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)

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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

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Hint

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

Draw a sketch to help you:

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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

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Find the midpoint of AB

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

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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

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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

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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

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

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

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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)

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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

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Hint

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

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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.

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Let \(f(x)=\frac{3x-2}{x-a},x\neq\ a\)

a) Find the inverse function \(f^{-1}(x)\) in terms of a

b) Find the value of a such that f is a self-inverse function

Hint

If f is a self inverse function, then \(f(x)=f^{-1}(x)\)

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The function f is defined by \(f(x)=\frac{6x+1}{2x-1},x\in\mathbb{R},x\neq\frac{1}{2}\)

a) Write f(x) in the form \(A+\frac{B}{2x-1}\) where A and B are constants

b) Sketch the graph of f(x) stating the equations of any asymptotes and the coordinates of any intercepts with the axes

Hint

The horizontal asymptote is at x = A

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Sketch the graph of \(f\left(x\right)=\frac{x^2+x-1}{x+2}\) giving the equations of any asymptotes and the coordinates of the x and y intercepts as well as any stationary points

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Hint

Before trying to sketch this function

• Find the equation of the vertical asymptote
• Check for 'the hole' by looking for common factpr in numerator and denominator
• Find the equation of the oblique asymptote
• Find the y intercept
• Find any x intercepts
• Find any stationary points

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Find the value(s) of a such that \(f(x)=\frac{x+1}{ax^2+3x+2}\) has only one vertical asymptote.

Hint

If f(x) has only one vertical asymptote, there are two possibilities

1) ax² + 3x + 2 has 2 equal factors

2) the numerator and denominator have a common factor

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In the quadratic equation \(px^{ 2 }−45x+25=0\quad ,\quad p\in \mathbb{Z}\) ,one root is two times the other.

Find the value of p.

Hint

Let the roots be \(\alpha, 2\alpha\)

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Let \(f(x)=2x^{ 4 }+x^{ 3 }−14x^{ 2 }+5x+6\quad ,\quad x\in \mathbb{R}\)

a. For the polynomial equation f(x) = 0 , find the value of

1. the sum of the roots
2. the product of the roots

b. A new polynomial is defined by g(x) = f(x - 2). Find the sum of the roots of the equation g(x) = 0

Hint

b. What transformation is represented by g(x) = f(x - 2)? What would happen to the position of the roots?

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Consider the equation \(8x^{ 3 }−42x^{ 2 }+px−27=0.\)

1. State
   1. the sum of the roots of the equation
   2. the product of the roots of the equation
2. The roots of this equation are three consecutive terms of a geometric sequence. Taking the roots to be \(\frac{\alpha}{\beta},\alpha,\alpha\beta\) show that one of the roots is \(\frac{3}{2}\)
3. Solve the equation.
4. Find the value of p.

Hint

b. What is the product of the three roots \(\frac{\alpha}{\beta},\alpha,\alpha\beta\)?

c. What is the sum of the three roots \(\frac{\alpha}{\beta},\alpha,\alpha\beta\)?

d. The equation is \(a(x-\frac{\alpha}{\beta})(x-\alpha)(x-\alpha\beta)=0\)

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One root of the equation z² + bz + c = 0 is 2+3i where \(b,c\in\mathbb{Z}\).

Find the value of b and the value of c.

Hint

If z=2+3i is a root of the equation, then the complex conjugate z=2-3i is also a root of the equation.

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The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).

a. Show that \(\alpha+\beta=\frac{1}{2}.\)

b. Find \(\alpha\beta\).

c. Hence find the two real roots α and β.

d. Find the values of p and q.

Hint

a. If γ=2−5i is a root…then δ = 2+5i is also a root. Work out γ+δ and sum of 4 roots

b. Work out γδ and the product of the 4 roots

c. Work out α and β using the two equations for \(\alpha+\beta\) and \(\alpha\beta\).

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