page break

University of New Zealand.

B.A. Pass Examination, 1885.

Mathematics.

Paper b. Algebra.

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

1. Show that is exactly divisible by a²+b²+c²−ab−ac−bc; and show that if x+y+z=0, then yz−x²=zx−y²=xy−z².

2. Find the Greatest Common Measure of 2x³−x²−4x+3 and 4x⁴−5x³−x+2, and the Least Common Multiple of (2x²−2, 3x³+3, 2x²−2x−4, x²−x+1.

3. Find whether greater or less than a/b, and whether is greater or less than it being given that c/d is a/b reduced to its lowest terms.

Simplify the following:—

page 2

4. Solve these equations:—

5. You can buy from a wine-merchant 10 dozen sherry and 12 dozen claret for £48, and 5 dozen more of sherry can be had for £45 than of claret for £50: required the price of each per dozen.

6. Define ratio, and explain why the equality of two ratios may be expressed by equating two fractions.

If a+b+c: −a+b+c:: a−b+c: a+b−c, show that a is a mean proportional between b+c and b−c.

7. Define a Geometrical Series, and find the sum of n terms of the series.

The 2nd and 5th terms of a Geometric Series are—1/3 and 8/81; write down the intermediate terms, and sum the series to n terms and to infinity.

8. Write down the Binomial Theorem, and find the middle term, when there is one, and the sum of the alternate coefficients.

Find the coefficient of x⁴ in the expansion of