MATHEMATICS - ORDINARY LEVEL

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1 M7 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 000 MATHEMATICS - ORDINARY LEVEL PAPER (300 marks) THURSDAY, 8 JUNE - MORNING, 930 to 00 Attempt SIX QUESTIONS (50 marks each) Marks may be lost if all necessary work is not clearly shown Page of 5

2 (a) Epress 400 grammes as a fraction of kilogramme Give your answer in its simplest form (b) euro = IR euro = DM Calculate the value of IR 00 in euro, correct to two places of decimals Hence, calculate the value of IR 00 in Deutschmarks (DM), correct to two places of decimals (c) A person has annual ta free allowances of IR 7400 The person pays income ta at the rate of 4% on the first IR 400 of taable income and at the rate of 46% on the remainder Calculate the amount of income ta paid on the first IR 400 of taable income Calculate the person's gross income if the total annual income ta paid is IR 538 (a) Find the value of 5 3y when = 5 and y = 3 (b) Solve for and y 3y = y = 0 (c) Write as a power of Hence, solve for the equation 43 3( 3 ) = 7 Page of 5

3 3 (a) Epress p in terms of t and k when tp k = 7k, t 0 3 (b) Show that = is a root of = 0 3 Find the other roots of = 0 (c) f ( ) = a + b 8, where a and b are real numbers If f() = 9 and f( ) = 3, find the value of a and the value of b Using your values of a and b from, find the two values of for which a + b = b + a 4 (a) Simplify 7( + i) + i( + 9i ) and epress your answer in the form + yi where, y R and i = (b) Let w = 3 i Plot w and w + 6i on an Argand diagram Calculate w + 6 i Epress w + 6i in the form u + vi where u, v R (c) Let z = +4i Epress z + 8 in the form p + qi where p, q R Solve for real k k( z + 8) = z ( + i) Epress your answer in the form a b where a, b N and a is a prime number Page 3 of 5

4 5 (a) The nth term of a sequence is given by T n = n + Write down the first three terms of the sequence Show that T + T + T3 = T4 (b) The first term of a geometric series is and the common ratio is 0 Write down the second, third and fourth terms of the series Calculate S 4, the sum of the first four terms Give your answer as a decimal (c) The first three terms of an arithmetic series are Find, in terms of n, an epression for T n, the nth term Find, in terms of n, an epression for S n, the sum to n terms Using your epression for S n, find the sum of the natural numbers that are both multiples of 5 and smaller than (a) Differentiate from first principles with respect to (b) f() The graph shows portion of a periodic function f : f ( ) which is defined for R Write down the period and the range of f() Complete the following table: f() (c) Let g( ) = ( + 3)( ) for R For what two values of is the slope of the tangent to the curve of g() equal to 0? Find the equations of the two tangents to the curve of g() which have slope 0 Page 4 of 5

5 7 (a) Differentiate with respect to (b) Find dy d when y 7 =, Find dy d when y = ( + ) 5 3 (c) A car, starting at t = 0 seconds, travels a distance of s metres in t seconds where 9 s = 30t t 4 Find the speed of the car after seconds After how many seconds is the speed of the car equal to zero? Find the distance travelled by the car up to the time its speed is zero 8 (a) Let p( ) = 3 For what values of is p( ) < 0 where is a positive whole number? (b) Draw the graph of g( ) = for 3 3, R and 0 Using the same aes and the same scales, draw the graph of h( ) = + for 3 3, R Use your graphs to estimate the values of for which = + 3 (c) Let f ( ) = 3 + a + for all R and for a R f() has a turning point (a local maimum or a local minimum) at = Find the value of a Is this turning point a local maimum or a local minimum? Give a reason for your answer Find the co-ordinates of the other turning point of f() Page 5 of 5

Mathematics (Project Maths Phase 2)

011. M7 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 011 Mathematics (Project Maths Phase ) Paper 1 Ordinary Level Friday 10 June Afternoon :00 4:30 300 marks