Everything you must know to achieve maximum marks on paper
1 Including an analysis of the proofs that have been asked
and the best approach to getting the best rests.
Advice
as to which questions to attempt and when.
What is the best kept secret on paper!. What questions can
you safely leave out?
Induction
Having problems with induction these files contain everything
you wanted to know about induction but were afraid to Ask!
Differential
Calculus
The
good, bad and the ugly. Here
are a questions which most students found very difficult,
the purpose of these questions was to filter out the A1,
and A2 students from the rest.
Solutions
to the 2000 paper fully explained
See
how  ,
makes proving the product rules and quotient rule in calculus
a lot easier
Find
out what is the best kept secret on the Higher Maths Paper
1.
Find
out which topics in your text books are not on the course.
Aalgebra
See
the word on algebra.
Sequence
& Series
|
Higher
Level
The good the bad and the ugly in differential
calculus.
The
following are a group of Questions most of them Question
6c or Question 7c on paper 1 of the higher Leaving Cert
maths which most students found very difficult, the purpose
of these questions was to filter out the A1, and A2 students
from the rest.
Example
1
Question 7c leaving Cert Higher Maths 2000.
If
 show
the max value of f(x) occurs at (e,1/e).
This
is a straightforward max/min problem, we
know the max or min occurs when dy/dx = 0, so find
dy/dx set it equal to 0 and solve for x.
so
we have a turning point at (e,1/e) To show this is a maximum
find f''(x) and show that f''(x)is negative.
so
we have a turning point at (e,1/e).
To
show this is a maximum find f''(x) and show that f''(x)is
negative.
.
max at (e,1/e). This was worth 10 marks
It
was the second part of this question, which for most students
was a total write off.
It
said hence show  .
Most students did not have a clue!
Solution:
We
know from above that the maximum value of 
**Here
we use the rule  ,
*** we use the rule  .
This
next question we will look at was asked in 1997 in fact
it is one of two questions, which were really over the top
on the '97 paper.
Question 6c 1997 paper 1.
If
 find
the value of a and the value of b if  .
There are many ways to do this, this is one of the better
ways, treat Siny as an implicit function
*Set
the top lines equal to each other this gives a = 3, b =
1.
You
can see from the above that this was a lot of hardship for
10 marks .
The
second question on the '97 was as follows :
Question 7 ©1997 Paper 1.
Let
 (i)
find the values of x for which dy/dx = 0.
(ii)for
x real show that y cannot have a real value between -2 and
2.
Solution:
(ii)
this part again proved to be very difficult, the key is
the words "for x real"
So
turn the equation from part (i) into a quadratic by multipling
everything by (x-1).
that
is y cannot lie between -2 and 2.
In 1996 Question 6c (i) they asked this little gem !
If
 if
a is a constant show that  .
This
was a question based on parametric differentiation that
is  do
this first
 .
Many
students got this far but getting from here to the end proved
very difficult and frankly was not worth the bother since
it was worth at most 5 marks!
The
trick was to realise that you needed  .To
change from  we
use Page 9 of the tables .
use the corresponding connection between A and A/2,
if .
The
problem here was that most students subbed in  into
 and
could not simplify it down !
Question
7c on the 1995 paper 1 was another question sent to try
us.
It
says let  show
 .
This looked worse than it actually was it really was just
an index equation .
**We
use the "+" form as We can only find ln of a positive number
.
Since
 .
The
Second part asked us to find dy/dx in the form 
The
algebra at the end of this was nice .
Question 7c 1999 paper1:
 show
f(x) has a local minimum at(0,-4) and a local maximum at
{ .
To find the Max and Min, just find dy/dx, set dy/dx = 0
The
next part ask to find the range of values of k for which
f(x)=0 has 3 real roots.
If
f(x) has 3 real roots (Ymax)(Ymin)<0
If
f(x) has two equal roots then Ymax.Ymin = 0 k= 3.
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