148
LEAVING CERTIFICATE, 1888.
No. 3.—ALGEBRA (I.).
Higher Grade and Honours.
Monday, 18th June. 3 p.m. to 5 p.m.
Give your working in full.
Endeavour rather to answer a moderate number of Questions accu¬
rately and fully than to answer a large number imperfectly.
1. Expand and arrange according to powers of x
{(b+c—d) *2+ (c+a—b) x+ (a+b—c)}
Expand, simplify and arrange symmetrically
{b + c—ay (6—c) + (c + a—5)2 (c—a)+ (a + 6—c)2 (g—b).
Find an integral function of x of the first degree whose value is
doubled when a? is doubled and which has the value 9 when x=3.
2. If popqo?-\-ra,-\-s—0, prove from first principles th&t pxa + qx2+
rx+s contains the factor x—a.
Resolve the following into factors of the first degree:—
22a;2+ 109#+45 («).
(a;+l)3+(2a?+l)3 (S).
a?->rxy+x—y—2 (y).
3. What is meant by a proper algebraic fraction ? Show that the sum
of two proper algebraic fractions is always a proper algebraic
fraction.
Simplify
(a;3+#+1 )2—(a;2—a; +1)2
(a? +a;+1)3— (ar*—# +1)3'
jn a^ + 3a; _Ax+B . Cx-\-D
(#2-l)2“(a;+l)2 + (^T)2’
where
A,
B,
numbers, determine the values of A, B, (7, D.
C, D, are mere
4. Explain carefully what is meant by a solution of a conditional
equation involving one or more variables.
Show that, if all the solutions of Ax+By+ C=0 be also solu¬
tions of A'x+B'y+ C'=0, then, in general A/A'—B/B'— CIC.
Find as many solutions as you can of the following :—
5. If three simultaneous equations, ax+by-\-cz=0,
a’x -\-b'y-\-dz=- 0,
a"x + b"y + c"z=0,
are consistent with any simultaneous values of x, y, z, except
zeroes, find the relation that must exist between the constants.
6, Solve (#+a) (y + i)=c3, (y+6) (z+c)=a2, (z + c) (# + «=)&