Higher Maths Paper 1 Sample Exam Question Explained & Solved

Higher Maths Paper 1 – Sample Exam Question Explained & Solved

QUESTION 1 - 10 marks

(a) Show

· Typical Question 1. Since this is worth 10 marks, it will involve (usually ) three steps, something similiar was asked in 1996 and 1997.

· As with all problems involving surds, the main focus of attention is to rationalise the bottom line (get rid of the square roots on the bottom line). This is done by multiplying above and below by the "conjugate "of the bottom line .

(b) Solve

· This is a very common type of inequality. Very popular with those who set the higher leaving cert. Inequalities of this type were asked in

· The standard way to solve the inequality was to muliply both sides of the inequality by the square of the bottom line. This turns the problem into a quadratic inequality.

· now tidy this up to get

· This is a quadratic inequality . To solve the inequiality (1) solve the equation (2) Plot the results on the number line (3)Plot zero on the number line (4) Sub zero into the inequality if zero works the zero is part of the solution if it does not work the zero is not part of the solution* remember the solution is either between the roots or not between the roots .

· now plot the results on the number line (don't forget zero)

...... -5/2...... 0...... 3......

· Now sub zero into which is true therefore 0 is part of thesolution so our solution is -5/2 <x < 3.

· This is a variation of the typical question based on the factor theorem . Since we know is a factor of x3-p the we know it will divide into it leaving no remainder and since the remainder is zero we can usually find the relationship between what we want an what is there. So divide x3 - p by x2+bx +c

· and we now require c in terms of p C3= b6 and p2= b6. Therefore C3= b2. A much nicer way to do this question is to use the fact that x 3 - p is in fact the difference of two cubes wher we could write p as say y3 .