z + az + bz + 12 = 0, where a and b are real numbers, is

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1 016 IJC H Math 9758 Promotional Examination 1 One root of the equation z = 1 i z + az + bz + 1 = 0, where a and b are real numbers, is. Without using a calculator, find the values of a and b and the other roots. Without using a calculator, solve the inequality ( x ) x [4] 1. [4] At the beginning of an experiment, an inverted circular cone is filled with sand. The base radius of the cone is 5 cm and the slant height is 1 cm. Sand is leaking through a small hole at the bottom of the cone at the rate of 7 cm s 1 at the instant when the depth of the sand in the cone is.5 cm. Find the rate of decrease of the depth of the sand at this instant. [5] [It is given that the volume of a circular cone with base radius r and vertical height h 1 is π r h.] 4 The diagram shows the graph of y = f ( x) with a maximum point at ( a, b) and a horizontal asymptote y = b where a > 0 and b > 0. The curve intersects the axes at ( 5 a,0 ), ( a,0) and ( 0, a). Sketch, on separate diagrams, the graphs of y = f x + b, [] (i) ( ) (ii) 1 y = f ( x), [] (iii) y = f ( x), [] showing in each case the equations of the asymptote(s) and the coordinates of any turning point(s) and point(s) of intersection with the axes whenever possible. 1

2 5 (a) Given that n n n ( n + 1) r =, find ( ) r= r + 5 n + 1 r= n in terms of n. Give your answer in fully factorised form. [4] 1 1 4n (b) (i) Verify that =. [1] ( n 1) ( n + 1) ( n 1) (ii) Find n r. (There is no need to express your answer as a single r = ( r 1) algebraic fraction.) [] (iii) Hence find the exact value of r. [] r = ( r 1) 6 A D E H E H B F n Fig. 1 G C F Fig. G Fig. Fig. 1 shows a piece of square card, ABCD, with sides n cm, where n is a positive constant. A rectangle EFGH, as shown in Fig., is being cut from the card such that EF = HG = x cm and BF = GC = x cm. The rectangle is rolled up to form a cylinder of height x cm, where EH and FG are the circumference of the circular ends of the cylinder, as shown in Fig.. The cylinder is placed on a horizontal table top and the volume of this open cylinder is V cm. (i) Show that V 1 ( 4 x 4 nx n x) = +. [] π (ii) Use differentiation to find, in terms of n, the stationary value of V as x varies. Determine whether the volume is a maximum or a minimum. [6]

3 7 The function f is defined by x + 5 f : x, for x R, x. 4x 4 (i) Define 1 f ( x). [] (ii) Deduce the rule of f ( x ) and state the range of f. [] (iii) Using part (i), describe the symmetry of the graph of f. [1] The function g is defined by g : x x + x, for x R, x 1. (iv) Explain why the composite function fg exists and find the exact value of fg(). [4] 8 (a) Show that, when x is sufficiently small for x and higher powers of x to be neglected, cos x a bx cx 1 sinx, where the values of a, b and c are to be determined. [] x (b) (i) Given that y = ( + 6e ), show that y dy x = e. d x By further differentiation of this result, find the Maclaurin series for x ( + 6e ) up to and including the term in x. [5] (ii) The second and third terms in the Maclaurin series for y = ( + 6e ) are equal to the first and second terms in the series expansion of e ax ln ( 1+ nx) respectively. Using appropriate expansions from the List of Formulae (MF6), find the constants a and n. [4] x

4 9 A bank has an account for investors. Interest is added to the account at the end of each year at a fixed rate of % of the amount in the account at the beginning of that year. Mary and Ben both invest money. (a) Mary decides to invest $x at the beginning of one year and then a further $x at the beginning of the second and each subsequent year. She also decides that she will not draw any money out of the account, but just leave it, and any interest, to build up. (i) (ii) (iii) How much will there be in the account at the end of 1 year, including the interest? [1] Show that, at the end of n years, when the interest for the last year has 10x been added, she will have a total of $ ( 1.0 n 1 ) in her account. [] If Mary starts investing at the beginning of 016 with an amount of $10 000, at the end of which year will she have, for the first time, at least $ in her account? [4] (b) Ben decides that, to assist him in his everyday expenses, he will withdraw the interest as soon as it has been added. He invests $y at the beginning of each year. Find the total amount of interest he will receive at the end of n years. [4] 10 A curve C has parametric equations x = asin 4θ, y = a cos θ, π where 0 < θ and a is a positive constant. (i) Sketch C, showing clearly the coordinates of any point(s) of intersection with the axes. [] π (ii) Find the equation of the tangent to C at the point P where θ =, leaving your 6 answer in exact form. [4] (iii) The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B. Find the exact area of triangle ABP. [6] 4

5 11 The curve C has equation x y = + 4x + λ 5, x + r where λ is a non-zero constant, and a vertical asymptote x = 1. (i) State the value of r and find the equation of the other asymptote of C. [] (ii) Draw a sketch of C for the case when λ < 0. [] (iii) By using an algebraic method, find the range of values of λ for which the line y = x and C have at least one point in common. [] It is now given that λ = 9. (iv) On a separate diagram from part (ii), sketch the graph of C, indicating clearly the coordinates of the stationary points. [] Another curve D has equation ( x 4) ( y 6) + = 1. 4 k (v) On the same diagram as part (iv), sketch D for the case when k =. [] (vi) Deduce the range of values of k for which C and D intersect more than once. [1] ~ ~ ~ End of Paper ~ ~ ~ 5

6 016 H Math Promo Exam Suggested Solution Q Solution 1 Since the coefficients are real, z 1 i is another root of the equation. Method 1 z1 i z1 i z 1 i z z 4 z az bz 1 0 z z z 4 0 (By inspection) Comparing coefficients of Comparing coefficients of z, The other roots are z, z 1 i a 1 b and z. Method Substitute z 1 i ( or z 1 i ) into the given eqn, 1 i a1 i b1 i 1 0 a b or equivalent 1 i 9 i 1 i 1 i 1 0 a b a b 4 i 0 Comparing the real parts, ab 4 Comparing the imaginary parts, ab 0 a 1, b See Method 1 for factorization to find the other roots. x x 1 x x x 4x 1x9 0 x x x x x < or x > or x = 1.5 +

7 Let the radius of the surface level of the sand be r cm and the depth of the sand be h cm at time t seconds. Height of the funnel = cm 5 r Using similar triangles, 1 r 5 5 r h h 1 h V r h h 4 Differentiating both sides wrt t, dv 5 dh h dt 144 dt When h.5, d V 7 dt d h 144 dt dh ( s.f.) dt the rate at which the depth of the sand falls in the funnel is.05 cm s 1 when the depth of the sand is.5 cm. 4(i) 4(ii)

8 4(iii) 5(a) n rn n n1 n 4 r r 5n 1 r 1 r1 rn n (n 1) n 1 ( n) 4 5n 1n n n 4(n 1) n 1 5n 1 ( n 1) n (n 1) n 1 (n 1) n 1 5n 1 ( n 1) 4r 5n1 n n 5n 1 5n 1 ( n 1) n 15n 1 ( n 1) 5(b)(i) LHS 1 1 ( n 1) ( n 1) ( n1) ( n1) = ( n1)( n1) 4n = RHS ( n 1)

9 5(b)(ii) n n r ( r 1) 4 ( r 1) ( r 1) r r (n) (n1) 1 1 (n ) ( n) 1 1 (n1) (n1) n (n1) 5(b)(iii) 1 1 As n, 0 4 n (n1) Thus, r r 5 ( r 1) = 144

10 6(i) 6(ii) Let r cm be the radius of the cylinder. r n x n x r V r h n x x n 4nx 4x x x 4 nx n x dv 1 1 x 8 nx n For stationary value of V, let 1x 8nx n 0 x n x n 6 0 dv 0 n n x or x (rejected nx 0) 6 Stationary value of V 1 n n n 4 4n n n 7 d V 1 4 x 8 n n When x, 6 d V 1 n n 4 8n 0 6 By nd derivative test, V is maximum when. n x. 6

11 7(i) Let y f ( x) x 5 y 4x 4xy y x 5 x(4y ) 5 y 5 y x 4y 5 x 4x 4 1 f ( x), x \ 7(ii) Since R f f f 1, \ 4 f ( x) ff ( x) x for x, x 4 1 7(iii) 7(iv) The graph of f is symmetrical about the line Rg [, ) and Df \ 4 Since R D, the function fg exists. g f y x. Method 1 fg() f ( ) = f 16 (16) 5 = 4(16) 5 = 61 You may use your GC to check the accuracy of your answer. Method fg( x) f x x = x 6x x x () 6() 5 fg() 4() 8() 5 = 61

12 8(a) 8(b)(i) 8(b)(ii) x 1 cos x 1 sin x 1 x x 1 1 x y y 1 x 1 1 x ( x) x x 6e 6e x x d y y x 6e d x y dy x e... d x Differentiat wrt x d y dy y When e x x 0 0, y 6e 9 dy 0 e 1 dy d y 0 d y 1 e d x y x x...! 1 x x... ax ax nx e ln 1 nx 1 ax... nx...! a x n x 1 ax... nx.. nx Comparing the terms, nx x n 1 nx anx n nx an x Alternative: y n 1 n 1 an x x an Sub n 1, a a e 1 dy 1 6e 6e x x x x 6e d y e y dy e d x x x

13 9(a) (i) 9(a) (ii) $1.0x Mary s account: Beginning of year End of year 1 st yr x 1.0x nd yr 1.0x + x 1.0(1.0x + x) = x( ) rd yr 1.0 x + 1.0x + x x( ) nth yr x( n ) (i) Amount at the end of nth year is n 1 10 x 1.0 n x (a) (iii) When x = 10000, n n nln(1.0) ln 1 10 n Miss Lee will have at least $ in her account at the end of 00. 9(b) Ben s account: Amt at the beginning of each year Interest received at the end of each year 1 st yr y 0.0y nd yr y 0.0(y) rd yr y 0.0(y) nth yr ny 0.0(ny) Total interest received 0.0y 1... n n 0.0y 1n y n(1 n)

14 10(i) x asin 4, y acos, where 0. 10(ii) x asin 4 4a cos 4 d dy 4acossin d asin y acos dy a sin 4a cos sin or 4a cos 4 4a cos 4 dy sin cos sin or cos 4 cos 4 When, 6 sin dy cos x asin a y a cos a 6 Equation of tangent at the point P y a x a y x a a 4 y x a 4

15 10(iii) Equation of normal at point P y a x a y x a a 5 y x a At point A, When y = 0, 0 x a 4 x a At point B, When y = 0, 5 0 x a x 5 4 a Area of Triangle ABP 1 a a a a 4 units

16 11(i) x 4x 5 y x r Since 1 r 0 r 1 x1 x 5 y x 5 x1 x1 The eqn of the oblique asymptote is x 1 is an asymptote, yx5. 11(ii) 11(iii) Let x 4x 5 x x 1 x 4x 5 x( x 1) x 6x 5 0 There are at least one common point b 4ac 0 6 4(5 ) Since 0, the range of values of is 4 0 or 0 (Accept 4 and 0 ). 11(iv) & (v) 11(vi) k > 6