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University of New Zeland.

B.A. Examination for Honours and for Senior Scholarships, 1885.

Mathematics.

Paper b. Algebra, Trigonometry, &c.

Examiner: Prof. C. Niven, F.R.S.

1. Determine when xⁿ+aⁿ is divisible by x + a.

Find the values of A and B that xⁿ−Ax²+B may be divisible without remainder by x²−x−2.

2. Assuming the Binomial Theorem true for a positive integral index, prove it true generally.

Prove that, squares and products of x, y above the second degree being neglected,

3. State and prove the principle of proportional parts for logarithmic tables.

Prove that

4. Sum the series

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5. Investigate the solution of the triangle in which A, a, c are given.

Find the distance between the centres of the circles inscribed in the two triangles satisfying the given conditions in terms of A, a, c.

6. Investigate Demoivre's Theorem, finding all the values of

7. Find any series suitable for the calculation of π Show that

8. Sum the series

9. Investigate the complete conditions for a maximum or minimum value of a function of one independent variable.

Examples:

10. Investigate the formula for integrating by parts. Integrate the following expressions:—