# The Pamphlet Collection of Sir Robert Stout: Volume 50

# University of New Zealand. — Mathematics. — Paper b. Algebra

## University of New Zealand.

## Mathematics.

##
Paper *b*. Algebra.

1. Show that is exactly divisible by *a*^{2}+*b*^{2}+*c*^{2}−*ab*−*ac*−*bc*; and show that if *x*+*y*+*z*=0, then *yz*−*x*^{2}=*zx*−*y*^{2}=*xy*−*z*^{2}.

2. Find the Greatest Common Measure of 2*x*^{3}−*x*^{2}−4*x*+3 and 4*x*^{4}−5*x*^{3}−*x*+2, and the Least Common Multiple of (2*x*^{2}−2, 3*x*^{3}+3, 2*x*^{2}−2*x*−4, *x*^{2}−*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 25. 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.