Leaving Cert 2001

Ballinteer Institute Leaving Higher Maths Paper Two

Question 1 : Circle : In this question parts (a) and (b( tend to quite easy ). Note the following are not on the course (i) Orthogonal Circles (ii)Radical Axis .Systems of Circles some of these topics are covered in the text books but will not be examined . Expect to be asked to find the equation of a circle/circles,remember if asked to find the equation of a circle you will be looking for three unknowns ,hence you need three equations ,therefore you need three bits of information!

Equations of tangents from a point to a circle is often asked .The proof came up in '97. The part © in 1999 was a bit of a hand full there was a nice way to do it which did not involve half the alphabet , if you would like to see it send us an email .

Question 2 : Vectors

Good use can be made of the formula for a point x on the line ab where x divides [ab] in the ratio r:s. Also expect to make ample use of the Scalar Product .This is usually a very easy question , don’t worry about the last part of part © on the 1999 it only received 5 marks ,again there is a lovely otherwise way to do it (it’s based on the bisector of an angle)

Question 3 : The Line /Transformations .

The co-ordinate geometry of the line did not feature at all in '94, but made a small appearance in part (a) of the 95. In '96 it featured in parts (a) and (b) . I t may be possible to get a question on a Co-ordinate Geometry proof here . Transformations appeared in ’99,none ’98, ’97!

So the two proofs that can be asked are (1) the angle between two lines (2) and the perpendicular distance from a point to a line!The parts (b) and (c) will involve a transformation and the affects of this transformation on a line or lines remember that for the transformation to be an Isometry the determinant of the matrix must be +or- 1. A line segment appeared in '96 and by all accounts was badly done so it may appear again.

Question 4 /5:Trigonometry :

Know how to use the Sine and Cosine Rules : They have shown in previous years they have asked questions concerning three poins on a horizontal plane with a vertical pole at one corner of the plane .A possible identity which may be asked to prove is Know how to use Sin(A+B)/Cos(A+B) . The proof of the Cosine Rule was asked in ,97 Be able to solve trig equations, and be able to use the fact that Lengths of arcs and areas of sectors may also be important . Q5c 1997 was a bit different could get another outing. In 1999 and 2000 both question 4 and question 5 could have done with the use of page 9 of the tables and very little knowledge of trigonometry!

Question 6/7 Statistics /Probability/Permutations/Combinations /Difference Equation . It is difficult to be very precise as to what topics will come up in each question . The difference equation and probability look likely in question 7 . Question 6 may contain a Mixture of Permutations/Combinations/probability and Statistics .I think there is plenty of scope to ask questions on the standard Deviation look for a repeat of '96 /'97in this question . .Maybe a Weighted mean could appear again .The properties of the mean and standard Deviation seem to flavour of the month at the moment !

Question 8 Further Calculus :Integration by parts is very likely ,the only certainty is the Max/Min part which will involve some form of figure although 97 was a bit different) Mac Lauren came up in 1995 whereas the Ratio test appeared in 1994 . No integration by parts in 1997 but was asked in 98,99,and 2000,should appear this year

The pattern for the last 4 years has been

1999 (a)Integration by parts,(b) Maclauren © Max/Min

1998 (a)Ratio Test(b)Integration by parts (c) Max /Min

1997(a)Ratio test (b) Maclauren (c) Max and Min

1996(a)Integration by Parts (b)Max/Min (c)Maclauren + ratio test

Which Questions to do ? .

Some questions select themselves because most students find them in general OK . Questions 1,2,3 and 8 will be done by most students . So the question is 4or 5 or 6or 7 ? . I do not think it is a good idea to total exclude either topics . Question 4 seems to be the more straightforward trig question while 7 has the least amount of probability . For those students hoping to get a C ,4 and 7 may solve the problem . For those hoping to get an A or B, 4 and 5 may be more straightforward as at least the will know if they are right or not . You will probably have time to do an extra question anyway if you have do . They will mark your best 6.

What about the proofs The history of the Exam so far indicates the following

Paper (1) Proof s Asked so far!

Differentiation from first principals including , UV, . The factor theorem, and Induction .(See paper 1 notes)

Paper (2) : Mainly trig identities of which identities like the following are most popular

,Difference equations,and the three Coordinate Geometry proofs

Proofs Asked On Paper 2

1999 (4c)

1998(3c) The perpendicular distance from a point to a line is

1997(3b)(5c) Proof of the Cosine Rule

1996 no proofs!

1995 (5c)

1994 (4a) proof involving difference equations

You can see from the above list that apart from 1997 , only one proof is asked !index.htm