Exam-Style Questions.

The diagram below is a sketch of \(y = f(x)\) where \(f(x)\) is a quadratic function.

The graph intersects the x-axis where \(x=-2\) and \(x = 0.5\).

Which of the following is the solution of \(f(x) \le 0\) ?

• (a) \( x \ge -2 \) or \(x \ge 0.5\)
  
  
• (b) \( -2 \ge x \ge 0.5\)
  
  
• (c) \( x \ge -2 \) and \( x \le 0.5\)
  
  
• (d) \( x \le -2 \) and \( x \ge 0.5\)
  
  

Worked Solution

\(3x^2+9x-12\) can be written in the form \(a(x+b)^2-c\).

Find the values of \(a\), \(b\) and \(c\).

Worked Solution

The volume of the larger red cube is 386cm³ greater than the volume of the smaller blue cube.

The lengths of the edges of the blue cube can be expressed as \(x\) cm.

The lengths of the edges of the red cube can be expressed as \(x+2\) cm.

(a) Show that \( x^2 + 2x - 63 = 0 \).

(b) Find the volume of the blue cube.

Worked Solution

The images below show a graphic display calculator screen with different functions displayed as graphs.

a) Which function is trigonometric?

b) Which function is inversely proportional to \(x\)?

c) Which function is exponential?

d) Which function is proportional to \(x^3\)?

Worked Solution

(a) Write \(x^2 - 6x + 5\) in the form \((x-a)^2 + b\).

(b) Write down the coordinates of the turning point of the graph of \(y = x^2 - 6x + 5\).

Worked Solution

Match the equation with the letter of its graph

Worked Solution

The graph of the curve A with equation \(y=f(x)\) is transformed to give the graph of the curve B with equation \(y=5-f(x)\).

The point on A with coordinates (3, 9) is mapped to the point W on B.

Find the coordinates of W.

Worked Solution

The graph of y = f(x) is drawn accurately on the grid.

(a) Write down the coordinates of the turning point of the graph.

(b) Write down estimates for the roots of f(x) = 0

(c) Use the graph to find an estimate for f(-5.5).

Worked Solution

(a) Show that the equation \(\frac{3}{x+1}+\frac{3x-9}{2}=1\) can be simplified to \(3x^2-8x-5=0\).

(b) Solve the equation \(3x^2-8x-5=0\) showing all of your working and giving answers to three significant figures.

(c) The total surface area of a cone with radius \(x\) and slant height \(8x\) is equal to the area of a circle with radius r. Show that \(r = 3x\).

[The curved surface area, \(A\), of a cone with radius \(r\) and slant height \(l\) is \(A=\pi rl\).]

Worked Solution

(a) By completing the square, solve \(x^2+8x+13=0\) giving your answer to three significant figures.

(b) From the completed square you found in part (a) find the minimum value of the curve \(y=x^2+8x+13\).

Worked Solution

In this question all measurements are in centimetres.

The height of a triangle is \(h\) and the height of a rectangle is \((h + 4)\).

The length of the base of the triangle is \(x\) and the length of the rectangle is \((x + 2)\).

The area of the triangle is \(21 \text{ cm}^2\) and the area of the rectangle is \(90 \text{ cm}^2\).

(a) Write down an expression, in terms of \(x\), for the height of the rectangle.

(b) Show that \(x^2 - 10x + 21 = 0\).

(c) By factorising and solving \(x^2 - 10x + 21 = 0\), find the two possible heights of the triangle.

Image not drawn to scale, nor to the proportions of the triangle and rectangle mentioned in the question. In fact, this image is for decoration only and has nothing to do with the question

Worked Solution

Consider the graph of the function \(f(x)=7-3x^2-x^3\)

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where \(f(x)>5\).

(d) Draw the tangent to the curve at \(x=-3\) on the graph.

(e) Write down the equation of the tangent at \(x=-3\).

Worked Solution

A red rug has a width of \(x-3\) cm and a length of \(4x\) cm.

(a) Write down an expression for the area, A, in cm², of the rug.

The area of the rug is 3240 cm².

(b) Calculate the value of \(x\).

(c) Hence, write down the value of the length and of the width of the rug in centimetres.

Worked Solution

The graph of the following equation is drawn and then reflected in the x-axis

(a) What is the equation of the reflected curve?

The original curve is reflected in the y-axis.

(b) What is the equation of this second reflected curve?

Worked Solution

In the diagram below, which is not drawn to scale, all dimensions are in centimetres and all angles are multiples of 90^o. If the shaded area is 698cm², work out the value of \(x\).

Worked Solution

Given that:

find the possible values of \(x\).

Worked Solution

The area of triangle ABC (not drawn to scale) is

If AB = \(x+2\) metres and AC = \(2x+1 \) metres, find the value of \(x\).

Worked Solution

(a) Write \(2x^2+8x+27\) in the form \(a(x+b)^2+c\) where \(a\), \(b\), and \(c\) are integers, by 'completing the square'

(b) Hence, or otherwise, find the line of symmetry of the graph of \(y = 2x^2+8x+27\)

(c) Hence, or otherwise, find the turning point of the graph of \(y = 2x^2+8x+27\)

Worked Solution

The function \(f\) is defined by:

(a) Write down the equation of the vertical asymptote of the graph of \(f\).

(b) Write down the equation of the horizontal asymptote of the graph of \(f\).

(c) Find the coordinates of the \(x\)-axis and \(y\)-axis intercepts.

(d) Sketch the graph of \(f\).

Worked Solution

A function \(f\) is defined by \(f(x) = 2 + \dfrac{1}{3-x}, \text{ where } x \in \mathbb{R}, x \neq 3.\)

The graph of \(y=f(x)\) has a vertical asymptote and a horizontal asymptote.

(a) Write down the equation of the horizontal asymptote;

(b) Write down the equation of the vertical asymptote;

Find the coordinates of the point where the graph of \(y\) intersects:

(c) the y-axis;

(d) the x-axis.

Worked Solution

A function is defined as \(f(x) = 2{(x - 3)^2} - 5\) .

(a) Show that \(f(x) = 2{x^2} - 12x + 13\).

(b) Write down the equation of the axis of symmetry of this graph.

(c) Find the coordinates of the vertex of the graph of \(f(x)\).

(d) Write down the y-intercept.

(e) Make a sketch the graph of \(f(x)\).

Let \(g(x) = {x^2}\). The graph of \(f(x)\) may be obtained from the graph of \(g(x)\) by the two transformations:

• a stretch of scale factor \(s\) in the y-direction;
• followed by a translation of \(\left( {\begin{array}{*{20}{c}} j\\ k \end{array}} \right)\) .

(f) Find the values of \(j\), \(k\) and \(s\).

Worked Solution

\(f\) and \(g\) are two functions such that \(g(x)=3f(x+2)+7\).

The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:

A vertical stretch by a factor of \(a\) , followed by a translation \(\begin{pmatrix}b \\c \\ \end{pmatrix}\)

Find the values of

(a) \(a\);

(b) \(b\);

(c) \(c\).

(d) Consider two other functions \(h\) and \(j\). Let \(h(x)=-j(2x)\). The point A(8, 7) on the graph of \(j\) is mapped to the point B on the graph of \(h\). Find the coordinates of B.

Worked Solution

The diagram shows part of the graph of \(f(x)=Ae^{kx}+2\).

The y-intercept is at \((0,10)\).

(a) Show that \(A=8\).

(b) Given that \(f(8)=3.62\) (correct to 3 significant figures), find the value of \(k\).

(c) (i) Using your value of \(k\), find \(f'(x)\).

    (ii) Hence, explain why \(f\) is a decreasing function.

    (iii) Find the equation of the horizontal asymptote of the graph \(f\).

Let \(g(x)=-x^2+7x+5\)

(d) Find the area enclosed by the graphs of \(f\) and \(g\).

Worked Solution

Let \(f(x) = \frac{9x-3}{bx+9}\) for \(x \neq -\frac9b, b \neq 0\).

(a) The line \(x = 3\) is a vertical asymptote to the graph of \(f\). Find the value of b.

(b) Write down the equation of the horizontal asymptote to the graph of \(f\).

(c) The line \(y = c\) , where \(c\in \mathbb R\) intersects the graph of \( \begin{vmatrix}f(x) \end{vmatrix} \) at exactly one point. Find the possible values of \(c\).

Worked Solution