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

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

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