Coimisiún na Scrúduithe Stáit State Examinations Commission

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David McGee
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M 9 Coimisiú a Scrúduithe Stáit State Examiatios Commissio LEAVING CERTIFICATE EXAMINATION, 006 MATHEMATICS HIGHER LEVEL PAPER 1 ( 00 marks ) THURSDAY, 8 JUNE MORNING, 9:0 to 1:00

1 M 9 Coimisiú a Scrúduithe Stáit State Examiatios Commissio LEAVING CERTIFICATE EXAMINATION, 006 MATHEMATICS HIGHER LEVEL PAPER 1 ( 00 marks ) THURSDAY, 8 JUNE MORNING, 9:0 to 1:00 Attempt SIX QUESTIONS (50 marks each) WARNING: Marks will be lost if all ecessary work is ot clearly show Aswers should iclude the appropriate uits of measuremet, where relevat Page 1 of 5

2 1 (a) Fid the real umber a such that for all x 9, x 9 = x + a x (b) f (x) = x + mx 17x +, where m ad are costats Give that x ad x + are factors of f (x), fid the value of m ad the value of (c) x t is a factor of x px qx + r (i) Show that pq = r Express the roots of x px qx + r = 0 i terms of p ad q (a) Solve the simultaeous equatios y = x 5 x + xy = (b) (i) Fid the rage of values of t R for which the quadratic equatio (t 1) x + 5tx + t = 0 has real roots Explai why the roots are real whe t is a iteger (c) x = 1 b ad f ( x) g( x) 1+ x = b, where b is a positive real umber Fid, i terms of b, the value of x for which f ( x) = g( x) Page of 5

3 (a) Give that z = + i, where i = 1, fid the real umber d such that d z + is real z (b) (i) Use matrix methods to solve the simultaeous equatios 4x y = 5 8x + y = 4 Fid the two values of k which satisfy the matrix equatio k ( 1 k ) = 11 (c) (i) Express 8 8 i i the form r ( cosθ + isiθ) Hece fid ( 8 8 ι& ) (iii) Fid the four complex umbers z such that 4 z = 8 8 i Give your aswers i the form a + bi, with a ad b fully evaluated 4 (a) (4 6) are the first terms of a arithmetic series S, the sum of these terms, is 160 Fid the value of 9 (b) The sum to ifiity of a geometric series is The secod term of the series is Fid the value of r, the commo ratio of the series (c) The sequece u 1, u, u, K, defied by u 1 = ad u + 1 = u +, is as follows:, 9, 1, 45, 9 (i) Fid u 6, ad verify that it is equal to the sum of the first six terms of a geometric series with first term ad commo ratio Give that, for all k, u k is the sum of the first k terms of a geometric series with first term ad commo ratio, fid u k = 1 k Page of 5

4 5 (a) Fid the value of the middle term of the biomial expasio of x y y x 8 (b) (i) Express ( r + )( r + ) 1 i the form A B + r + 1 r + Hece fid ( r + 1 )( r + ) r = 1 (iii) Hece evaluate r = 1 1 ( r + )( r + ) (c) (i) Give two real umbers a ad b, where a > 1 ad b > 1, prove that log a log b b 1 1 Uder what coditio is + = log a log b a b a 6 (a) Differetiate x ( x + ) with respect to x (b) The equatio of a curve is y = x x 9x + 8 (i) Show that the curve has a local maximum at the poit (0, 8) 4 Fid the coordiates of the two local miimum poits o the curve (iii) Draw a sketch of the curve d (c) Prove by iductio that ( x ) dx = x 1, 1, N Page 4 of 5

5 7 (a) Takig x as the first approximatio to the real root of the equatio 1 = x + x 9 = 0, use the Newto-Raphso method to fid x, the secod approximatio (b) The parametric equatios of a curve are: x = cosθ cos θ si si π y = θ θ, where 0 < θ < dy dx (i) Fid ad dθ dθ dy 1 Hece show that = dx ta θ (c) Give l, + x dy a y = fid ad express it i the form 9 x dx b x 8 (a) Fid (i) xdx x e dx + 1 (b) Evaluate (i) ( x ) π 4 x 1 dx 0 si5θ cosθ d θ (c) The diagram shows the graphs of the y = f x ad y = g( x), curves ( ) where f ( x) = 1 x ad g( x) = 9x y y = g(x) (i) Calculate the area of the regio eclosed by the curve y = f ( x) ad the x-axis Show that the regio eclosed by the curves f ( x) y = g x has half that area y = ad ( ) x y = f(x) Page 5 of 5

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De Moivre s Theorem - ALL

De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.