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Assessment Report
New Zealand Scholarship
Calculus 2018
Standard 93202
Part A: Commentary
Successful candidates in 2018 were well prepared and had a sound grasp of both the fundamental mathematical skills required at this level as well as skills in problem solving and worked to an acceptable level of accuracy. Candidates who gained Scholarship with Outstanding Performance possessed mastery of the knowledge embodied within all the Mathematics Learning Objectives underpinning the Scholarship paper; were able to clearly establish a strategy to solve a problem as well as communicate that strategy through well developed and laid out solutions; utilise rigorous and correct mathematics to a high and exacting standard.
Some unsuccessful candidates were let down by the poor layout of their solutions which they struggled to bring to closure, having lost their way somewhere within a selfcreated maze. Many of these candidates lost valuable time pursuing false arguments rather than either reflecting upon their strategy and redirecting their effort or moving on to another question. Others possessed a very restricted knowledge of the topics spanned by the Scholarship examination. Candidates need to be made aware of the requirements of Scholarship Calculus and especially the assessment specifications.
Successful candidates were able to integrate their knowledge for the development of correct solutions to problems. While some of the questions tested narrowly defined skills pertinent to one or two Learning Objectives, many required students to draw on knowledge from multiple areas of the curriculum. Students who were able to do this, and successfully interlace these skills and knowledge, certainly earned the higher grades.
Part B: Report on performance standard
Candidates who were awarded Scholarship with Outstanding Performance commonly:
 understood and were proficient in the general discipline of mathematical proof
 proved a trigonometric identity with clear setting out and flair
 had a broad mathematics vocabulary at their disposal
 established a mathematical model to solve problems within an unfamiliar context
 established a differential equation involving related rates of change and then solved the equation by separating the variables
 solved nonlinear systems of equations using elegant algebra and often useful substitutions
 understood the domains and ranges of logarithmic functions
 manipulated complicated algebraic expressions in exact form
 laid out their working logically and clearly throughout the assessment
 established strategies upon which to build their solutions rather than charge blindly into opportunistic algebraic routines
 integrated skills and knowledge from multiple Learning Objectives in the curriculum to develop successful solutions.
Candidates who were awarded Scholarship commonly:
 manipulated trigonometric expressions using basic identities
 determined the integrals of functions using the substitution method
 generalised a pattern beyond its initial terms
 considered the relevance of a domain restriction when solving problems
 applied similar triangle relationships correctly to establish patterns
 manipulated and rearranged multiterm equations and expressions correctly
 established optimisation problems by forming an algebraic function and differentiating it to find its minimum
 rationalised complex numbers correctly.
Other candidates
Candidates who were not awarded Scholarship commonly:
 possessed little or no understanding of the meaning of mathematical proof
 failed to deploy fundamental algebraic skills such as factorisation and expansion when manipulating expressions involving trigonometric terms
 showed little understanding of related rates of change
 were unfamiliar with the definition of the logarithm
 were unable to change from logarithmic to exponential form
 were unfamiliar with the algebraic laws of logarithms
 were deficient in the vocabulary of mathematics
 were unaware of the definition of a function
 lacked understanding about the domain of a function
 could not differentiate a function implicitly
 lacked the ability to assess the validity of their answers to real life problems
 oversimplified or trivialised problems
 made careless algebraic errors
 were incompetent in solving rational equations
 falsely equated the real and imaginary parts of a complex number, z, to the reciprocals of the real and imaginary parts of z^{1}
 were unable to solve simultaneous equations involving quadratic terms
 were unable to solve an integral using a simple substitution
 produced multiple answers when asked for a point of tangency
 misunderstand the concept of discontinuous function
 attempted to bluster their way through proofs rather than find their minor algebraic errors.
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