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WAIKATO PATHWAYS COLLEGE\WINTEC Bridging Mathematics and Statistics (FOPR008\CUPR008-10A) Assignment 2 DUE DATE: Wednesday, 12\05\10 For full marks

Question 1
a) A(2,4), B(-2,2) and C(4,6) are vertices of a triangle.
i) Find the length of AB ii) Find the coordinates of P, the midpoint of AC.
iii) What is the equation of BC? iv) Find the equation of the median from vertex B to
AC.
b) Find the equation of the line (in the form y = mx + c) with gradient = -
5
3
and passing through
point (-3,-4).
c) Find the angle which the line 2x + 5y + 6 = 0 makes with the positive direction of the x-axis.
Question 2
a) Determine the equation of the line passing through (-1,-3) which is
i) Parallel to the line y = -3x + 17
ii) Perpendicular to the line 2x + y = 8
b) Consider the points P(-2,2) and Q(4,-6). Find the gradient of:
i) the line PQ; ii) the line parallel to PQ;
iii) the line perpendicular to PQ.
Question 3
a) Find the equation (in the form ax + by + c = 0) of the perpendicular bisector of
PQ if P(-3,-4) and Q(7,-8).
b) Find the perpendicular distance of the origin from the line 3x  2y 8  0.
c) Find the equation of the straight line which makes an angle of 40º with the x-axis
and bisects the line segment joining (0, 2) and (-4, 6).
Question 4
a) Write down the domain and range of the following functions:
i) y  3x 5 (ii) 2 3 2 y  x 
ii) 2 f (x)  (x  2) (iv) f (x)  x  4
b) Write down the domain of each of these functions:
i)
3
2
( )

x
f x (ii)
x
f x
5
( )

 (iii)
( 3)( 2)
2 3
( )
 

x x
x
f x
Question 5
a) Find the inverse of
i)
5
2
( )

x
f x ii) y  2x  3 iii)
1
3 4
( )

x
x
f x
b) Two functions f (x)  2x 3 and ( ) 3 1 2 g x  x  x  are defined over real
numbers. Find i) g  f (x) and f  g(x) (Simplify your answers)
ii) g[ f (3)].
Question 6
Find
dx
dy
if: a) 4 3 5 3 3 2      y x x x b) 5 6
5
3
5
3
4 x
x
y  x  
c)  2 7 y  3x  4 d)
3
5 4 5
x
x x
y
 
 e)
2 1
4 2 3

x
x x
y
Question 7
a) Find
2
2
d y
dx
if: i) 5
2
1
3
1 3 2 y  x  x  x 
ii) ( 5)( 4) 3 2 y  x  x  iii) 2 4 y  (9x 1)(3x  4)
b) If ( ) 2 3, 2 f x  x  x  solve f (x)  f (x).
c) Find the coordinates of all stationary points on the given curves, and determine the
nature of each one i) 3 4
3
1 3 2 y  x  x  x  ; ii) 4 2 f (x)  x  9x .
d) The curve y  x  px  qx  r 3 2 has a minimum turning point at x = 4 and a point of
inflection at (1, 2). Find the values of p, q and r.
e) A particle moves on the circumference of the semi-circle 2 y  4  x . Find
dt
dy
when
x 1, if  4.
dt
dx

WAIKATO PATHWAYS COLLEGE\WINTEC Bridging Mathematics and Statistics (FOPR008\CUPR008-10A) Assignment 2 DUE DATE: Wednesday, 12\05\10 For full marks show all essential working. Question 1 a) A(2,4), B(-2,2) and C(4,6) are vertices of a triangle. i) Find the length of AB ii) Find the coordinates of P, the midpoint of AC. iii) What is the equation of BC? iv) Find the equation of the median from vertex B to AC. b) Find the equation of the line (in the form y = m x + c) with gradient = - 5 3 and passing through point (-3,-4). c) Find the angle which the line 2 x + 5 y + 6 = 0 makes with the positive direction of the x-axis. Question 2 a) Determine the equation of the line passing through (-1,-3) which is i) Parallel to the line y = -3 x + 17 ii) Perpendicular to the line 2 x + y = 8 b) Consider the points P(-2,2) and Q(4,-6). Find the gradient of: i) the line PQ; ii) the line parallel to PQ; iii) the line perpendicular to PQ. Question 3 a) Find the equation (in the form a x + by + c = 0) of the perpendicular bisector of PQ if P(-3,-4) and Q(7,-8). b) Find the perpendicular distance of the origin from the line . 0 8 2 3 y x c) Find the equation of the straight line which makes an angle of 40º with the x-axis and bisects the line segment joining (0, 2) and (-4, 6). Question 4 a) Write down the domain and range of the following functions: i) 5 3 x y (ii) 3 2 2 x y ii) 2 ) 2 ( ) ( x x f (iv) 4 ) ( x x f b) Write down the domain of each of these functions:
i) 3 2 ) ( x x f (ii) x x f 5 ) ( (iii) ) 2 )( 3 ( 3 2 ) ( x x x x f Question 5 a) Find the inverse of i) 5 2 ) ( x x f ii) 3 2 x y iii) 1 4 3 ) ( x x x f b) Two functions 3 2 ) ( x x f and 1 3 ) ( 2 x x x g are defined over real numbers. Find i) ) ( x f g and ) ( x g f (Simplify your answers) ii) )] 3 ( [ f g . Question 6 Find dx dy if: a) 3 5 3 4 2 3 x x x y b) 5 6 5 3 5 3 4 x x x y c)   7 2 4 3 x y d) 3 5 5 4 x x x y e) 1 2 2 4 3 x x x y Question 7 a) Find 2 2 dy dx if: i) 5 2 1 3 1 2 3 x x x y ii) ) 4 )( 5 ( 2 3 x x y iii) 4 2 ) 4 3 )( 1 9 ( x x y b) If , 3 2 ) ( 2 x x x f solve ). ( ) ( x f x f c) Find the coordinates of all stationary points on the given curves, and determine the nature of each one i) 4 3 3 1 2 3 x x x y ; ii) 2 4 9 ) ( x x x f . d) The curve r qx px x y 2 3 has a minimum turning point at x = 4 and a point of inflection at (1, 2). Find the values of p , q and r . e) A particle moves on the circumference of the semi-circle 2 4 x y . Find dt dy when , 1 x if . 4 dt dx

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