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:—
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
x
4 in the expansion of