 4 re perpendiculr. 93. Sttement : There re no common tngents etween circle = 0 nd prol =. Sttement : Eqution of tngents to the prol = 4 is = m + /m where m denotes slope of tngent. 94. Sttement : Three distinct normls of the prol = cn pss through point (h,0) where h > 6. Sttement : If h > then three distinct nromls cn pss through the point (h, 0) to the prol = Sttement : The normls t the point (4, 4) nd, 4 of the prol = 4 re perpendiculr. Sttement : The tngents to the prol t the nd of focl chord re perpendiculr. 96. Sttement : Through (λ, λ + ) there cnnot e more thn onenorml to the prol = 4 if λ <. Sttement : The point (λ, λ + ) lines out side the prol for ll λ. 97. Sttement : Slope of tngents drwn from (4, 0) to prol = 9 re /4, 9/4 Sttement : Ever prol is smmetric out its is. 98. Sttement : If prol is defined n eqution of the form = + + c where,, c R nd > 0, then the prol must possess minimum. Sttement : A function defined n eqution of the form = + + c where,, c R nd 0, m not hve n etremum Sttement : The point (sin α, cos α) does not lie outside the prol + = 0 when,, α π 6 Sttement : The point (, ) lies outside the prol = 4 if 4 > Sttement : The line = + touches the prol = 4( + ). Sttement : The line = m + c touches = 4( + ) if c = m + /m. 30. Sttement : If PQ is focl chord of the prol = 3 then minimum length of PQ = 3. Sttement : Ltus rectum of prol is the shortest focl chord. 30. Sttement : Through (λ, λ + ), there cn t e more thn one norml to the prol = 4 if λ <. Sttement : The point (λ, λ + ) lies outside the prol for ll λ R ~ {} Sttement : Perpendiculr tngents to prol = 8 meets on + = 0 Sttement : Perpendiculr tngents of prol meets on tngent t the verte. 8 of 9
2 DOWNLOAD FREE FROM PH.: , Let = 4 nd = 4 e two prols Sttement: The eqution of the common tngent to the prols is + + = 0 Sttement: Both the prols re reflected to ech other out the line = Let = 4 ( + ) nd = 4 ( + ) re two prols Sttement : Tngents re drwn from the locus of the point re mutull perpendiculr Stte.: The locus of the point from which mutull perpendiculr tngents cn e drwn to the given com is + + = 0 ELLIPSE 306. Tngents re drwn from the point (3, 4) to the curve = 44. STATEMENT : The tngents re mutull perpendiculr. STATEMENT: The locus of the points from which mutull perpendiculr tngents cn e drwn to the given curve is + = Sttement : Circle + = 9, nd the circle ( 5) ( 3) + ( ) = 0 touches ech other internll. Sttement : Circle descried on the focl distnce s dimeter of the ellipse = 36 touch the uilir circle + = 9 internll 308. Sttement : If the tngents from the point (λ, 3) to the ellipse = re t right ngles then λ is equl to ±. Sttement : The locus of the point of the intersection of two perpendiculr tngents to the ellipse + =, is + = Sttement : 5 = 0 is the eqution of the tngent to the ellipse = 44. Sttement : The eqution of the tngent to the ellipse + = is of the form = m ± m Sttement : At the most four normls cn e drwn from given point to given ellipse. Sttement : The stndrd eqution + = of n ellipse does not chnge on chnging nd. 3. Sttement : The focl distnce of the point ( 4 3, 5 ) on the ellipse = 600 will e 7 nd 3. Sttement : The rdius of the circle pssing through the foci of the ellipse its centre t (0, 3) is Sttement : The lest vlue of the length of the tngents to Sttement : If nd e n two positive numers then + = nd hving = intercepted etween the coordinte es is Sttement : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnces of n point on it. Sttement : The eccentricit of n ellipse is less thn. 34. Sttement : An chord of the conic + + =, through (0, 0) is isected t (0, 0) Sttement : The centre of conic is point through which ever chord is isected. 35. Sttement : A tngent of the ellipse + 4 = 4 meets the ellipse + = 6 t P & Q. The ngle etween the tngents t P nd Q of the ellipse + = 6 is π/ Sttement : If the two tngents from to the ellipse / + / = re t right ngle, then locus of P is the circle + = of 9
3 DOWNLOAD FREE FROM PH.: , Sttement : The eqution of the tngents drwn t the ends of the mjor is of the ellipse = 0 is = 0, = 7. Sttement : The eqution of the tngent drwn t the ends of mjor is of the ellipse / + / = lws prllel to is 37. Sttement : Tngents drwn from the point (3, 4) on to the ellipse + = will e mutull perpendiculr 6 9 Sttement : The points (3, 4) lies on the circle + = 5 which is director circle to the ellipse + = Sttement : For ellipse + =, the product of the perpendiculr drwn from focii on n tngent is Sttement : For ellipse + =, the foot of the perpendiculrs drwn from foci on n tngent lies on the circle = 5 which is uilir circle of the ellipse. 39. Sttement : If line + = 3 is tngent to n ellipse with foci (4, 3) & (6, ) t the point (, ), then = 7. Sttement : Tngent nd norml to the ellipse t n point isects the ngle sutended foci t tht point. 30. Sttement : Tngents re drwn to the ellipse + = t the points, where it is intersected the line + 3 =. 4 Point of intersection of these tngents is (8, 6). Sttement : Eqution of chord of contct to the ellipse + = from n eternl point is given + = 0 3. Sttement : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnces of n point on it. Sttement : The eccentricit of n ellipse is less thn. 3. Sttement : The eqution + + λ = 0 cn never represent hperol Sttement : The generl eqution of second degree represent hperol it h >. 33. Sttement : The eqution of the director circle to the ellipse = 36 is + = 3. Sttement : The locus of the point of intersection of perpendiculr tngents to n ellipse is clled the director circle. 34. Sttement : The eqution of tngent to the ellipse = 36 t the point (3, ) is =. 3 Sttement : Tngent t (, ) to the ellipse + = is = 35. Sttement : The mimum re of PS S where S, S re foci of the ellipse + = nd P is n vrile point on it, is e, where e is eccentricit of the ellipse. Sttement : The coordintes of pre ( sec θ, tn θ). 36. Sttement : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnce of n point on it. Sttement : The eccentricit of ellipse is less thn. 37. Let Y = ± 9 3 [3, ) nd Y = ± HYPERBOLA 9 3 e (, 3] two curves. Sttement : The numer of tngents tht cn e drwn from Y = ± 9 3 is zero 0 5, 3 to the curve 84 of 9
4 DOWNLOAD FREE FROM PH.: , Sttement : The point 5, 3 lies on the curve Y = ± Sttement : If (3, 4) is point of hperol hving focus (3, 0) nd (λ, 0) nd length of the trnsverse is eing unit then λ cn tke the vlue 0 or 3. Sttement : S P SP =, where S nd S re the two focus = length of the trnsverse is nd P e n point on the hperol. 39. Sttement : The eccentricit of the hperol = 0 is 5 4. Sttement : The eccentricit of the hperol = is equl to 330. Let,, α R {0}, where, re constnts nd α is prmeter. Sttement : All the memers of the fmil of hperols smptotes = hve the sme pir of α Sttement : Chnge in α, does not chnge the slopes of the smptotes of memer of the fmil + =. α 33. Sttement : The slope of the common tngent etween the hperol = nd + = m e or. Sttement : The locus of the point of inteeersection of lines = m nd + = is m hperol (where m is vrile nd 0). 33. Sttement : The eqution + + λ = 0 cn never represent hperol. Sttement : The generl eqution of second degree represents hperol if h > Sttement If point (, ) lies in the region II of =, shown in the figure, then < 0 Y = I III II II I = Sttement If (P(, ) lies outside the hperol =, then < 334. Sttement Eqution of tngents to the hperol 3 = 6 which is prllel to the line = is = 3 5 nd = Sttement = m + c is tngent to / / = if c = m Sttement : There cn e infinite points from where we cn drw two mutull perpendiculr tngents on to the hperol = 9 6 III X 85 of 9
5 DOWNLOAD FREE FROM PH.: , Sttement : The director circle in cse of hperol = will not eist ecuse < nd director circle is = Sttement : The verge point of ll the four intersection points of the rectngulr hperol = nd circle + = 4 is origin (0, 0). Sttement : If rectngulr hperol nd circle intersect t four points, the verge point of ll the points of intersection is the mid point of linejoining the two centres Sttement : No tngent cn e drwn to the hperol =, which hve slopes greter thn Sttement : Line = m + c is tngent to hperol =. If c = m 338. Sttement : Eccentricit of hperol 3 3 = 0 is 4/3 Sttement : Rectngulr hperol hs perpendiculr smptotes nd eccentricit = 339. Sttement : The eqution + + λ = 0 cn never represent hperol Sttement : The generl eqution of second degree represent hperol it h > Sttement : The comined eqution of oth the es of the hperol = c is = 0. Sttement : Comined eqution of es of hperol is the comined eqution of ngle isectors of the smptotes of the hperol. 34. Sttement : The point (7, 3) lies inside the hperol 9 4 = 36 where s the point (, 7) lies outside this. Sttement : The point (, ) lies outside, on or inside the hperol = ccording s < or = or > Sttement : The eqution of the chord of contct of tngents drwn from the point (, ) to the hperol 6 9 = 44 is = 44. Sttement : Pir of tngents drwn from (, ) to = is SS = T S = = S = 343. Sttement : If PQ nd RS re two perpendiculr chords of = e, nd C e the centre of hperol = c. Then product of slopes of CP, CQ, CR nd CS is equl to. Sttement : Eqution of lrgest circle with centre (, 0) nd ling inside the ellipse is = 0. Answer 86. C 87. B 88. A 89. C 90. B 9. B 9. B 93. C 94. A 95. A 96. B 97. A 98. C 99. B 300. A 30. A 30. B 303. C 304. B 305. A 306. A 307. A 308. A 309. A 30. B 3. C 3. B 33. A 34. A 35. A 36. C 37. A 38. B 39. A 30. D 3. A 3. A 33. A 34. C 35. C 36. A 37. A 38. D 39. A 330. A 33. B 33. D 333. D 334. C 335. D 336. A 337. A 338. D 339. A 340. A 34. A 34. B 343. B Solution 86. Option (C) is correct. = m + m 0 = 4m. 9 / 4 m 6m 40m + 9 = 0 m =, m 9 = Ever m =,m = Ever prol is smmetric out its is Option (B) is correct An norml to = 4 is 86 of 9
6 DOWNLOAD FREE FROM PH.: , Y + t = t + t 3 If this psses through (λ, λ + ), we get λ + + λ = t + t 3 t 3 + t(  λ)  λ  = 0 = f(t) (s) If λ <, then f (t) = 3t + (  λ) > 0 f(t) = 0 will hve onl one rel root. So A is true. Sttement is lso true coz (λ + ) > 4λ is true λ. The sttement is true ut does not follow true sttement For the prol =, eqution of norml with slope  is = () 3 () 3 + = 9, k = 9 Ans. (A). 89. SP = + t = ( + t ) ( + t SQ = + /t ) = = t ( + t ) + = = SP SQ ( + t ),, re in A.P. SP SQ is H.M. etween SP & SQ Hence + = = k = / = k k 90. = 4 eqution of tngent of slope m = m + m Ans. (C) If it touches = 4 then = 4 (m + /m) 4 4m  = 0 will hve equl roots m D = m + 0 m = m 3 =  m =  So = = 0 (, ) & (, ) lies on it B is correct. 9. ( ) ( t ) + ( t) = 0 Solve with = 0 t + ( t) = 0 t + t = 0 If it touches is then ove qudrtic must hve equl roots. SO, D = 0 4 t 4 t = 0 which is correct. B is correct. S(, 0) (t, t) 96. (B) An norml to the prol = 4 is + t = t + t 3 It this psses through (λ, λ + ) t 3 + t(  λ)  λ  = 0 = f(t) s) λ < thn f (t) = 3t + (  λ) > 0 f(t) = 0 will hve onl one rel root A is true The sttement is lso true since (λ+ ) > 4λ is true for ll λ. The sttement is true ut does not follow true sttement. 87 of 9
7 DOWNLOAD FREE FROM PH.: , = m + m 0 = 4m + 9 / 4 m 6m 40m + 9 = 0 m = /4, m = 9/4 Ever prol is smmetric out its is. 98. (C) Sttement is true ut Sttement is flse. 99. (B) If the point (sin α, cos α) lies inside or on the prol + = 0 then cos α + sin α 0 sin α( sin α ) 0 sin α 0, or sin α (A) = ( + ) + is of the form = m( + ) + /m where m =. Hence the line touches the prol. 30. An norml to the prol = 4 is + t = t + t 3 If this psses through (λ, λ + ). We get λ + + λ t = t + t 3. t 3 + t (  λ) (λ + ) = 0 = f(t) (let) if λ <, then, f (t) = 3t + (  λ) > 0 f(t) = 0 will hve onl one rel root. sttement I is true. Sttement II is lso true since (λ + ) > 4λ is true for ll λ R ~ {}. Sttement I is true ut does ot follow true sttement II. Hence () is the correct nswer (B) Becuse the common tngent hs to e perpendiculr to =. Its slope is Ellipse is + = focus ( 5,0), e = 3, An point n ellipse ( 3, eqution of circle s the dimeter, joining the points ( 3/, / ) nd focus ( 5,0) is ( 5 ) ( 3) + (. ) = 0 (A) is the correct option () (λ, 3) should stisf the eqution + = 3 λ = ± (A) Here = 4, = 3 nd m = eqution of the tngent is = ± = ± Sttement I is true s it is known fct nd sttement II is oviousl true. However sttement II is not true resoning for sttement I, s coordinte sstem hs nothing to do with sttement I. 3. Given ellipse is + = = 64; 3 = 00 e = ( < ) 5 Now, focl distnce of (, ) on ellipse will e 7 nd 3. Now, for ellipse 7 6, 9, e = = = =. Focus is (e, 0) or ( ) Now rdius of the circle = Distnce etween ( 7, 0 ) nd (0, 3) = 4. 7, of 9
8 DOWNLOAD FREE FROM PH.: , Hence (c) is the correct nswer. 33. Option (A) is correct Sum of the distnce etween foci = e Sum of the focl distnces = e e < e coz e <. Both re true nd it is correct reson. 34. Option (A) is true. Let = m e n chord through (0, 0). This will meet conic t points whose coordintes re given + m + m = ( + m + m ) = = 0 = 0 Also = m, = m + = m ( + ) = 0 + = 0 midpoint of chord is (0, 0) m. 35. Eqution of PQ (i.e., chord of contct) to the ellipse + = 6 h + k =... () 6 3 An tngent to the ellipse + 4 = 4 is i.e., / cosθ + sinθ =... () () & () represent the sme line h = 3cosθ, k = 3sinθ Locus of R (h, k) is + = 9 Ans. (A) 36. /5 + (3) /9 = Ends of the mjor is re (0, 6) nd (0, 0) Eqution of tngent t (0, 6) nd (0, 0) is = 6, nd = 0 Anc. (C) = will hve director circle + = = 5 nd we know tht the locus of the point of intersection of two mutull perpendiculr tngents drwn to n stndrd ellipse is its director circle. is correct. 38. B formul p p = = 3. lso foot of perpendiculr lies on uilir circle of the ellipse. B is correct. 3. Sum of distnces etween foci = e sum of the focl distnces = /e e < /e since e <. (A) 3. The sttement is flse. Since this will represent hperol if h > λ > λ > 4 Thus reson R eing stndrd result is true. (A) 33. () Both Sttement nd Sttement re True nd Sttement is the correct eplntion of Sttement (C) Required tngent is 3 = or = of 9
9 DOWNLOAD FREE FROM PH.: , (C) re of PS S = e sin θ clerl its mimum vlue is e. P( cos θ, sin θ) S ( e, 0) S (e, 0) 37. Tngents cnnot e drwn from one rnch of hperol to the other rnch. Ans. (A) 38. (d) ( λ 3) = λ = 0 or (A) Hperol is ( 4 ) ( 3 ) e = =. 6 4 = Both sttements re true nd sttement II is the correct resoning for sttement I, s for n memer, semi trnsverse nd semi conjugte es re α nd α Hence () is the correct nswer respectivel nd hence smptoters re lws = ±. 33. If = m + c e the common tngent, then c = m... (i) nd c =  m +... (ii) on eliminting c, we get m = m = ±. Now for sttement II, On eliminting m, we get =, Which is hperol. Hence () is the correct nswer. 33. Option (D) is correct. The sttement is flse coz this will represent hperol if h > λ > λ > 4 The sttmenet, eing stndrd result, is true The sttement is flse coz points in region II lie elow the line = / The region is true (stndrd result). Indeed for points in region II 0 < <. > / / = if c = m c = 3.3 = 5 c = ± 5 90 of 9
10 DOWNLOAD FREE FROM PH.: , rel tngents re = Ans (C) 335. The locus of point of intersection of two mutull perpendiculr tngents drwn on to hperol 336. circle whose eqution is + =. For So director circle does not eist. So d is correct = = 0 =, + = So (0, 0) is verge point which is lso the mid point of line joining the centres of circle & rectngulr hperol is correct The sttement is flse. Since this will represent hperol if h > λ > λ > 4 Thus reson R eing stndrd result is true. (A) 340. () Both Sttement nd Sttement re True nd Sttement is the correct eplntion of Sttement. = is its director 34. (A) nd 7 ( 3) > < (B) Required chord of contct is = 44 otined from =. for 39 Yrs. Que. of IITJEE & 5 Yrs. Que. of AIEEE we hve distriuted lred ook 9 of 9
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 informationLesson5 ELLIPSE 2 1 = 0
Lesson5 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 informationEllipse. 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/ 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 informationCONIC 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 informationHYPERBOLA. 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 informationDrill 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 ltusrectum is 8 nd conjugte
More informationMATHEMATICS 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 informationP 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 informationELLIPSE. Standard equation of an ellipse referred to its principal axes along the coordinate axes is. ( a,0) A'
JMthemtics LLIPS. STANDARD QUATION & DFINITION : Stndrd eqution of n ellipse referred to its principl es long the coordinte es is > & = ( e ) = e. Y + =. where where e = eccentricit (0 < e < ). FOCI
More informationNORMALS. 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 informationPARABOLA 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 informationCET 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 informationR(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
More informationk ) 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 informationTime : 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 information1. 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)
More informationMathematics. 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 informationSketch graphs of conic sections and write equations related to conic sections
Achievement Stndrd 909 Sketch grphs of conic sections nd write equtions relted to conic sections Clculus.5 Eternll ssessed credits Sketching Conics the Circle nd the Ellipse Grphs of the conic sections
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationKEY 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 informationEigen 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( β ) touches the xaxis 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 informationSCORE JEE (Advanced)
SLUTIN. ns. (D) L : x + y 0 S L : x + y 0 L : x + y 7 0 Point of intersection of L 0 & L 0 is (,9) Point of intersection of L 0 & L 0 is (0,) line perpendiculr to L nd pssing through (, 9) isx y + 0...
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationFP3 past questions  conics
Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions  conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd
More informationAlgebra 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 informationMATHEMATICS 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
More informationA quick overview of the four conic sections in rectangular coordinates is presented below.
MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is
More information1. 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 informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
More informationSECTION 94 Translation of Axes
94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More information15  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 informationS56 (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 informationMATH 115: Review for Chapter 7
MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol
More information1 CONIC SECTIONS While cutting crrot ou might hve noticed different shpes shown b the edges of the cut. Anlticll ou m cut it in three different ws, nmel (i) (ii) (iii) Cut is prllel to the bse (see Fig.1.1)
More informationPrerna 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 informationCoimisiún 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
More informationLinear 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
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationGEOMETRY 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 informationPolynomials 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 informationMATHEMATICS (Part II) (Fresh / New Course)
Sig. of Supdt... MRDXII(A) MATHEMATICS Roll No... Time Allowed : Hrs. MATHEMATICS Totl Mrks: 00 NOTE : There re THREE sections in this pper i.e. Section A, B nd C. Time : 0 Mins. Section A Mrks: 0 NOTE
More informationMATHEMATICS IV 2 MARKS. 5 2 = e 3, 4
MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log ()  log (3) + log () Ans. ()  log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn  in sec  in
More informationParabola Exercise 1 2,6 Q.1 (A) S(0, 1) directric x + 2y = 0 PS = PM. x y x y 2y 1 x 2y Q.2 (D) y 2 = 18 x. 2 = 3t. 2 t 3 Q.
Prbol Exercise Q. (A) S(0, ) directric x + y = 0 PS = PM x y x y 5 5 x y y x y Q. (D) y = 8 x (t, t) t t = t t 8 4 8 t,t, 4 9 4,6 Q. (C) y 4 x 5 Eqution of directrix is x + = 0 x 0 5 Q.4 y = 8x M P t,t
More informationSUBJECT: 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  + +
More information10.5. ; 43. The points of intersection of the cardioid r 1 sin and. ; Graph the curve and find its length. CONIC SECTIONS
654 CHAPTER 1 PARAETRIC EQUATIONS AND POLAR COORDINATES ; 43. The points of intersection of the crdioid r 1 sin nd the spirl loop r,, cn t be found ectl. Use grphing device to find the pproimte vlues of
More informationThomas 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 informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationPrecalculus 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 informationIntroduction. Definition of Hyperbola
Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rellife problems. Clssif
More informationEdexcel 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 informationJEE(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 CODED. Let, b nd c be three nonzero vectors such tht no two of them re colliner nd, b c b c. If is the ngle
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationTriangles 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 informationTHIS FILE CONTAINS TWO DIMENSIONAL GEOMETRY (COLLECTION # 2) Very Important Guessing Questions For IIT JEE 2011 With Detail Solution
Teko Clsses IITJEE/AIEEE Mts b SUHAAG SIR, Bopl, P (0755) 00 000 www.tekoclsses.co THIS FILE CONTAINS TWO DIMENSIONAL GEOMETRY (COLLECTION # ) Ver Iportnt Guessing Questions For IIT JEE 0 Wit Detil Solution
More informationOn 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 informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationPrerequisite 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( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationA 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 informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following suquestions: i. (D) ii..p
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bordpproved clcultors my be used A tble of stndrd
More information1Preliminary topics FINAL PAGES. Chapter 1. Objectives
1Preliminr topics jectives To revise the properties of sine, cosine nd tngent. To revise the sine rule nd the cosine rule. To revise geometr in the plne, including prllel lines, tringles nd circles. To
More informationMEP 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 informationPROPERTIES 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 informationC Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson
JUST THE MATHS UNIT NUMBE 13.1 INTEGATION APPLICATIONS 1 (Second moments of n re (B)) b A.J.Hobson 13.1.1 The prllel xis theorem 13.1. The perpendiculr xis theorem 13.1.3 The rdius of grtion of n re 13.1.4
More informationDA 3: The Mean Value Theorem
Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHSIB
` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHSIB 78 FIITJEE KUKATPALLY CENTRE: # 97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd  5 7 Ph: 66 Regd
More informationGEOMETRICAL 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 informationAnswers for Lesson 31, 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 ; smeside int. '. nd with
More informationQUADRATIC 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 informationJUST THE MATHS SLIDES NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 13.12 INTEGRATION APPLICATIONS 12 (Second moments of n re (B)) b A.J.Hobson 13.12.1 The prllel xis theorem 13.12.2 The perpendiculr xis theorem 13.12.3 The rdius of grtion
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More informationYear 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. Bordpproved clcultors my be used A tble of
More informationSimple Harmonic Motion I Sem
Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.
More informationMathematics 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 informationELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000]
ELLIPSE. If the ltus rectum of ellipse e equl to hlf of its mior is, the its eccetricit is [Krtk CET 000] / / / d /. The legth of the ltus rectum of the ellipse is [MNR 7, 0, ] / / / d 0/. Eccetricit of
More informationUS01CMTH02 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
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationCalculus 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 informationA toolbox. Objectives. Defining sine, cosine and tangent. 1.1 Circular functions
C H P T E R 1 toolo Ojectives To revise the properties of sine, cosine nd tngent To revise methods for solving rightngled tringles To revise the sine rule nd cosine rule To revise sic tringle, prllel
More informationSet 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 informationAnswers 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
More informationBRIEF 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 informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationLoudoun 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 information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationCHAPTER 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 informationy = 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 informationAlg. 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 informationTHREE DIMENSIONAL GEOMETRY
MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es
More informationNat 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 informationAPPLICATIONS 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 informationMinnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017
Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.
More informationCalculus 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