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Assessment Report
New Zealand Scholarship
Calculus 2019
Standard 93202
Part A: Commentary
It was refreshing to see the number of candidates who provided innovative and insightful solutions to problems. These candidates had mastered the foundational principles of algebra, trigonometry and calculus and could draw on this competence to approach questions holistically, integrating that knowledge rather than being reliant on a suite of single skill algorithms.
There were, however, some candidates who failed to make any progress in solving even the easiest problem. They were not adequately prepared, particularly lacking algebraic and trigonometric skills needed at this level. A troubling number of candidates were awarded no marks at all.
Unsuccessful candidates were often let down by the poor layout of their solutions, not knowing when to abandon a strategy or failing to recognise how close they were to a solution and did not provide the final steps. However, the same can be said of good students not reaching the outstanding category of achievement. Careless notation and disorganised presentation resulted in good talent not being fully showcased.
Mention must be made about the Assessment Specifications. The Scholarship examination draws on the Mathematics Curriculum of New Zealand, not a subset of Achievement Standards. This was particularly evident in the very poor attempt made at question 2(d). While this is a challenging question, its solution lies on ratio and area. Technically, no more than Level 1 knowledge is required to solve the problem. However, when students have a very granular view of the mathematics curriculum, it becomes extremely challenging. At scholarship level, knowing how to differentiate and integrate a polynomial is not sufficient. Implicit differentiation and integration using substitution (the socalled reverse chain rule) are essential skills.
In closing, tribute must be paid to those candidates who earned outstanding scholarships. They displayed a mastery of mathematics, flair and creativity in their solutions. Overall, those candidates worthy of scholarship and outstanding scholarship were made clearly visible by this examination paper.
Part B: Report on performance standard
Candidates who were awarded Scholarship with Outstanding Performance commonly:
 produced clear and logical working
 demonstrated elegance and flair when manipulating algebraic and trigonometric expressions
 planned their strategies and executed the plan
 understood when to preserve and when to abandon an approach
 use implicit differentiation to find the derivative of arctan
 applied mathematical knowledge in unfamiliar context
 established an equation and differentiated it to find an optimal solution
 determined whether an optimised solution was a minimum or maximum
 solved a first order differential equation by separating variables
 demonstrated understanding of the 11 relationship between and over the appropriate domain.
Candidates who were awarded Scholarship commonly:
 established and differentiated an equation to find an optimal solution
 rationalised the denominator of a complex number
 considered the relevance of a domain restriction when solving problems
 checked the validity of solutions
 understood the ramifications of a quadratic discriminant on the nature of the solutions
 analysed and manipulated expressions with absolute value symbols and inequalities
 integrated functions using the reversechain rule or a simple substitution
 solved nonlinear systems of equations using substitutions
 differentiated a function using first principles
 analysed correctly the narrative of a related rate of change problem
 applied a conjugate to simplify an expression involving complex numbers
 understood the mathematical notion of generalised proof
 applied trigonometric identities and formulae to demonstrate proof.
Other candidates
Candidates who were not awarded Scholarship commonly:
 made careless algebraic errors
 did not check the validity of their solutions
 lacked consistency in algebra manipulations
 abandoned their efforts too quickly
 attempted to “fudge” proofs rather than find algebraic errors in their working
 provided calculator only solutions where algebraic working was clearly expected
 did not know that absolute value symbols are algebraic operators
 did not know the basic log rules
 displayed serious algebraic misconceptions such as:
 If (x – 2)^{2} > 0, then x > 2 is the only solution.
 If a^{2} – b^{2} = c^{2}, then a – b = c^{}
 If xy < 0, then x < 0 and y <0
 could not use substitution or the reverse chain rule to find an antiderivative
 could not differentiate using first principles or work with limit notation
 were unfamiliar with Leibniz notation for derivatives,
 did not understand the significance of the differentials in the Leibniz notation
 could not form an equation using one variable.
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