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 from Section A nd ONE question from Section B. Ech question crries 50 mrks. WARNING: Mrks will e lost if ll necessry work is not clerly shown. Answers should include the pproprite units of mesurement, where relevnt. Pge of 7
SECTION A Answer FIVE questions from this section.. () Circles S nd K touch externlly. Circle S hs centre (8, 5) nd rdius 6. Circle K hs centre (, 3). Clculte the rdius of K. K S () Prove tht the eqution of the tngent to the circle t the point (, y ) x is xx + yy =. r x + y = r Hence, or otherwise, find the two vlues of such tht the line 5 x + y = 69 is tngent to the circle x + y = 69. A circle psses through the points (7, ) nd (7, 0). The line x = is tngent to the circle. Find the eqution of the circle.. () Copy the prllelogrm oc into your nswerook. Showing your work, construct the point d such tht d = + c, where o is the origin. o c () p = 3 i + 4 j. q is the unit vector in the direction of p. Express q nd q in terms of i nd j. Express i j in the form k q + l q, where k, l R. u = i + 5 j nd v = 4 i + 4 j. Find cos uov, where o is the origin. = ( k) u + k v, r where k R nd k 0. Find the vlue of k for which uov = vor. Pge of 7
3. () The line L : 3x y + 7 = 0 nd the line L : 5x + y + 3 = 0 intersect t the point p. Find the eqution of the line through p perpendiculr to L. () The line K psses through the point ( 4, 6) nd hs slope m, where m > 0. Write down the eqution of K in terms of m. Find, in terms of m, the co-ordintes of the points where K intersects the xes. The re of the tringle formed y K, the x-xis nd the y-xis is 54 squre units. Find the possile vlues of m. f is the trnsformtion ( x, y) ( x, y ), where x = 3x y nd y = x + y. Prove tht f mps every pir of prllel lines to pir of prllel lines. You my ssume tht f mps every line to line. oc is prllelogrm, where [ o ] is digonl nd o is the origin. Given tht f () c = (, 9), find the slope of. 4. () Evlute sin4θ lim θ 0 3θ. () Using cosa = cos A sin A, or otherwise, prove cos A = ( + cosa). Hence, or otherwise, solve the eqution + cosx = cosx, where 0 x 360. o o S is circle of rdius 9 cm nd S is circle of rdius 3 cm. S nd S touch externlly t f. A common tngent touches S t point nd S t. Find the re of the qudrilterl cd. Give your nswer in surd form. Find the re of the shded region, which is ounded y [] nd the minor rcs f nd f. S e d f c S Pge 3 of 7
5. () The re of n equilterl tringle is 4 3 cm. Find the length of side of the tringle. () In the tringle xyz, xyz = β nd xzy = β. x xy = 3 nd xz = 5. 3 5 Use this informtion to express sin β in the form sinβ, where, N. y β β z c Hence express tnβ in the form, where c, d N. d qrst is verticl rectngulr wll of height h on level ground. s p is point on the ground in front of the wll. The ngle of elevtion of r from p is θ nd the ngle of elevtion of s from p is θ. t pq = 3 pt. θ x Find θ. θ p 3x q r h 6. () How mny three-digit numers cn e formed from the digits,, 3, 4, 5, if the three digits re ll different the three digits re ll the sme? () Solve the difference eqution u n+ 4u n+ 8u n = 0, where n 0, given tht u = nd u. 0 0 = Verify tht your solution gives the correct vlue for u. Nine crds re numered from to 9. Three crds re drwn t rndom from the nine crds. Find the proility tht the crd numered 8 is not drwn. Find the proility tht ll three crds drwn hve odd numers. Find the proility tht the sum of the numers on the crds drwn is greter thn the sum of the numers on the crds not drwn. Pge 4 of 7
7. () How mny different groups of four cn e selected from five oys nd six girls? How mny of these groups consist of two oys nd two girls? () There re sixteen discs in ord-gme: five lue, three green, six red nd two yellow. Four discs re chosen t rndom. Wht is the proility tht (iv) the four discs re lue the four discs re the sme colour ll four discs re different in colour two of the discs re lue nd two re not lue? On st Septemer 003 the men ge of the first-yer students in school is.4 yers nd the stndrd devition is 0.6 yers. One yer lter ll of these students hve moved into second yer nd no other students hve joined them. Stte the men nd the stndrd devition of the ges of these students on st Septemer 004. Give reson for ech nswer. A new group of first-yer students egins on st Septemer 004. This group hs similr ge distriution nd is of similr size to the first-yer group of Septemer 003. Stte the men ge of the comined group of the first-yer nd second-yer students on st Septemer 004. Stte whether the stndrd devition of the ges of this comined group is less thn, equl to, or greter thn 0.6 yers. Give reson for your nswer. Pge 5 of 7
SECTION B Answer ONE question from this section. 8. () Use integrtion y prts to find x lnxdx. () Derive the Mclurin series for f x) = ln( + x) 3 contining x. Use those terms to find n pproximtion for ln. 0 ( up to nd including the term Write down the generl term of the series f (x) nd hence show tht the series converges for < x <. A cone hs rdius r cm, verticl height h cm nd slnt height 0 3 cm. h 0 3 Find the vlue of h for which the volume is mximum. r 9. () z is rndom vrile with stndrd norml distriution. Find ( < z < ) P. () During mtch John tkes numer of penlty shots. The shots re independent 4 of ech other nd his proility of scoring with ech shot is. 5 Find the proility tht John misses ech of his first four penlty shots. Find the proility tht John scores exctly three of his first four penlty shots. If John tkes ten penlty shots during the mtch, find the proility tht he scores t lest eight of them. A survey ws crried out to find the weekly rentl costs of holidy prtments in certin country. A rndom smple of 400 prtments ws tken. The men of the smple ws 30 nd the stndrd devition ws 50. Form 95% confidence intervl for the men weekly rentl costs of holidy prtments in tht country. Pge 6 of 7
0. () Show tht {0,, 4} forms group under ddition modulo 6. You my ssume ssocitivity. () R 90 o nd S M re elements of D 4, the dihedrl group of squre. List the other elements of the group. Find C ( S M ), the centrlizer of S M. d o L c M K N A regulr tetrhedron hs twelve rottionl symmetries. These form group under composition. The symmetries cn e represented s permuttions of the vertices,, c nd d. c d c d X =,, o is sugroup of this tetrhedrl group. c d d c Write down one other sugroup of order. d Write down sugroup of order 3. Write down the only sugroup of order four. c. () Find the eqution of n ellipse with centre (0, 0), eccentricity 6 5 nd one focus t (0, 0). () f is similrity trnsformtion hving mgnifiction rtio k. A tringle c is mpped onto tringle c under f. Prove tht c = c. g is the trnsformtion ( x y) ( x, y ),, where x = x nd y = y nd > > 0. C is the circle x + y =. Show tht g(c) is n ellipse. L nd K re tngents t the end points of dimeter of the ellipse g(c). Prove tht L nd K re prllel. Pge 7 of 7
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