The Pamphlet Collection of Sir Robert Stout: Volume 50
University of New Zealand. — Mathematics. — Paper b. Algebra
University of New Zealand.
Paper b. Algebra.
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
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.