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1 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 pper contins two sections. Section contins ten objective response questions. Section contins six free response questions. All questions my be ttempted. Bord pproved clcultors my be used A tble of stndrd integrls is provided t the bck of this pper. All necessry working should be shown in every question. Answer ech question in the booklets provided nd clerly lbel your nme nd techer s nme. Section -ll questions mrk ech Section - Q-6, 5 mrks ech This pper hs been prepred independently of the Bord of Studies NSW to provide dditionl exm preprtion for students. Although references hve been reproduced with permission of the Bord of Studies NSW, the publiction is in no wy connected with or endorsed by the Bord of Studies NSW.

2 This pge is left blnk

3 Section 0 Mrks Answer on sheet provided.. Wht is the exct vlue of tn 0? (A) (B) (C) (D). Wht is the eqution of the norml to the curve y = x 4x t (, )? (A) x + y 7 = 0 (B) x y 7 = 0 (C) x y 5 = 0 (D) x + y + 5 = 0. Wht is the vlue of n = 4 n? (A) 576 (B) 0 (C) 0 (D) 6 4. Wht is the size of ech interior ngle in regulr octgon? (A) (B) 80 (C) 5 (D) 80

4 5. Which of the following is the point of intersection of the two lines x 4y + 6 = 0 nd x y = 0? (A) (0,0) (B) (-, ) (C) (0,9) (D) (,0) 6. Wht re the solutions of the eqution 4 x 5 x + 4 = 0? (A) x = 0, (B) x =, (C) x =,4 (D) x = 4, 5 7. Consider the series z = 5 5. How mny terms re there in this series? (A) 5 (B) 6 (C) 5 (D) 6 8. Which of the following is equl to sin? (A) (B) (C) (D) tn 90 cos 90 sin 80 sin 60 4

5 9. Which of the following describes the re given in the grph bove? (A) (B) sin x cos x dx sin x cos x dx (C) 4 cos x sin x dx (D) 4 4 cos x sin x dx 5

6 0. Which of the following describes the region given in the grph bove? (A) y e -x, x + y 4 (B) y e x, x + y 4 (C) y e -x, x + y 4 (D) y e x, x + y 4 6

7 Section Question (Strt new Booklet) Mrks () Fctorise completely4x. (b) Solve x + 6 = (c) Solve 0 x = 78, correct to 4 deciml plces (d) Drw the grph of x + 4x + y = 0 (e) A(-,4) nd B(6, ) re points on the number line. (i) Clculte the grdient of the line AB. (ii) Hence show tht the eqution of the line AB is x + 4y = 0 (iii) Find the distnce between the x nd y intercepts of the line AB. (iv) On the sme grph show the region described by x + 4y > 0, x 0, y 0 7

8 Question (Strt new Booklet) Mrks () Differentite the following: (i) x e x (ii) + sin x 4 (b) (i) Evlute e 5 x dx (ii) Evlute 0 x + x dx (c) An AP hs first term of nd lst term of 6. If there re terms in the series, find the sum of the series. (d) The ngle of elevtion of the top of tree BT when viewed from point P is 0. Not To Scle T After wlking 00m directly towrds the tree one rrives t Q where the ngle of elevtion is 4 8. P 0' 00m 48' Q B Find the height of the tree to the nerest centimetre. 8

9 (e) Copy the following grph into your nswer booklet nd on the sme grph drw the function y = f (x) 9

10 Question (Strt new Booklet) Mrks () Solve cos = in the domin 0 (b) The grph given is in the form y = A sin (x + ). Find the vlues of A nd. (c) Given the prbol x + (m )x + 4 = 0, find the vlues of m for which the prbol hs no rel roots. (d) If nd re the roots of the qudrtic eqution x + 4x 8 = 0, clculte: (i) + (ii) (iii) + (iv) + (e) In the tringle ABC, M is the midpoint of AC. Prove tht M is equidistnt from ll three vertices of the right ngle tringle. 0

11 Question 4 (Strt new Booklet) Mrks () Find the eqution of the tngent to the curve y = x e x t the point (,e). (b) Consider the prbol y = x + (i) Find the coordintes of the vertex nd focus of the prbol. (ii) The re between the prbol nd the line y = 6 is rotted bout the y-xis. Clculte the volume of the solid formed by this rottion leving your nswer in terms of. (c) Clculte the pproximte re (to two deciml plces) between the curve y = ln x, the x-xis nd the line x =, using the Trpezoidl Rule with four function vlues. 4 Question 4 continues on pge

12 Question 4 continued (d) (i) Clculte 0 4 f(x) dx (ii) Explin why 4 8 f(x) dx = 0 (iii) Wht is the vlue of if f(x) dx = -6

13 Question 5 (Strt new Booklet) Mrks () Rdioctive mteril is decying ccording the function R = R 0 e kt. There is initilly kg of the mteril nd fter 0 yers there is 0.95 kg of the mteril remining. (i) Clculte the vlue of R 0 nd k in exct form (ii) Determine the hlf-life of the mteril (b) A prticle is trveling with the ccelertion in terms of time given by the expression.. x = 4 e -t. The prticle is initilly t rest. (i) Explin why the prticle moves in positive direction for t > 0 (ii) Find n expression for the velocity of the prticle. (iii) Find the vlue of the velocity s the ccelertion pproches zero. (c) A couple is wishing to buy home for \$ They tke out lon t % p.. interest compounded monthly. The term of the lon is 5 yers, with repyments pid monthly. (i) Show tht the fter the second repyment hs been mde, the mount outstnding is given by the expression. A = (.0) M(.0) M where M is the mount of the monthly repyment. (ii) Clculte the vlue of M. (iii) Insted of pying the mount in (ii) for the lon repyment, the couple pys \$50 more on their lon so tht they will py the mount in less time. By pying this extr money per month, how mny months does the couple sve on their home lon?

14 This pge is left blnk 4

15 Question 6 (Strt new Booklet) Mrks () Consider the curve y = x 5 x x +. (i) Find the sttionry points of the curve nd determine their nture. (ii) Show tht there is n inflexion point t x = 4 (iii) Sketch grph of the function for the domin - x 6. (b) A lot of lnd hs the form of right tringle, with perpendiculr sides 60 nd 80 metres long. (i) Show tht r = 4 x nd s = 4 x (ii) Show tht y = 00 5 x (iii) Find the length nd width of the lrgest rectngulr building tht cn be erected, fcing the hypotenuse of the tringle. (c) The centres of two circles re 7 cm prt, with one circle hving rdius of 5 cm nd the other rdius of cm. Find the re of their intersection. 5

16 STANDARD INTEGRALS x n n dx x, n ; x 0, if n 0 n dx x ln x, x 0 e x x dx e, 0 cos x dx sin x, 0 sin x dx cos x, 0 sec x dx tnx, 0 sec x tnxdx sec x, 0 x dx tn x, 0 x dx x sin, 0, x dx lnx x, x 0 x dx lnx x x NOTE : ln log x, x 0 x e 6

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