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Tropical Practice Chap 2

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100 views12 pages

Tropical Practice Chap 2

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32423dfds
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TOPICAL PRACTICE PAPER 1: 14 Questions [80 marks) TION A: 12 Questions [64 marks} ‘Time: 2 hours 1. Given that the roots of a quadratic equation x(2x—5)=x48 are 4-2 and 4k, Find the values of hand k. ; re {4 marks} ae : hedey 4d be? Re ee ik: 4 pecan Wefig -kek 2. Given that 4 and p +1 arethe roots of the quadratic equation x” + (¢—3)r-+24=0, where p and q are constants. 4) fle) h Pern y +3 5). mesieewen zh [let Find the values of p and q. [4 marks] SOR: he PoR : #8 Sub one lay | nS Sqe-d ul pet dy LAD nT 1: pte ® peo 998 ‘ pes oy > -2ay 3. Given a quadratic fimetion f(x) =x? +2x-+3 where h is a constant. (a) Express f(x)in the form of f(x)=a(x-p) +q in terms of h. (2 marks] (b) If h <0, state the type of tuning point and find the coordinates of the turning point in terms of h. (2. marks] (b) Find values of p and q. 9) FOd= Ler P- (2)? -7 ele $y £7, DIAGRAM | ay. 22 PUSAT TUISYEN TELITI © 2022 Diagram 1 shows the graph of function f(x)=x°+px—7 with axis of symmetry of x= and y=q isa tangent to the curve. (a) Express f(x) in the form of f(x)=(x-b) +c. (©) State what happens to the graph if p=5. [4 marks} D The gp tee | a er om te WH se of ens a a a 5 en that quadratic equation x? + 3k =x=12 has a positive and a negative root, Find the range of values of aiuto Leen ay at aed [2 marks} Y a Por de detk 22. Uy ye Oe. Weae cikeee atk oo Sub OY my QO), nee Meek = (en)? he x drake P-lea] * W-Ie-dnt|-k se | eal -kee | bea 6 ‘ [3 marks} 7. (a) Solve the quadratic equation (x~ 2)’ = ke+ 7 correct to 3 decimal places. leeays due wT Be dpe + fT 1-0 tee wb -dee Lye 4 ia ue Hos x: tvsfar 2) | | | (b) The graph of quadratic function y= m+3—h(x—2k)' , where h, k and m are constants, has a maxiraum point at (AAs). Find {i) the range of values of h, (ii) the values of & and m B marks] Job LAN) na Jno ste, ke Jk aS Uk, mes) = (4, Is) hoo, ee onze is me. meh 8. (a) Given that f(x)=9—2x—3x". Find the range of values of x for which f(x) >1. [B marks] 4-m-3e >) Op Be--8 NY Z x te In F 40 3 (3x-4) bea) <0 eas (b) On the given axes below, sketch the graph of f(x) =~3(x-1)°-2. LB marks} Ps The 2-3) uel yroris, Reo le) = -3(0-1 2-f DITIONAL, MATHEMATICS FORM 4 KSAT TOPICAL PRACTICE 2 QUADRATIC FUNCTIONS 9% (a) Quadratic equation x? + eg Ohas roots of «rand /7. Find in terms of p,q and x, a quadratic equation with Teste roots of and 7 Given your answer in the form of ax? + e+e 0 where a, b and ¢ are constants, [4 marks] eA ag Mew Pek Por ’ 1 New 5 Noy 4 (b) Quadratic equation Ax® +(2K =1)x+4=3=0 has roots of a and # wherec # f. Find the range of values of k. [2 marks} D-H Se 0b) AOVUEY> c 8 - HEAD MEE. ak > © SHEER Hk «1 > 0 Th+! oc Tho>-] I Set 10.(a) Given that f(x)=k—4x—2" has a maximum value of +1. Express / in terms of k. [B marks] £00 =~ te tk TOS nex? Hak sh ~ Dos Weak i Cosa sak (ea) a Hak (b) Find the range of values of x for (2x+1)? <19x-8. [3 marks] One om 8 Waa | Mu -8 Wd te Me tlt 0 Ya - Wn e160 C3) (4x -3) £0 1@ 3 Diagram 2 shows the graph of function f(x) = p—(x-2)' where p is a constant, Find (the vi ofp and g. (ii) the equation of the on when it is reflected in the x-axis. [3 marks] U) (D0 -De-a)?4 p cn ‘ G {One ep, x09 hey x le)=p-(e-2) be 4 DIAGRAM 2 4 x: Ef Sub (6,0) o: -[b-y)'+ p a ee a (b) Given that a quadratic function f(x) =(1—p)x? —4x+2., where pis a constant. By finding the range of values of pp, determine whether the function f (x) can be always negative. [4 marks} ay We Mae < 6 aN mae’ C4) = 4 U-p)OL< © By ...u)| 6 ae are, 4? - 4a < 0 : rt <0 Ne scl Fs bp co FU) amet beghuey: v Bp<-8 e<-l 12. (a) A quadratic equation has roots of « and . Given that the sum of the roots is 2 and the sum of squares of the roots is 10. Express the quadratic equation in the form of ax’ +bx-+c=Owhere a, b and c are integers. [3 marks] Or eae b= Jop wap? sic ape -3 Wray = J 7 ' ‘ Codes (a) 6 4 B24 Va, F 6 2 Wen Beso o> 04 Dap . 4s 104 Mp (b) Quadratic function y = x* +2px—p+4, where p is a constant, has a minimum value of —2. Find the possible values of p. (4 marks] ge Gere) pp a 4 Yuin cpt pa Henn PP+p 4-220 Ptp bro Up-al(pszdze P26 peace Betis ps3 13.(a) Kamal wants to plant some vegetable on a rectangular piece of land with an area 10m", Given that the length of the land is 3 m longer than twice the length of its breadth, Calculate the fength and breadth of the piece of land, in m correct to 2 decimal places, [4 marks} any i 0 4 7 ant de 1. Gon £108 mee Ath = 1. 608 cle J lowgth © OCI Len) 43 Par or) eee ee) () Given that f(x) =x? +/+ and f(x) <0 when p We dyna YP <0 | Sub Li) Tal 4 0 | . d heures eh golet)? =k he - 3, v2 Pea ai o | Jp eee 14.(a) Quadratic equation 3x” +(p—1)x+2q=0 where p and q are constants, has roots ofS and a where a+ B= 6andaf =8. Find the values of p and g. [4 marks] Sop): + ea) aie fea : dtp pal ae 8 2 rae t pal a et epeerels ty (b) A ball is thrown vertically upwards from a point 2 m vertically above point O on the ground, Its height, in m above ground after / seconds can be modelled by the quadratic function h(t). The ball reaches a maximum height of 6 mafter 1 second. (i) Express /h(t) in the form of h(t)=a(t—p) +q , where a, p and q are constants. iid Find the time taken forthe ball to reach the ground. [4 marks] () Melee 26, te J neso, 4a 20 NU) nen = 4. tp | Yl4-1)= & ab pel | Geet hlt) = alt-I eh | t-1 =f Sub (0,3) 2a (6-N*+ | thE ae Ths KA) = 44-46 , PAPER 2: 0. SECTION A: 7 Questions [50 marks} 1. Quadratic equation 3(r=2)' = 7—4p , where p is a constant, has the roots of a and J. I a—/=6, find (a) the values of a and 2 (b) the value of p. (la) 50 Hes 47 hp QO) mks stip ‘Time: 2 hours [4 ma [2 me 2. (a) Quadratic equation x(x+2k) 1k -10 where kis a constant has two equal roots, Find the possible values 0 (b) Hence, using the positive value of k, find the root of the quadratic equation. b [@ xGar): 3k =I G) ked Xt 4 Dhe = 3-6 Sus in U)) x? + Mex - 3k + 10 = 0 x490)x - 3D)4 100 bee Hac 20 tte = b 4 le = 6 Oh)*- 40) (Bele) = 6 Xt Heo CH + Dk 2 4050 977 OG) O44) 0 Ry 2k odo 20) (3) Ue sje | k-2:0 kes | kd ke-S | 3. A quadratic function is given by y= 2x? 44x45. (a) Express y in the form y= a(x—p)’ +g. Hence, state the coordinates of the maximum or the minimum point of the curve. [3 marks} (b) Hence, sketch the graph of y = 2x? + 4x45 showing clearly the turning point and x- intercept and y- intercept if any. (2 marks} (c) Find the range of values of x given that_y 211. [2 marks} ©) dese? one ce ‘ de ty ‘pe Bee Tee rn ni GEN eH 26 Xe} sad oe F xS-2 x21 ¥ POs, ey ye 4. Given that the minimum value for quadratic function (x)=? —2kx+k* —3k ish—S where h and & are constan (a) By completing the square, show that h+3k=5. B ma (b) Find the values of h and kif the equation of axis of symmetry is x= /—1. [4 ma ) 40) = ONY Bs PSE i Seah | FOdmin s -3k 2 hee | kearo.. Zo [G)_x=k Ee | keh). OD [sre Sub om 0) he 3-1 es k= he Bh 2S > eae hedh = S23 ary ‘ ted S$. (a) Given that quadratic equation 3x?) (i) Find the values of e and /? (ii) Hence, expression of p in terms of g ® G) &xt O-?) ware New ay eg uasiew 4 2 ' dl the quadratic equation with the roots of a! and 0 has the roots of a and) where a > fi [5 marks} (b) Given that quadratic equation, p(y? 4 1)= dye where p and q are constants, has two equal roots. Find an [3 marks\ et de Cy Pak Mat dug a hole in his garden and poured a layer of gravel and alayer of soil in it. Diagram 1 shows the cross-section of the hole in the shape of a parabola. The depth, in m of the hole, d(x) can be ue. modelled by a quadratic function d(x —x, where x is the horizontal distance, in m from point O on the ground level. (a) Find the length, in m of the surface of the hole. (2 marks} (b) By completing the square, find the maximum depth, in m of the hole. 1B marks} DIAGRAM 1 (c) Given that the layer of gravel is at least 0.75 m below the ground. Find the range of values of x. [3 marks} (a) tW=6 (4) 800 = =) Wid = O15 4 ] C4? - x26 =p Le-n) 47 Woes -2 cal x= tne i 3) =| R= 3 £0 — HOC 4) = 6 063) Ux-1) £6 4c KY Ma Leo = 5 a 7. (a) Given that one the roots of quadratic equation 2x! = k(x 1) where & is a constant is half the value of the other root, Find (i) the roots of the equation, (ii) the value of k {5 marks] (b) Given that the curve y= 3x" 4 2mx + m4 6, where m isa. constant, touches the x-axis at one point, Find the possible values of m, G) yoo dw Date kn =k Dehua ks 0 OR: 4 dot = 3h ke be r f 5 [@ GO) Say me bs 0G) as 4 —— GQ) tena Cm) = 413) Uae) 2 oo Cit - dm = 108 OYE f “ mee By IF =O 5 13 marks] SECTION B: 3 Questions [30 marks| 8. (a) Quadratic equation 3x? —2x-+6=0 has the roots of « and f’, Form the quadratic equation with the roots of 1-2a and 1-2. [5 marks} (b) Find the range of values of k if straight line y=x-+k , where k is a constant does not meet the curve 2x? +y? +3x+ yt+1=0. [5 marks] ©) Seas dap @) Se 0) sO), Dros Gk) 4 Serer ka] = 6 :F Pek ap £ Deas bal ee a cal mA 3x7t Dhet He 4 ea kel s 0 New So = |-Jmsl = 2p at Bet Ok etkt eke} 2 e re: = 1-24) Okeay 4 (et katy co eT) 4le + Ibk + |b - Bk - Hk -11 <0 22 C-Bho + 4k 44 <0) CD New PeR = U-34)(I-3p) | Aaa aT Ide - lat Hoe Okst) Uke-) > @ = [-A(pta) + Map = I-03) + 40) z = New equation G2 - Fx 422 0)? 32- Ie +8 = 6 9. (2) (0,9) ve s(x) DIAGRAM 2 [DFO nae Diagram 2 shows the graph of y= f(x) -The graph has a ‘maximum point at P and intercepts the y-axis at Q- (a) Express f(x) in the form of f(x) = a(x= A) and k are constants, (b) Find the range of values of x for which 0< f(* (©) State what happens to the graph if a= =I +k, where a, [3 marks) <9 [6 marks) [1 mark] ass = [toy = aun) | Subilae _4ea(onyst | aan Seer aes =e. ST ary incuuns £09 = -3 OCH)? + [a i =a o> G) FO) 26 so [301+ be I COE “3ex¢-0 os keel | Oey 4 wal 2-2 ea gd | x2-3 ay | 3 x 10. Given a quadratic fiction, f (x)= p+12x+gx* =10-2(x+r)' where p, q and are constants. (a) Find the values ofp, q and r. (b) Solve the equation f (x) =0 giving your answer in the form of a/b where a and b are constants. (c) Sketch the graph of y= f(x), with the turning point and intercepts clearly indicated. (d) Hence, find the range of values of k for which the equation 2f(x)+&=Lhas real roots. [4 marks] [2 marks] [2 marks} [2 marks} ©) {09+ [o-2 G84 Ie +P) G) Je ale-3y = 0 = 10 ~ Biche Ai = Dr? 20a)" = Io =n dete Ht 4 le = Oe rat = F 5 ge + Ihet p ean le é rueae X= 32 fe i ei Me = ih tee OO flxy = 2063)4 Ie Constant + p= lo - 21-3)” fl) mex = 1G KES = 0-18 ywis Keo ye -F 2-8 ye le - 2(x-3)° L (4) 9 fO9= I-k foxy = EK rk z ne Ss a a Ink < )¢ k In A \ ne

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