Expert: Paul Klarreich - 12/7/2008

Question
1) Solve the simultaneous equations:

  x – 3y + 1 = 0
  x² - 3xy + y² = 11



2) f(x) = x² - kx + 9, where k is a constant.

a) Find the set of values of k for which the equation f(x)= 0 has no real solutions.

Given that k= 4,

b) express f(x) in the form (x – p) ² + q, where p and q are constants to be found



3) Giving your answers in the form a + b sqrt 2, where a and b are rational numbers, find

a) 1/ 4 – sqrt 8



4) The points A and B have co-ordinates (1, 2) and (5, 8) respectively.

a) Find the co-ordinates of the mid-point of AB.



5) The points A (-1 , -2), B (7 , 2) and C (k , 4), where k is a constant, are the vertices of triangle ABC. Angle ABC is a right angle.

a) Find the gradient of AB.
b) Calculate the value of k
c) Show that the length of AB may be written in the form p sqrt 5, where p is an integer to be found.
d) Find the exact value of the area of triangle ABC.


6) Given that f(x) = 15 – 7x – 2x²

a) Find the co-ordinates of all points at which the graph of y = f(x) crosses the co-ordinate axes.
b) Sketch the graph of y = f(x)


Answer
Questioner:   omar
Category:  Advanced Math
Private:  No
 
Subject:  As-level maths
Question:  
.....................................
Hi, Omar,

this is not a 'we'll do your homework for you' site.  My instructions tell you to indicate what you already tried.  Since you didn't, I assume you just need hints for each one.




1) Solve the simultaneous equations:

x - 3y + 1 = 0
x² - 3xy + y² = 11

Substitute  x = 3y - 1

.......................

2) f(x) = x² - kx + 9, where k is a constant.

a) Find the set of values of k for which the equation f(x)= 0 has no real solutions.


Use the discriminant  b^2 - 4ac, and set that < 0, and solve.


Given that k= 4,

b) express f(x) in the form (x - p) ² + q, where p and q are constants to be found

Just do completing the square.


3) Giving your answers in the form a + b sqrt 2, where a and b are rational numbers, find

a) 1/ 4 - sqrt 8

simplify sqrt(8) = sqrt(4) sqrt(2).

4) The points A and B have co-ordinates (1, 2) and (5, 8) respectively.

a) Find the co-ordinates of the mid-point of AB.

Look up the standard midpoint formula.



5) The points A (-1 , -2), B (7 , 2) and C (k , 4), where k is a constant, are the vertices of triangle ABC. Angle ABC is a right angle.

a) Find the gradient of AB.

>> This is illogical.

b) Calculate the value of k


m(AB) * m(BC) = -1.  That should do it.

c) Show that the length of AB may be written in the form p sqrt 5, where p is an integer to be found.

Use the standard distance formula and simplify.


d) Find the exact value of the area of triangle ABC.

Find length(AB), length(BC) and use the standard formula for the area of a triangle.


6) Given that f(x) = 15 - 7x - 2x²

a) Find the co-ordinates of all points at which the graph of y = f(x) crosses the co-ordinate axes.


>> Set  f(x) = 0, and also  find f(0).

b) Sketch the graph of y = f(x)

I'll leave that to you.