Expert: Scott A Wilson - 1/19/2009

Question
1) Simplify:

a) 10/sqrt5   + sqrt 20
b) 1/sqrt2 (2sqrt2 – 1) + sqrt2 (1-sqrt8)



2) Find the greatest or the least values of these functions:

a)   f(x) = x² - 3x + 5
b)   g(x) = 3 – 2x - x²


3) The sequence U1, U2, U3… Where U1 is a real number, is defined by Un+1 = Un² - 1. If U1 = 3, find U2, U3 and U4



4) The sum of the first two terms of an A.P. is 181 and the sum of the first four terms is 52. Find the sum of the first 8 terms.



5) Solve the simultaneous equations:
  
  2x + 3y = 5
  x² + 3xy = 4



6) Solve the inequality (x + 1)(x – 2)(x – 3) > 0



7) Find the derived function of :

a)   f(x) = 3x² + 2x
b)   f(x) = 1/x²



8) Find the equation of the line passing through the points A (3, 4) and B (-1, 2). Also, find :

i)   The distance AB
ii)   The equation of the normal at A  

Answer
1) Simplify:
a) 10/sqrt5   + sqrt 20
note that 10 = 2√5√5, so one of √5 can be cancelled
note that 20=2*2*5 and a 2 can be taken outside the squareroot

b) 1/sqrt2 (2sqrt2 – 1) + sqrt2 (1-sqrt8)
Multiply out the terms; note that 8=2*2*2, so √8=2√2;
eliminate any square roots in the denominator by multiplying the top and bottom by the root.

2) Find the greatest or the least values of these functions:

a) f(x) = x² - 3x + 5: this one can't be factored, so the answer is 1
b) g(x) = 3 – 2x - x²

3) The sequence U1, U2, U3… Where U1 is a real number, is defined by Un+1 = Un² - 1. If U1 = 3, find U2, U3 and U4
U2 = U1² - 1, os U2 = 3² - 1 = 8;
find U3 = U2²-1 in the same way
and then find U4 = U3² - 1.

4) The sum of the first two terms of an A.P. is 181 and the sum of the first four terms is 52. Find the sum of the first 8 terms.
What kind of series is A.P.?
Since the sum of the first four terms is less, there must be some negatives involved...

5) Solve the simultaneous equations:

2x + 3y = 5
x² + 3xy = 4
Solve the first one for x, rewrite the second as
x(x+3y) = 4 and put in the solution for x from the first equation.
Hint: solution is x = (5 -3y)/2

6) Solve the inequality (x + 1)(x – 2)(x – 3) > 0
Write a table with four columns.
Make division line going down the side at 3, 2, and -1.
Label the tops of each of the columns with x+1, x=2, x=3, and result.
In each of the cells, put in the sign of the function (+ or -).
Multiply the first three cells (- * - = +) together and put in result column.
The sign in the result column will say whether it is + or - in the interval between the lines -1, 2, and 3 as well as above the top line and below the bottom line.

7) Find the derived function of :
a) f(x) = 3x² + 2x
b) f(x) = 1/x²

What is the 'derived function'?

8) Find the equation of the line passing through the points A (3, 4) and B (-1, 2). Also, find :

The slope is m=(Ay-By)/(Ax-Bx) = (4-2)/(3+1).

i) The distance AB
The distance between A and B is √[(Ay-By)²+(Ax-Bx)²]

ii) The equation of the normal at A
The normal line has slope =1/m, so the basic form is
y-yA = -(x-xA)/m.