Ego EAVING CERT SOLUTIONS 2022 TO 2018

LEAVING CERT HIGHER MATHS PAPER 1 SOLUTIONS

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atett BO TRUE (iii) The circle c passes through the points z, iz, and 0, as shown in the diagram below (not to scale). z and iz are endpoints of a diameter of the circle. Find the area of the circle c in terms of m. There is space on the next page for your solution.

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= la3a 0 5 est, 4* se ) 3 s , 3 I5_ = Se 1S 153 eS S3 -get. 'rice tied A-s (ii) Use the equation in y in part (b)(i) to find the two possible values of k. Give each value in the form In p or in p, where p E N. 1 Is .1v ° 111) r ic Cs, asz • US $ •Iss I 1 K k4 3 — 5 t stAse, L.2 fM645 ED s3 a 1 .•,- 0 rests 0 I 7.:.• &ri. sj E.,- t ‘,#1 „ems 3 a) eL—• c 1lc. •:- • 613 Kw:. 1 4‘, et,...

1 is a factor of f (x) . Find the three values of x for which f(x) = 0.

00 Find the range of values of x for which f' (x) is negative, correct to 2 decimal places. 3 °IL tit -1•471/4i i- ca o fe—ik2,6 far 14;e: est* 3t-ggeb4--Iszo •RD1-4eiRe s) 441 — 240 ar— c) I g IC ss • su ISO

1 fr 449-sefie iTh .3('44) fts 60 ED/ 60.h) f(*) Lp( -F 14 11 1 e 0 4•6‘4A1)*-- f(6) ec 164) tots is (b) A rectangle is expanding in area. Its width is x cm, where x 6 R and x > 0. Its length is always four times its width. Find the rate of change of the area of the rectangle with respect to its width, x, when the area of the rectangle is 225 cm 2 . There is space for more work on the next page. 44)( 0 LOC ZetS 14 an x 4 42C cel :Cs (8 zr (f• go.is CD el: cites :41•

(c) The graph of a cubic function p (x) is shown in the first diagram below, for 0 C x 5. 4, x C IR. The maximum value of p' (x) in this domain is 1, and pi(0) = -3, where pi(x) is the derivative of p (x) Use this information to draw the graph of pi(x) on the second set of axes below, for 0 < x < 4, x E R.

I D S ( 1 . ( ) (b) Find h'(x). esI (c) Find h'(2), and explain what this value means in the context of Hannah's heart-rate. h12) : Hi- ÷ S Explanation: frn cas 10-e CCIASIa

The graph below shows y = Mx), where 0 C xC 8, x E R. (d) Find the least value and the greatest value of Mx), for 0 < x < 8, x E R. Use calculus in your solution. You may also use information from the graph above, which is to scale. anolP )11--% 5 ege o • k 60 salt c>1 re 5 l.st) 3yr az I tr S caNicr iliessre X 4.1111.1 411e t v 4-loS )e 4- 3- in—sah Se / zi6(-3 -Hos asse6 es) '7• -41-3Pro Least value of h(x) S gs Greatest value of h(x): k

I S Oft yr° eie Bruno, Karen, and Martha start a training session at the same time as Hannah. All of their heart-rates are measured in BPM. (i) For the first 8 minutes of the session, Bruno's heart-re ate, b (x), is always 15 BPM more than Hannah's heart-rate. Use this information to write bix) in terms of h' (x) , where 0 < x < 8, x E R. Ic 'es 6)11- CgAleisi 2,0 (ii) For the first 8 minutes of the session, Karen's heart-rate Hannah's heart-rate. , k (x), is always 10% less than Use this information to write kix) in terms of (x) , where 0 < x < 8, x E ))1° • 9 4iO4.2-- sex-4146 Adi

d where a, b, ci, d E IR, for 0 < x < 10. Sc ) 44 t oS

QUESTION 8 LCHM N 2022 APPLIED ALGEBRA WITH GRAPHS A Ferris wheel has a diameter of 120 m. When it is turning, it completes exactly 10 full rotations in one hour. The diagram above shows the Ferris wheel before it starts to turn. At this stage, the point A is the lowest point on the circumference of the whee height of 12 m above ground level. I, and it is at a The height, h, of the point A after the wheel has been turning for t minutes is given by: h(t) = 72 60 cos Er t) 3 where h is in metres, t E R, and n radians. Jr at 1St 3 (a) Complete the table below. The value of h(1) is given. vie CvsCo) 42-be 6) 42-4017 12_ 72- 400 A Cz) Ati

140 120 100 80 60 40 20 1 2 3 7 8 (b) Draw the graph of y = h(t) for 0 5. t 5_ 8, t E R. (c) Find the period and range of h(t). Period Range = [ 1 ,1202- (d) During a 50-minute period, could be higher than 42 m? what is the greatest number of minutes for which the point A

(e) By solving the following equation, find the second time (value of t height of 110 m, after it starts turning: ) that the point A is at a 72 60 cos (Iir= 110 Give your answer in minutes, correct to 2 decimal places. e 4 Gy:ct i 1 j 4 3 2, sz. / 4 3 MINESIISS.11.1110 6 V 6

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QUESTION 9 LCHM P1 2022 SEQUENCES Alex gets injections of a medicinal drug. Each injection has 15 mg of the drug. Each day, the amount of the drug left in Alex's body from an injection decreases by 40%. So, the amount of the drug (in mg) left in Alex's body t days after a single injection is given by: 15(0•6) t where t E lit (a) Find the amount of the drug left in Alex's body 2•5 days after a single 15 mg Give your answer in mg, correct to 2 decimal places. injection. :2*Sas Is (b) How long after a single 15 mg injection will there be exactly 1 mg of the drug left in Alex's body? Give your answer in days, correct to 1 decimal place. • (6 ) asz At.4 Os. esi 3 es. Itik 'MSS S42 8 PA- • 061.

15(0-6) 3 ceelvvo 0-• rita-&t C es 524,4100 Oicrt fri1/40 4411)4AA 1 bak ateeatectotes ti 4D /e ekia%, 00V tht Anovi- itcaj "'ay° de' (d) Find the total amount of the drug in Alex's body immediately after the 10th injection. Give your answer in mg, correct to 2 decimal places. Lf) 15 (.* G j ---- apvit 7 n ® s n, a cfrr i at. lc i-" 117 r: • ‘ ___----------e t I.) a 31°.; issi:±•„•*‘ s , i fr.:, %to i -.— 6 8 , a' • It :7 -51")2 _sesso (e) Use the formula for the sum to infinity of a geometric series to estimate the amount of the drug (in mg) in Alex's body, after a long period of time during which he gets daily injections. M° •\,%- Jr •r.. is , 34-ts ‘ az e_m-c-c„ 441 % This question continues on the next page. 1 id

• (0 Jessica also gets daily injections of a medicinal drug at the same time every day. She gets d mg of the drug in each injection, where d E I. Each day, the amount of the drug left in Jessica's body from an injection decreases by 15%. (I) Use the sum of a geometric series to show that the total amount of the drug (in mg) in Jessica's body immediately after the nth injection, where n E N, is: 20c/(1 0•85) 3 rim d• •tm ci/(15) 4-i‘• cl(ss) -2!) S -us a(br a l 45 d 6 0 ta, 1 en •gs WNW sens *is 76 0 miza /cad /I- effsm) r is 2oci

(if) Immediately after the 7th injection, there are 50 mg of the drug in Jelsica's body. Find the amount of the drug in one of Jessica's daily injections. Give your answer correct to the nearest mg.

1) for 0 < t < 12 and t E R. (a) Find the proportion of the digits recalled correctly after 3 hours, according to this model. Give your answer correct to 2 decimal places. • 12. r • b2, --"yaLl • -4= (b) After how many hours would exactly 55% of the digits be recall this model? Give your answer correct to 2 decimal places. ed correctly, according to .55 •82 •ss •SL aposse• L4) 42415 72. Lee q•4I 84?

ei6) _sof 2 •0(g ets , II (ii) P' is always negative for 0 < t < 12, t E R. What does this tell you about the proportion of digits recalled correctly after t hours, according to this model? me se.4 fe4-•) is IAA"' a-sato& von of‘red-44 (A-6 (d) Use calculus to show that the graph of y = P(t) has no points of inflection, for 0 < t < 12, t E R. • f'(E C ota thero(tH ED p"(c.) tie az =Liu: ckET I-19 Fs " 1.1 4 Po id 1424 (4.0 a eeLck r e as a? ase le 2, sa ¬ pria efrituf ..*. No pliabt opettlesitt I This question continues on the next page.

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Question 3 QUESTION 3 LCHM Pit 2021 ALGEBRA (30 marks) The diagram shows a cuboid with dimensions x, y and z cm. The areas, in cm 2, of three of its faces are also shown. (a) Find the volume of the cuboid in the form afb- cm3, where a, b 6 N. vcemAt‘t tm_ x.3 n7_0\ri • arC - 611644-bjege-22\rie Go z< g4j lc\ ( Zysar ic% olfirs is% t(i) zudis 343 4;%44-- Cit Kt =Ada %A:AFL CIO x F 46)\it n Craire ess ser 4'5ACIAR)

1= 35 •— 8(3m), where in C R. Give your answer in the form in = log 3 p q, where p,q C N. ED 3cLAIVIS Ill IN) 333-9/ re) 1.0 5 J 3 .4-Egeg mesi SA-7C 0Q,efeesa3 itt• tA•0653.3 `jdeb -L) doideig;ij . 411)gat1 34 Yr\ 6) ?e hex k,Jj.'

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21, ... .... is an arithmetic sequence, where p E N. (I) Find the nth term, 71n, in terms of n and p, where n E N. ern sesi a -fre/Od it/ Me-ol (c/7-_,07i 000110 owe (ii) Find the smallest value of p for which 2021 is a term in the sequence. p-F4n gai 9r• 4eap tot) #1-e n 4-t Lots( 1 v On% dufictoto 8d1 1 e-%9 cfr- cst Since 1 ••••

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(c) The graph of h(x) passes through the point (0, —2). Find the equation of h(x). te6s) st=e r O, • 0 rias C. 'La) tfr OM% AP' 4111%, zcs/ \ic

Section B Contexts and Applications 100 marks Answer any two questions from this section. Question 7QUESTION 7 LCHM Pt 2021 SEQUENCES (50 marks) The tip of the pendulum of a grandfather clock swings initially through an arc length of 45 cm. On each successive swing the length of the arc is 90% of the previous length. Swing 1 45 cm Swing 2 45(0-9) cm Complete the table below by filling in the missing lengths. Swing •1 2 3 4 5 Length of Arc (cm) 45 4o'5 729 20 4316 5 ale S D )e • = Co•13 cLt 948t4SY• r}2PX0N4 9 47-4 23cq V(0 S (ii) T= 450-9r-1is the arc length of swing n. Find the arc length of swing 25, correct to 1 decimal place. &At. w s Peel n:Le QfS)(144.7 51‘

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(b) if the length of the pendulum is 1 m, show that the angle, 0, of swing 1 of the pendulum is 26°, correct to the nearest degree. (ii) Hence, find the total accumulated angle that the pendulum swings through (i.e. the sum of all the angles it swings through until it stops swinging). Give your answer correct to the nearest degree. • GO r e••••00s**0,ear tr° Sue 01.111•■•■■1510 I Plato • ■

(iii) Hence, or otherwise, find the total distance travelled by the tip of the pendulum when it has moved through half of the total accumulated angle. Give your answer, in cm, correct to the nearest integer. piea---Q,Ca ,

5. 3° 7- ta it *stop efab S • 30 -rs lag •4-1 P toe ele.7.,C p 00 Complete the table below and hence draw the graph of h(x) in the domain 0 C x C 75 on the grid below. x 0 10 20 30 40 50 60 70 75 h(x) 0) 30 51s 21 0 5 21'875 taw air asp air adi Nut %0 %or %.10 %if

(b) The function h(x) can be used to model the height above leve I ground (in metres) of a section of the path fo S the horizontal distance from a fixed point. Vowed by a rollercoaster track, where x i (I) Find h' (x), the derivative of h(x). eqb aTh ••141/44 (ii) Show that this section of the track reaches its maximum height above level ground when x = 20. est "cis * 003 x ftase **DI/J. 4-* 2og• G Pees I/OW(4 14 11.1 th 4 0 en6 Yks an is 4 bC44 4. 3 4 ne 3 4 CI tkitrna A 2) pets dmiese Lacet a (Vac a s's Saw ALE o • Y-N eo o 8b4x 4 w itou c (s 4O, Lo gs c 2 el et ep ti Q:6) .004x. flit •06 G (0). • (us) • This question continues on the next page.

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