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1 NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence (a)(i) (x 5) (3x + ) 6(x + 3 )(x 5 ) Correctly factorised. Correct decimal / rounding. (a)(ii) x = 3 ( 0.67) or x = 5 (.5) Complete correct solutions found. Any rounding / truncation. Consistency with a(i) but not trivial (coefficients of x > 1) (b) b 4ac = 0 16m = 64 m = 4 Perfect square. 4(x 1)(x 1) = 0 or equivalent x = 1 m = 4 Recognising the discriminant must = 0. CRO. Calculating the value of m. (c) (x + )(x ) (x + )(x 4) Factorised. Correctly simplified. = (x ) (x 4) accept (x 4) (x 4) Factorised and simplified with one error in factorising. (d) a 3 a 4 = 0 (a 4)( a + 1) = 0 a = 4 or a = 1 (x + ) = 4 or ( x + ) = 1 (not a solution) x = 14 ((x+) 4) = (3 (x + )) x 4x + 4 = 9(x + ) x 13x 14 = 0 (x + 1)(x 14) = 0 x = 1 and x = 14 Equation rearranged and factorised Solved using either method. (RANW= n) Solved for x. x = 14 and 1 but not disregarding x = 0. Recognition that x = 1 is not a solution. (e)(i) (ii) x mx + nx mn = (x nx + mx mn) x + 3mx 3nx mn = 0 x + (3m 3n)x mn = 0 3(m n) ± x = 9(m n) + 4mn = 3(m n) ± 9m 14mn + 9n Hence 9m 14mn + 9n > 0 9(m n) + 4mn > 0 Cross multiplication and collection of like terms. Mei for one incorrect simplification step. Correct substitution into the quadratic formula, not necessarily simplified. No roots given but correct inequality in (ii). Roots and inequality found.

2 NCEA Level Mathematics (9161) 013 page of 5 N 1 of u A3 of u M6 of r E8 of t

3 NCEA Level Mathematics (9161) 013 page 3 of 5 TWO Expected Coverage Merit Excellence (a) 64a 6 /64a 10 = 1/ a 4 (or a 4 ) Correct simplification. (b)(i) x 0.5 ( or x, x 1 Correct simplification. (ii) x 0.5 3x 1.5 = 6x Correct simplification. Consistent with (b) (i). (c) 6x + 1x 48 = 0 x + x 8 = 0 (x + 4) (x ) = 0 x = 4 and x = Correct equation = 0 CAO or guess and check. If answer correct & x = 4 is eliminated. Equation solved showing equation. (d) x log a = (x 1)log5 = x log5 log5 x(log a log5) = log5 Expression written in log form and expanded. Expression for x correct. equivalent. x = log5 log a log5 (e) (3x + n)(x ) = 0 3x + (n 6)x n =0 n 6 = 4 n =10 (3x + n) =0 root is n / 3 = 10 / 3 k = n = 0 Establishing the relationship. CRO of k and other root. One value for n or k or the other root found with algebraic working. Solutions found for k and the other root with algebraic working. N 1 of u A3 of u M6 of r E8 of t

4 NCEA Level Mathematics (9161) 013 page 4 of 5 THREE Expected Coverage Merit Excellence (a)(i) x 3 = 64 x = 4 (ii) x+1 = 3 3x = 5 x+1 3x = 5 1 x = 5 x = Correctly solved. CRO Whole equation in powers of two. Use of log, with exponents eliminated. eg: (x + 1)log = log3 +x log8 Correctly solved. (b)(i) n Expression correct. (ii) 100 > n 0.6 n < 1 18 nlog0.6 < log 1 18 n > years equivalent. In/Equation rearranged in index form. CRO. Solved by guess and check. 5.7 years with working. Consistent use of 0.4 n give n = 3.15 Number of years found as a whole number (n = 6) Consistent use of 0.4 n using logs give n = 4 years (whole number). (c) 9x + 6x + 8 = 0 b 4ac = 5 therefore no real roots. Graph of the parabola does not cut the x- axis. A squared term can never be negative hence there is no solution therefore the graphs do not intersect each other. Quadratic expression rearranged=0 Explanation of no x intercepts because the discriminant is less than zero, without 5. A squared term can never be negative hence there is no solution. Discriminant found and therefore no real roots but no x axis analysis. Quadratic expression rearranged = 0. AND Explanation of no x intercepts because the discriminant is less than zero, without 5. Discriminant calculated and explanation of no x intercepts given. Full explanation of the two graphs not intersecting. (d) x = (mx) x(m x 1) = 0 therefore either m x = 1 Equation given in index form. One solution found. x = 1 m Correctly solved with x = 0 disregarded. x = 1 if x 0. m x = 0 But log 0 is undefined Therefore x = 1 m Both solutions and x = 0 not disregarded. Use of log properties to solve completely. X = 0 still needs to be disregarded.

5 NCEA Level Mathematics (9161) 013 page 5 of 5 N 1 of u A3 of u M6 of r E8 of t Judgement Statement Not Achieved with Merit with Excellence Score range

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