Expert: Sherry Wallin - 12/13/2008

Question
1.   Solve the inequality:

  10 + x² > x(x – 2)


2.   The fifth term of an arithmetic series is 14 and sum of the first three terms of the series is -3.

a)   Use algebra to show that the first term of the series is -6 and calculate the common difference of the series.
b)   Given that nth term of the series is greater than 282, find the least possible value of n.


3.   The fourth term of an arithmetic series is 3k, where k is a constant, and the sum of the first six terms of the series is 7k + 9.

a)   Show that the first term of the series is 9 – 8k

b)   Find an expression for the common difference of the series in terms of k.

Given that the seventh term of the series is 12, calculate:
c)   The value of k
d)   The sum of the first 20 terms of the series.


4.   

a)  Show that eliminating y from the equation 2x + y = 8  and  3x² + xy = 1. Produces
    the equation:  
      
                        x² + 8x – 1 = 0

b) Hence solve the simultaneous equations  2x + y = 8   and   3x² + xy = 1. Giving your answer in the form a + b sqrt 17, where ‘a’ and ‘b’ are integers.


5.   The diagram shows a sketch of the curve y = f(x). The point B (0, 0) lies on the curve and the point A (3, 4) is a maximum point. The line y = 2 is an asymptote.

Sketch the following and in each case give the co-ordinates of the new positions of A and B and state the equation of the asymptote:

a)  f (2x)        b)  ½ f(x)     c)  f(x) -2     d)   f(x+3)     e)  f(x) + 1


6.   The line l has equation 2x - y – 1 = 0. The line m passes through the point A (0, 4) and is perpendicular to the line l.

a)   Find an equation of m and show that the lines l and m intersect at the point P (2, 3). The line n passes through the point B (3, 0) and is parallel to the line m.

b)   Find an equation of n and hence find the co-ordinates of the point Q where the lines l and n intersect.  

Answer
1. Solve the inequality:

10 + x² > x(x – 2)expand right hand side
10 + x² > x²-2x
subtract 10 from both sides x² > x²-2x-10
x² > x² – 2x -10
the x² cancel each other out, add 2x to both sides
2x >-10 divide both sides by 2
x > -5
You can check your answer by choosing a value of x greater than -5 say 0, is 0 > -5 yes it is correct. You can also choose 0 and put it into the original inequality is 10 + 0 > 0(0-2)? Is 10 > 0? yes

4.

a)  Show that eliminating y from the equation 2x+y = 8 and 3x²+xy = 1. Produces the equation:  
                       x² + 8x – 1 = 0

Solution:  You can solve the 2nd equation for y getting: y = (1- 3x²)/x. Now substitute y into the first equation: 2x+(1- 3x²)/x = 8
mutiply both sides by x getting 2x²+1-3x² = 8x. Move the rest to the right hand side: x² + 8x – 1 = 0


b) Hence solve the simultaneous equations  2x + y = 8   and   3x² + xy = 1. Giving your answer in the form a + b sqrt 17, where ‘a’ and ‘b’ are integers.

Use the quadratic formula to find x: [-8+-sqrt((-8)^2-4(1)(-1))]/2(1)
[-8+-sqrt(64+4)]/2 =[-8+-sqrt(68)]/2 = [-8+-sqrt(4 *17)]/2
= [-8+-2sqrt(17)]/2 = 2(-4+-sqrt(17)]/2 = -4+-sqrt(17)

5) I don't see the sketch so I don't know what type of curve the problem is referring to.

6. The line l has equation 2x - y – 1 = 0. The line m passes through the point A (0, 4) and is perpendicular to the line l.

a) Find an equation of m and show that the lines l and m intersect at the point P (2, 3). The line n passes through the point B (3, 0) and is parallel to the line m.


a) first find the slope of line l by solving for y: y = 2x -1 so m of line l is 2 so m of line m is -1/2 (the negative reciprocal of the line l perpendicular to line m). Using the point on m of (0,4) and the slope -1/2 put the info into y = mx + b. Note b = 4 so line m has equation: y = (-1/2)x+4. To show lines l and m intersect at the point (2,3) set the two equations equal to each other and find x and y.
2x-1 = (-1/2)x+4 add 1/2 x to both sides: (5/2)x -1 = 4 add 1 to both sides: (5/2)x = 5. Multiply both sides by (2/5) getting x = 2. Now put 2 for x back into either equation using y = 2x-1 we get y = 2(2)-1
=3 so the two lines l and m intersect at (2,3).

b) Find an equation of n and hence find the co-ordinates of the point Q where the lines l and n intersect.

The line n parallel to line m ha the same slope (-1/2) and using the point (3,0) and the point slope form of an equation:
y-0 = (-1/2)(x-3) simplify: y = (-1/2)x +3/2 which is the equation to line n. To find where lines l and n intersect, set n and l equal to each other: 2x-1=(-1/2)x - 3/2 add (1/2)x to both sides:
(5/2)x - 1 = -3/2: add 1 to both sides:(5/2)x = -1/2 and finally multiply both sides by 2/5 getting x = -1/5. Substitute x = -1/5 into either of l or n I choose to use y = 2x - 1 so y = 2(-1/5) -1 = -7/5 so the point lines l and n intersect is (-1/5, -7/5).

I didn't do 2 and 3, if you still need help send the question again.

Math Prof