The point A has coordinates (a , 3) and the point B has coordinates (7 , b).

The line AB has equation 2x + 3y = 11.

a) Find the values of a and b

The line AC is perpendicular to the line AB.

b) Find the equation of the line AC in the form ax + by + d = 0, where a, b and d are constants

c) Given that C lies on the x axis, find its coordinates.

Hint

b) the line AC is perpendicular to AB and passes through the point A (found in part a)

Full Solution

Print from here

 

The line \(l_1 \) passes through the point P(3k , 2k) with gradient = -2.

\(l_1 \) meets the x axis at A and the y axis at B.

a) Find the equation of the line \(l_1 \) and show that A(4k , 0)

b) Find the area of the triangle AOB in terms of k

The line \(l_2\) passes through P and is perpendicular to \(l_1 \)

c) Find the equation of \(l_2\)

\(l_2\)meets the x axis at C

d) Show that the midpoint of PC lies on the line y = x

---

Hint

The key to this type of question is to be able to visualize it.

Draw a sketch to help you:

Full Solution

Print from here

 

Point A has coordinates (a , 6) and point B has coordinates (5 , b).

The line 8x - 6y + 3 = 0 is the perpendicular bisector of AB.

Find a and b.

Hint

Find the gradient of AB in terms of a and b.

Gradient of AB is perpendicular to line 8x - 6y + 3 = 0

---

 

Find the midpoint of AB

Midpoint should lie on line 8x - 6y + 3 = 0

Full Solution

Print from here

 

Let f and g be the functionsLet f and g be the functions

\(f(x)=ln(1-2x),x<\frac{1}{2}\)

\(g(x)=\frac{4}{x-2},x\neq2\)

a) Find the exact value of fg(-2)

b) Find f^-1(x), stating its domain

c) Show that \(gg(x)=\frac{4x-8}{8-2x}\)

Hint

b) domain of f^-1 is the range of f

Full Solution

Print from here

Use the graph to find values of

a) \(ff(-2)\)

b) \(f^{-1}(0)\)

Let g be the function such that g(x) = 4x + 2

c) Work out fg(1)

d) Draw a sketch of \(y=f^{-1}(x)\)

Hint

Think of this question in terms of inputs and outputs

f(-2) is the output from the function f when we input 2. From the graph, what is the y value, if the x value is -2?

f(-2) = 1

ff(-2) is the value that we get when the output from f(-2) becomes the new input into the function f

ff(-2) = 2

Full Solution

Print from here

 

Let \(f(x)=\frac{ax-1}{x-a},\ x\neq\ a\)

a) Show that \(f^{-1}(x)=f(x)\)

b) Hence, find ff(b)

c) The graph of y = g(x) is shown below. Find the values of a if fg(3) = a + 5

Hint

b) Notice the word 'hence'. This means that you should use what you found out in part a to answer this question

Full Solution

Print from here

 

Let the function f be defined by

\(f(x)=\frac{x-k}{2-x},x\neq2\)

a) Find \(f^{-1}(x)\) in terms of k

b) Find k if \(y=f(x)\) and \(y=f^{-1}(x)\) intersect at (-2 , -2) and (3 , 3)

Hint

Solve \(f(x)=f^{-1}(x)\)

\(\frac{x-k}{2-x}=\frac{2x+k}{1+x}\)

Full Solution

Print from here

 

Let f(x) = 2x²+12x +11

The function can also be expressed in the form f(x) = a(x-h)² +k

a) Find the equation of the axis of symmetry

b) Write down the value of h

c) Write down the value of k

Hint

h and k give the coordinates of the vertex (h,k)

Full Solution

Print from here

 

Let f(x) = x² + 2px + (3p+4)

Find the value of p so that f(x)=0 has two equal roots.

Hint

You can use the discriminant to solve this question

\(\triangle =b^ 2-4ac\)

For two equal roots , b² - 4ac = 0

Full Solution

Print from here

 

Hint

If f and g intersect at two distinct points, then f(x)=g(x) has two roots (discriminant >0)

Full Solution

Print from here

 

A quadratic function f can be written in the form f(x) = a(x - h)² + k . The graph of f has vertex (-1 , 4) and has y-intercept at (0 , 5).

a) Find the value of a , h and k.

b) The line y = mx + 1 is a tangent to the curve f. Find the value of m.

Hint

b) If the line y = mx + 1 is a tangent to the curve to the curve of f, then the solution to

mx+1=a(x-h)²+k

has one repeated root.

Full Solution

Print from here

 

The graph of f(x) has a local maxima at \((1 - a , 2b)\) and a local minima at \((3a,b-3)\).

a) Find the coordinates of the local maxima of \(f(x+a)-2b\)

b) Find the coordinates of the local minima of \(2f(3x)\)

Hint

It helps to sketch a graph of what the function might look like then mark on the local maxima and minima

a) Consider the 2 transformations for \(f(x+a)-2b\) and apply them to the point \((1 - a , 2b)\)
b) Consider the 2 transformations for \(2f(3x)\)and apply them to the point \((3a,b-3)\)

Full Solution

Print from here

 

Consider the function f(x) = x³ - 4x² - x + 6 , \(x \in \mathbb{R}\)

The graph of f is translated two units to the left and 3 units up to form the function g(x). Express g(x) in the form ax³ + bx² + cx + d where \(a,b,c,d \in \mathbb{Z}\)

Hint

g(x) = f(x + 2) + 3

Full Solution

Print from here

 

Hint

Use the properties of indices to re-write the equation of the graph

\(m^x\times m^y=m^{x+y}\)

Full Solution

Print from here