Leaving Cert Maths Samples

_________________________________________ leavingcertsolutions.com ________________________________________ 2018 Question 9 – Higher Paper 1 Solution (a) Step 0 1 2 3 Number of black triangles 1 3 9 27 Fraction left 1 3/4 9/16 27/64 ( 10 marks ) (b) (i) The sequence of black triangles is 1, 3, 9, 27 Step 0 is 3 0 , step 1 is 3 1 ∴ ( ) 3 0,1, 2,3 n n t n = = ( 5 marks ) (ii) We are asked to find k if 9 3 10 k k t = > Get log base 10 of both sided 9 10 10 log 3 log 10 log3 9 k k ⇒ > ⇒ > 10 9 18.86 19 log 3 k k k ⇒ > ⇒ > ⇒ = ( 10 marks ) (c) (i) We are asked to find h if 0.01 h t < The pattern for the remaining fraction is 1, 3/4, 9/16, 27/64 A geometric series 3 1, 4 a r = = . 3 , 0,1, 2 4 n n t n   = =     3 0.01 4 h   <     (take logs of both sides) 3 3 ln ln 0.01 ln ln 0.01 4 4 h h     < ⇒ <         (we must reverse the symbol as 3 ln 0 4   <     ) ( ) ( ) ln 0.01 16.008 17 ln 3 / 4 h h h ⇒ > ⇒ > ⇒ = ( 10 marks ) (ii) 3 lim 0 4 n n →∞   =     (fraction of original area or all the area) ( 5 marks ) (d) (i) ( 5 marks ) Step 0 1 2 3 Perimeter 3 9/2 27/4 81/8 (ii) The sequence for the perimeter is 3, 9/2, 27/4, 81/8. 3 3, 2 a r = = 33 , 0,1, 2 2 n n t n   = =     34 35 33 2,912, 219 2 t   = =     ( 5 marks ) (iii) From part (c)(ii), as n → ∞ , the remaining area is zero. The total area of the triangles ( )( ) 1 3 1 1 sin 60 2 4 = = (area of 1 st black triangle). From part (d)(ii), the perimeter of the black triangles is infinite. 3 lim3 2 h n →∞   = ∞     ( 5 marks ) Comment: Very tricky question. This is the first time fractals have been asked.