Leaving Cert Maths Samples

_________________________________________ leavingcertsolutions.com ________________________________________ 2018 Question 7 – Ordinary Paper 1 Solution (a) This is a question based on sequences and series. Row 1, 28 seats; row 2, 29 seats; row 3, 30 seats The rows form a sequence 28, 29, 30, an arithmetic sequence ( ) ( ) 1 28 1 1 27 n n T a n d n n T = + − = + − = + = Number of seats in row 10 is 27 + 10 = 37 = T 10 (1 0 marks ) (b) We are told there are 50 seats in the last row 50, 27 50 50 27 23 n T n n n = + = ⇒ = − ⇒ = ∴ 23 rows in the theatre ( 10 marks ) (c) We are looking for the number of seats in the theatre We want S 23 of the aritmetic sequence 1 28, 1 T d = = ( ) { } 2 1 2 n n S a n d = + − ( ) ( )( ) { } 23 23 2 28 23 1 1 2 S ⇒ = + − { } { } 23 23 23 56 22 78 897 2 2 S = + = = ( 10 marks ) (d) ( ) ( ) { } 2 28 1 1 600 2 n n S n = + − = { } { } 56 1 600 55 600 2 2 n n n n + − = ⇒ + = ( ) 2 55 1200 55 1200 0 n n n n + = ⇒ + − = Use 2 4 2 b b ac n a − − = ∓ ( ) ( ) ( ) ( ) 2 55 55 4 1 1200 , 16.73, 71 2 1 n n n − − − = = = − ( ) ( ) ( ) ( ) 16 16 16 16 16, 2 28 15 1 , 56 15 568 2 2 n S S = = + = + = , (600 – 568) = 32 ∴ 32 in last row (e) Total income 276 €25 212 €12 €9444 × + × = ( 5 marks ) (f) Now 752 tickets sold. Adult:child = 3:1 (i) Add up ratio: 3 1 4 + = . Now find 752 188 4 = children ∴ Adult tickets 3 188 564 × = , child tickets 188 1 188 × = ( 5 marks ) (ii) If x is the price of a child ticket, the price of an adult ticket = 2.5 x ∴ ( ) ( ) €17,578 2.5 564 188 €17, 578 1598 €17, 578 €11 1598 x x x x + = ⇒ = ⇒ = = Price of child ticket is €11 ( 5 marks ) Comment : Parts (e) and (f would have caused problems for weaker students.