Past Exam Questions

Express \(4x^2 - 12x + 13\) in the form \((2x + a)^2 + b\), where \(a\) and \(b\) are constants.

The function \(f\) is defined by \(f(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\). Express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\).

Express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants.

Express \(x^2 + 6x + 2\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.

The function \(f\) is defined for \(x \in \mathbb{R}\) by \(f(x) = x^2 + ax + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(f(x) = 0\) are \(x = 1\) and \(x = 9\). Find:

1. the values of \(a\) and \(b\),
2. the coordinates of the vertex of the curve \(y = f(x)\).

Express \(2x^2 - 12x + 7\) in the form \(a(x+b)^2 + c\), where \(a, b\) and \(c\) are constants.

Express \(2x^2 - 10x + 8\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants, and use your answer to state the minimum value of \(2x^2 - 10x + 8\).

Express \(4x^2 - 12x\) in the form \((2x + a)^2 + b\).

A curve is described by the equation \(y = 2x^2 - 3x\). Express \(2x^2 - 3x\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\), and \(c\) are constants, and determine the coordinates of the vertex of the curve.

The function \(f\) is defined as \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\). Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\).

Rewrite the expression \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\). Then, find the coordinates of the minimum point on the curve.

Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).

Express \(2x^2 - 4x + 1\) in the form \(a(x + b)^2 + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2x^2 - 4x + 1\).

The equation of a curve is \(y = 4x^2 + 20x + 6\).

1. Express the equation in the form \(y = a(x + b)^2 + c\), where \(a, b,\) and \(c\) are constants.
2. Hence solve the equation \(4x^2 + 20x + 6 = 45\).
3. Sketch the graph of \(y = 4x^2 + 20x + 6\) showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the \(x\)- and \(y\)-axes.

(a) Express \(x^2 - 8x + 11\) in the form \((x + p)^2 + q\) where \(p\) and \(q\) are constants.

(b) Hence find the exact solutions of the equation \(x^2 - 8x + 11 = 1\).

Express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\).

Rewrite the expression \(5y^2 - 30y + 50\) in the form \(5(y + a)^2 + b\), where \(a\) and \(b\) are constants.

Express \(16x^2 - 24x + 10\) in the form \((4x + a)^2 + b\).

Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.

The equation of a curve is given by \(y = 2x^2 + kx + k - 1\), where \(k\) is a constant. Given that \(k = 2\), express the equation of the curve in the form \(y = 2(x + a)^2 + b\), where \(a\) and \(b\) are constants. Also, state the coordinates of the vertex of the curve.

A line has equation \(y = 6x - c\) and a curve has equation \(y = cx^2 + 2x - 3\), where \(c\) is a constant. The line is a tangent to the curve at point \(P\).

Find the possible values of \(c\) and the corresponding coordinates of \(P\).

The line x + 2y = 9 intersects the curve xy + 18 = 0 at the points A and B. Find the coordinates of A and B.

The equation of a curve is \(y = 4x^2 - kx + \frac{1}{2}k^2\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac{3}{4}\), find the values of \(k\) and \(a\).

A line with equation \(y = mx - 6\) is a tangent to the curve with equation \(y = x^2 - 4x + 3\).

Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.

The diagram shows the curve \(y = 7\sqrt{x}\) and the line \(y = 6x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).

For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).

The equation of a curve is \(y^2 + 2x = 13\) and the equation of a line is \(2y + x = k\), where \(k\) is a constant. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.

The diagram shows the line \(2y = x + 5\) and the curve \(y = x^2 - 4x + 7\), which intersect at the points \(A\) and \(B\). Findthe \(x\)-coordinates of \(A\) and \(B\),

The equation of a curve C is \(y = 2x^2 - 8x + 9\) and the equation of a line L is \(x + y = 3\).

(i) Find the x-coordinates of the points of intersection of L and C.

(ii) Show that one of these points is also the stationary point of C.

The equation of a curve is \(xy = 12\) and the equation of a line \(l\) is \(2x + y = k\), where \(k\) is a constant.

In the case where \(k = 11\), find the coordinates of the points of intersection of \(l\) and the curve.

Find the coordinates of the points of intersection of the line \(y + 2x = 11\) and the curve \(xy = 12\).

The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer.
2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\).

Points A and B lie on the curve \(y = x^2 - 4x + 7\). Point A has coordinates (4, 7) and B is the stationary point of the curve. The equation of a line L is \(y = mx - 2\), where \(m\) is a constant.

(i) In the case where L passes through the mid-point of AB, find the value of \(m\).

(ii) Find the set of values of \(m\) for which L does not meet the curve.

A curve has equation \(y = x^2 - x + 3\) and a line has equation \(y = 3x + a\), where \(a\) is a constant.

(i) Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x^2 - 4x + (3 - a) = 0\). [1]

(ii) For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is \(-1\). Find the \(x\)-coordinate of the other point of intersection. [2]

(iii) For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\). [4]

A straight line has equation \(y = -2x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac{2}{x - 3}\).

(i) Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2x^2 - (6 + k)x + (2 + 3k) = 0\). [1]

(ii) Find the two values of \(k\) for which the line is a tangent to the curve. [3]

The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).

(iii) Find the coordinates of \(A\) and \(B\) and the equation of the line \(AB\). [6]

The diagram shows the curve \(y = 2x^5 + 3x^3\) and the line \(y = 2x\) intersecting at points \(A, O\) and \(B\).

(i) Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2x^4 + 3x^2 - 2 = 0\).

(ii) Solve the equation \(2x^4 + 3x^2 - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form.

The equation of a curve is \(y = x^2 - 3x + 4\).

(i) Show that the whole of the curve lies above the \(x\)-axis.

(ii) Find the set of values of \(x\) for which \(x^2 - 3x + 4\) is a decreasing function of \(x\).

The equation of a line is \(y + 2x = k\), where \(k\) is a constant.

(iii) In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.

(iv) Find the value of \(k\) for which the line is a tangent to the curve.

Solve the equation \(8x^6 + 215x^3 - 27 = 0\).

Use an appropriate substitution to solve the equation:

\((2x - 3)^2 - \frac{4}{(2x - 3)^2} - 3 = 0\).

Solve the equation \(4x - 11x^{\frac{1}{2}} + 6 = 0\), showing all necessary steps.

Determine the intersection points of the curves given by the equations:

\(y = x^{\frac{2}{3}} - 1\) and \(y = x^{\frac{1}{3}} + 1\).

Determine the real roots of the equation \(\frac{18}{x^4} + \frac{1}{x^2} = 4\).

The function \(f\) is defined for \(x \in \mathbb{R}\) by \(f(x) = x^2 - 6x + c\), where \(c\) is a constant. It is given that \(f(x) > 2\) for all values of \(x\). Find the set of possible values of \(c\).

Solve the equation \(3x + 2 = \frac{2}{x - 1}\).

The function \(f\) is defined by \(f(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\). Find the set of values of \(x\) for which \(f(x) < 9\), giving your answer in exact form.

A curve is described by the equation \(y = 2x^2 - 6x + 5\). Determine the range of \(x\) values for which \(y > 13\).

Find the set of values of \(x\) for which \(x^2 + 6x + 2 > 9\).

The function \(f\) is defined by \(f : x \mapsto 6x - x^2 - 5\) for \(x \in \mathbb{R}\). Find the set of values of \(x\) for which \(f(x) \leq 3\).

Find the set of values of \(x\) satisfying \(4x^2 - 12x > 7\).

A curve is defined by the equation \(y = 2x^2 - 3x\). Determine the set of \(x\) values for which \(y > 9\).

The function \(f\) is defined by \(f: x \mapsto x^2 - 3x\) for \(x \in \mathbb{R}\). Find the set of values of \(x\) for which \(f(x) > 4\).

A line has equation \(y = 2x + 3\) and a curve has equation \(y = cx^2 + 3x - c\), where \(c\) is a constant.

Showing all necessary working, determine which of the following statements is correct.

A. The line and curve intersect only for a particular set of values of \(c\).

B. The line and curve intersect for all values of \(c\).

C. The line and curve do not intersect for any values of \(c\).

The equation of a curve is \(y = (2k - 3)x^2 - kx - (k - 2)\), where \(k\) is a constant. The line \(y = 3x - 4\) is tangent to the curve.

Find the value of \(k\).

A line has equation \(y = 3x + k\) and a curve has equation \(y = x^2 + kx + 6\), where \(k\) is a constant.

Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.

A curve has equation \(y = 3x^2 - 4x + 4\) and a straight line has equation \(y = mx + m - 1\), where \(m\) is a constant.

Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.

The equation of a curve is \(y = 2x^2 + m(2x + 1)\), where \(m\) is a constant, and the equation of a line is \(y = 6x + 4\).

Show that, for all values of \(m\), the line intersects the curve at two distinct points.

Find the set of values of m for which the line with equation \(y = mx - 3\) and the curve with equation \(y = 2x^2 + 5\) do not meet.

Find the set of values of m for which the line with equation \(y = mx + 1\) and the curve with equation \(y = 3x^2 + 2x + 4\) intersect at two distinct points.

The equation of a curve is \(y = 2x^2 + kx + k - 1\), where \(k\) is a constant. Given that the line \(y = 2x + 3\) is a tangent to the curve, find the value of \(k\).

The equation of a line is \(y = mx + c\), where \(m\) and \(c\) are constants, and the equation of a curve is \(xy = 16\).

(a) Given that the line is a tangent to the curve, express \(m\) in terms of \(c\).

(b) Given instead that \(m = -4\), find the set of values of \(c\) for which the line intersects the curve at two distinct points.

A line has equation \(y = 3kx - 2k\) and a curve has equation \(y = x^2 - kx + 2\), where \(k\) is a constant.

(i) Find the set of values of \(k\) for which the line and curve meet at two distinct points.

(ii) For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the x-axis.

A straight line has gradient m and passes through the point (0, -2). Find the two values of m for which the line is a tangent to the curve y = x^2 - 2x + 7 and, for each value of m, find the coordinates of the point where the line touches the curve.

Find the set of values of p for which the equation \(4x^2 - 24x + p = 0\) has no real roots.

The line \(4y = x + c\), where \(c\) is a constant, is a tangent to the curve \(y^2 = x + 3\) at the point \(P\) on the curve.

(i) Find the value of \(c\).

(ii) Find the coordinates of \(P\).

A curve has equation \(y = 2x^2 - 3x + 1\) and a line has equation \(y = kx + k^2\), where \(k\) is a constant.

(i) Show that, for all values of \(k\), the curve and the line meet. [4]

(ii) State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve. [4]

The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.

Find the set of values of \(k\) for which the line does not meet the curve.

A line has equation \(y = x + 1\) and a curve has equation \(y = x^2 + bx + 5\). Find the set of values of the constant \(b\) for which the line meets the curve.

The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.

(i) Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis.

(ii) Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve.

Find the set of values of a for which the curve \(y = -\frac{2}{x}\) and the straight line \(y = ax + 3a\) meet at two distinct points.

Find the set of values of k for which the equation \(2x^2 + 3kx + k = 0\) has distinct real roots.

Find the set of values of k for which the curve y = kx^2 - 3x and the line y = x - k do not meet.

A curve has equation \(y = 2x^2 - 6x + 5\). Find the value of the constant \(k\) for which the line \(y = 2x + k\) is a tangent to the curve.

The function \(f\) is defined by \(f : x \mapsto 6x - x^2 - 5\) for \(x \in \mathbb{R}\).

Given that the line \(y = mx + c\) is a tangent to the curve \(y = f(x)\), show that \(4c = m^2 - 12m + 16\).

The line with equation \(y = kx - k\), where \(k\) is a positive constant, is a tangent to the curve with equation \(y = -\frac{1}{2x}\).

Find, in either order, the value of \(k\) and the coordinates of the point where the tangent meets the curve.

Find the values of the constant m for which the line y = mx is a tangent to the curve y = 2x^2 - 4x + 8.

A line has equation \(y = 2x - 7\) and a curve has equation \(y = x^2 - 4x + c\), where \(c\) is a constant. Find the set of possible values of \(c\) for which the line does not intersect the curve.

The function \(f\) is defined by \(f : x \mapsto 2x^2 - 6x + 5\) for \(x \in \mathbb{R}\).

Find the set of values of \(p\) for which the equation \(f(x) = p\) has no real roots.

Find the set of values of k for which the line y = 2x - k meets the curve y = x^2 + kx - 2 at two distinct points.

Find the set of values of k for which the equation \(2x^2 - 10x + 8 = kx\) has no real roots.

A line has equation \(y = 2x + c\) and a curve has equation \(y = 8 - 2x - x^2\). For the case where the line is a tangent to the curve, find the value of the constant \(c\).

The straight line \(y = mx + 14\) is a tangent to the curve \(y = \frac{12}{x} + 2\) at the point \(P\). Find the value of the constant \(m\) and the coordinates of \(P\).

A curve has equation \(y = x^2 - 4x + 4\) and a line has equation \(y = mx\), where \(m\) is a constant.

Find the non-zero value of \(m\) for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

The line \(y = \frac{x}{k} + k\), where \(k\) is a constant, is a tangent to the curve \(4y = x^2\) at the point \(P\). Find

1. the value of \(k\),
2. the coordinates of \(P\).

Find the set of values of k for which the line 2y + x = k intersects the curve xy = 6 at two distinct points.

A line has equation \(y = 3x - 2k\) and a curve has equation \(y = x^2 - kx + 2\), where \(k\) is a constant.

Show that the line and the curve meet for all values of \(k\).

Find the value of k for which y = 6x + k is a tangent to the curve y = 7/√x.

(i) A straight line passes through the point (2, 0) and has gradient m. Write down the equation of the line.

(ii) Find the two values of m for which the line is a tangent to the curve \(y = x^2 - 4x + 5\). For each value of m, find the coordinates of the point where the line touches the curve.

The equation of a curve is \(y^2 + 2x = 13\) and the equation of a line is \(2y + x = k\), where \(k\) is a constant. Find the value of \(k\) for which the line is a tangent to the curve.

A line has equation \(y = kx + 6\) and a curve has equation \(y = x^2 + 3x + 2k\), where \(k\) is a constant. Find the two values of \(k\) for which the line is a tangent to the curve.

Find the set of values of m for which the line y = mx + 4 intersects the curve y = 3x^2 - 4x + 7 at two distinct points.

The equation \(x^2 + px + q = 0\), where \(p\) and \(q\) are constants, has roots \(-3\) and \(5\).

(i) Find the values of \(p\) and \(q\).

(ii) Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x^2 + px + q + r = 0\) has equal roots.

A curve has equation \(y = kx^2 + 1\) and a line has equation \(y = kx\), where \(k\) is a non-zero constant.

(i) Find the set of values of \(k\) for which the curve and the line have no common points. [3]

(ii) State the value of \(k\) for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve. [4]

The equation of a curve is \(y = \frac{9}{2-x}\).

Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points.

The function \(f : x \mapsto 2x^2 - 8x + 14\) is defined for \(x \in \mathbb{R}\). Find the values of the constant \(k\) for which the line \(y + kx = 12\) is a tangent to the curve \(y = f(x)\).

Determine the set of values of k for which the line 2y = x + k does not intersect the curve y = x^2 - 4x + 7.

Find the set of values of k for which the equation \(8x^2 + kx + 2 = 0\) has no real roots.

Find the set of values of k for which the line y = kx - 4 intersects the curve y = x^2 - 2x at two distinct points.

Determine the set of values of the constant k for which the line y = 4x + k does not intersect the curve y = x².

Find the value of the constant c for which the line y = 2x + c is a tangent to the curve y² = 4x.

The equation of a curve is \(xy = 12\) and the equation of a line \(l\) is \(2x + y = k\), where \(k\) is a constant.

Find the set of values of \(k\) for which \(l\) does not intersect the curve.

The equation of a curve is \(y = 4x^2 - kx + \frac{1}{2}k^2\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

Given instead that \(a = -\frac{7}{2}\), find the values of \(k\) for which the line is a tangent to the curve.

A curve has equation \(y = x^2 + 2cx + 4\) and a straight line has equation \(y = 4x + c\), where \(c\) is a constant.

Find the set of values of \(c\) for which the curve and line intersect at two distinct points.

A curve has equation \(y = kx^2 + 2x - k\) and a line has equation \(y = kx - 2\), where \(k\) is a constant.

Find the set of values of \(k\) for which the curve and line do not intersect.

It is given that the equation \(16x^2 - 24x + 10 = k\), where \(k\) is a constant, has exactly one root.

Find the value of this root.

The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
2. Find the median and the interquartile range of the heights of the players in the Aces.
3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.

The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
2. Find the median and the interquartile range for the heights of the Anvils.

The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.

104, 88, 82, 65, 44, 38, 35, 34, 28, 28, 18, 18, 17, 17, 14, 13, 13, 12, 12, 10, 10, 10, 9, 6, 5, 2, 2

Draw a stem-and-leaf diagram to illustrate the data.

The masses, in grams, of components made in factory A and components made in factory B are shown below.

Factory A

Factory B

(i) Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the two factories.

(ii) Find the median and the interquartile range for the masses of components made in factory B.

(iii) Make two comparisons between the masses of components made in factory A and the masses of those made in factory B.

The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.

1. Draw a back-to-back stem-and-leaf diagram to represent this information.
2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.

The weights, in kilograms, of the 15 rugby players in each of two teams, A and B, are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team A on the left-hand side of the diagram and team B on the right-hand side.
2. Find the interquartile range of the weights of the players in team A.
3. A new player joins team B as a substitute. The mean weight of the 16 players in team B is now \(93.9\ \text{kg}\). Find the weight of the new player.

A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.

23, 19, 32, 14, 25, 22, 26, 36, 45, 42, 47, 28, 17, 38, 15, 46, 18, 26, 22, 41, 19, 21, 28, 24, 30

Draw a stem-and-leaf diagram to represent the data.

Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the data.
2. Make two comparisons between the estimates of the adults and the children.

The following are the annual amounts of money spent on clothes, to the nearest $10, by 27 people.

1. Construct a stem-and-leaf diagram for the data.
2. Find the median and the interquartile range of the data.
3. An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. List the outliers.

Prices in dollars of 11 caravans in a showroom are as follows.

16 800, 18 500, 17 700, 14 300, 15 500, 15 300, 16 100, 16 800, 17 300, 15 400, 16 400

(i) Represent these prices by a stem-and-leaf diagram.

(ii) Write down the lower quartile of the prices of the caravans in the showroom.

The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.

Flat screen: 0.85, 0.94, 0.91, 0.96, 1.04, 0.89, 1.07, 0.92, 0.76

Conventional: 0.69, 0.65, 0.85, 0.77, 0.74, 0.67, 0.71, 0.86, 0.75

1. Represent this information on a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.

The times taken, in minutes, to complete a cycle race by 19 cyclists from each of two clubs, the Cheetahs and the Panthers, are represented in the following back-to-back stem-and-leaf diagram.

Key: \( 7 \mid 9 \mid 1 \) means 97 minutes for Cheetahs and 91 minutes for Panthers.

(a) Find the median and the interquartile range of the times of the Cheetahs.

The median and interquartile range for the Panthers are 103 minutes and 14 minutes respectively.

(b) Make two comparisons between the times taken by the Cheetahs and the times taken by the Panthers.

Another cyclist, Kenny, from the Cheetahs also took part in the race. The mean time taken by the 20 cyclists from the Cheetahs was 99 minutes.

(c) Find the time taken by Kenny to complete the race.

The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows:

Sugar: 1.961, 1.983, 2.008, 2.014, 1.968, 1.994, 2.011, 2.017, 1.977, 1.984, 1.989

Flour: 1.945, 1.962, 1.949, 1.977, 1.964, 1.941, 1.953

(i) Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand side.

(ii) Find the median and interquartile range of the weights of the bags of sugar.

The numbers of people travelling on a certain bus at different times of the day are as follows.

17, 5, 2, 23, 16, 31, 8, 22, 14, 25, 35, 17, 27, 12, 6, 23, 19, 21, 23, 8, 26

(i) Draw a stem-and-leaf diagram to illustrate the information given above. [3]

(ii) Find the median, the lower quartile, the upper quartile and the interquartile range. [3]

(iii) State, in this case, which of the median and mode is preferable as a measure of central tendency, and why. [1]

The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.

115, 120, 158, 132, 125, 104, 142, 160, 145, 104, 162, 117, 109, 124, 134

Draw a stem-and-leaf diagram to represent the data.

The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\).

Stem-and-leaf diagram:

Key: \( 1 \mid 5 \) represents 15 hits.

(i) Find the median and lower quartile for the number of hits each day.

(ii) The interquartile range is 19. Find the value of \(x\).

The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9-year-olds and by the members of a class of eight 16-year-olds are shown below.

9-year-olds: 13.0, 16.1, 16.0, 14.4, 15.9, 15.1, 14.2, 13.7, 16.7, 16.4, 15.0, 13.2

16-year-olds: 14.8, 13.0, 11.4, 11.7, 16.5, 13.7, 12.8, 12.9

(i) Draw a back-to-back stem-and-leaf diagram to represent the information above.

(ii) A new pupil joined the 16-year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil’s time to swim the distance.

(i) The diagram represents the sales of Superclene toothpaste over the last few years. Give a reason why it is misleading.

(ii) The following data represent the daily ticket sales at a small theatre during three weeks.

52, 73, 34, 85, 62, 79, 89, 50, 45, 83, 84, 91, 85, 84, 87, 44, 86, 41, 35, 73, 86.

(a) Construct a stem-and-leaf diagram to illustrate the data.

(b) Use your diagram to find the median of the data.

The Lions and the Tigers are two basketball clubs. The heights, in cm, of the 11 players in each of their first team squads are given in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Lions on the left.
2. Find the median and the interquartile range of the heights of the Lions first team squad.
3. It is given that for the Tigers, the lower quartile is 183 cm, the median is 190 cm and the upper quartile is 195 cm.
4. Make two comparisons between the heights of the players in the Lions first team squad and the heights of the players in the Tigers first team squad.

Lakeview and Riverside are two schools. The pupils at both schools took part in a competition to see how far they could throw a ball. The distances thrown, to the nearest metre, by 11 pupils from each school are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Lakeview on the left-hand side.
2. Find the interquartile range of the distances thrown by the 11 pupils at Lakeview school.

The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below.

1. State one advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information.
2. Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram.
3. Find the interquartile range of the heights of the players in the Amazons.
4. Four new players join the Amazons. The mean height of the 15 players in the Amazons is now \(191.2\) cm. The heights of three of the new players are \(180\) cm, \(185\) cm and \(190\) cm. Find the height of the fourth new player.

The following table gives the weekly snowfall, in centimetres, for 11 weeks in 2018 at two ski resorts, Dados and Linva.

1. Represent the information in a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range for the weekly snowfall in Dados.
3. The median, lower quartile and upper quartile of the weekly snowfall for Linva are 12, 9 and 32 cm respectively. Use this information and your answers to part (b) to compare the central tendency and the spread of the weekly snowfall in Dados and Linva.

The annual salaries, in thousands of dollars, for 11 employees at each of two companies A and B are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with company A on the left-hand side of the diagram.
2. Find the median and the interquartile range of the salaries of the employees in company A.
3. A new employee joins company B. The mean salary of the 12 employees is now $38,500. Find the salary of the new employee.

The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.

1. Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.
2. Find the interquartile range of the times for pupils at Thaters School.

The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Dolphins on the left-hand side of the diagram and Sharks on the right-hand side.
2. Find the median and interquartile range for the Dolphins.

The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

Draw a histogram to represent this information.

The speeds, in km h^-1, of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest km h^-1. The results are summarised in the following table.

1. On the grid, draw a histogram to illustrate the data in the table. [4]
2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker. [2]

The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table.

The lengths, t minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

1. Find the value of a.
2. Calculate an estimate of the mean length of these phone calls.
3. On the grid, draw a histogram to illustrate the data in the table.

The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.

\(\begin{array}{|c|c|} \hline \text{Time (} t \text{ seconds)} & \text{Number of people} \\ \hline 0 \leq t < 20 & 320 \\ 20 \leq t < 40 & 280 \\ 40 \leq t < 60 & 220 \\ 60 \leq t < 100 & 220 \\ 100 \leq t < 140 & 100 \\ \hline \end{array}\)

(i) On the grid, draw a histogram to illustrate this information.

(ii) Calculate an estimate of the mean of \(t\).

The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.

1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10,000 on the horizontal axis.
2. Calculate an estimate of the mean capacity of these 130 stadiums.
3. Find which class in the table contains the median and which contains the lower quartile.

A survey was made of the journey times of 63 people who cycle to work in a certain town. The results are summarised in the following cumulative frequency table.

1. State how many journey times were between 25 and 45 minutes.
2. Draw a histogram on graph paper to represent the data.
3. Calculate an estimate of the mean journey time.

The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

(i) Draw a histogram on graph paper to illustrate the data.

(ii) Calculate estimates of the mean and standard deviation of these heights.

Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).

\(\begin{array}{|c|c|c|c|c|} \hline \text{Time (t minutes)} & 60 - 62 & 63 - 64 & 65 - 67 & 68 - 71 \\ \hline \text{Frequency (number of days)} & 3 & 9 & 6 & b \\ \hline \text{Frequency density} & 1 & a & 2 & 1.5 \\ \hline \end{array}\)

(i) Find the values of \(a\) and \(b\).

(ii) On graph paper, draw a histogram to represent Robert’s times.

The table summarises the lengths in centimetres of 104 dragonflies.

1. State which class contains the upper quartile.
2. Draw a histogram, on graph paper, to represent the data.

The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.

(a) Draw a histogram to represent this information.

(b) Write down the class interval which contains the median for this information.

(c) Find the greatest possible value of the interquartile range for the populations of the 150 villages.

A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.

1. Draw a histogram on graph paper to represent this information.
2. Calculate an estimate of the mean number of typing errors for these 111 people.
3. State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.

The distance of a student’s home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table.

Dominic is asked to draw a histogram to illustrate the data. Dominic’s diagram is shown below.

Give two reasons why this is not a correct histogram.

The following histogram summarises the times, in minutes, taken by 190 people to complete a race.

(i) Show that 75 people took between 200 and 250 minutes to complete the race.

(ii) Calculate estimates of the mean and standard deviation of the times of the 190 people.

(iii) Explain why your answers to part (ii) are estimates.

In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

(i) Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.

(ii) Draw, on graph paper, a histogram to illustrate the information in the table.

The table summarises the times that 112 people took to travel to work on a particular day.

1. State which time interval in the table contains the median and which time interval contains the upper quartile.
2. On graph paper, draw a histogram to represent the data.
3. Calculate an estimate of the mean time to travel to work.

The weights of 220 sausages are summarised in the following table.

1. State which interval the median weight lies in.
2. Find the smallest possible value and the largest possible value for the interquartile range.
3. State how many sausages weighed between 50 g and 60 g.
4. On graph paper, draw a histogram to represent the weights of the sausages.

The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower.

(i) Copy and complete the following frequency table for the data.

(ii) Calculate an estimate of the mean time to take a shower.

The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table.

A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The 1–10 rectangle has height 3 cm.

(i) Calculate the value of \( x \) and the height of the 51–70 rectangle.

(ii) Calculate an estimate of the mean weight of the stones.

The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

(i) Draw a histogram on graph paper to represent these results.

(ii) Calculate estimates of the mean mark and the standard deviation.

As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.

(i) Draw, on graph paper, a histogram to illustrate this information.

(ii) Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.

The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.

(a) Draw a histogram to represent this information.

From the data, the estimate of the mean time taken by these 250 employees is \( 43.2 \) minutes.

(b) Calculate an estimate for the standard deviation of these times.

The weights of 30 children in a class, to the nearest kilogram, were as follows:

50, 45, 61, 53, 55, 47, 52, 49, 46, 51, 60, 52, 54, 47, 57, 59, 42, 46, 51, 53, 56, 48, 50, 51, 44, 52, 49, 58, 55, 45

Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 41.5 kg and the fifth class having an upper boundary of 61.5 kg.

Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information.

1. What is the modal age group?
2. How many fathers were between 25 and 30 years old when their first child was born?
3. How many fathers were in the sample?

The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.

(i) Find the value of \( a \).

(ii) Draw a histogram on graph paper to represent the data.

The floor areas, \( x \) m\(^2\), of 20 factories are as follows:

150, 350, 450, 578, 595, 644, 722, 798, 802, 904, 1000, 1330, 1533, 1561, 1778, 1960, 2167, 2330, 2433, 3231

Represent these data by a histogram on graph paper, using intervals:

• \( 0 \le x < 500 \)
• \( 500 \le x < 1000 \)
• \( 1000 \le x < 2000 \)
• \( 2000 \le x < 3000 \)
• \( 3000 \le x < 4000 \)

A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.

(i) Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.

(ii) On graph paper, draw a fully labelled histogram to represent the data.

The times taken to travel to college by 2500 students are summarised in the table.

(a) Draw a histogram to represent this information.

From the data, the estimate of the mean value of \( t \) is \( 31.44 \).

(b) Calculate an estimate of the standard deviation of the times taken to travel to college.

(c) In which class interval does the upper quartile lie?

It was later discovered that the times taken to travel to college by two students were incorrectly recorded. One student’s time was recorded as \( 15 \) instead of \( 5 \) and the other’s time was recorded as \( 65 \) instead of \( 75 \).

(d) Without doing any further calculations, state with a reason whether the estimate of the standard deviation in part (b) would be increased, decreased or stay the same.

At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.

(a) Draw a histogram to represent this information.

(b) State which class interval contains the median.

(c) Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.

The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.

(a) Draw a histogram to represent this information.

(b) Calculate an estimate of the mean time taken by an employee to travel to work.

The times taken by 200 players to solve a computer puzzle are summarised in the following table.

1. Draw a histogram to represent this information.
2. Calculate an estimate of the mean time taken by these 200 players.
3. Find the greatest possible value of the interquartile range of these times.

A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.

1. Draw a histogram to represent this information.
2. State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
3. Calculate an estimate for the mean number of incorrect notes.

The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.

(a) Draw a histogram to represent this information.

(b) What is the greatest possible value of the interquartile range for the data?

(c) Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.

The weights, x kg, of 120 students in a sports college are recorded. The results are summarised in the following table.

(a) Draw a cumulative frequency graph to represent this information.

(b) It is found that 35% of the students weigh more than W kg. Use your graph to estimate the value of W.

Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.

(a) Draw a cumulative frequency graph to illustrate the data.

(b) 40% of these fish have a length of d cm or more. Use your graph to estimate the value of d.

The mean length of these 150 fish is 15.295 cm.

(c) Calculate an estimate for the variance of the lengths of the fish.

Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below.

(i) Estimate how many leaves have a length between 14 and 24 centimetres.

(ii) 10% of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).

(iii) Estimate the median and the interquartile range of the lengths.

Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below.

(iv) Compare the central tendency and the spread of the two sets of data.

The Mathematics and English A-level marks of 1400 pupils all taking the same examinations are shown in the cumulative frequency graphs below. Both examinations are marked out of 100.

Use suitable data from these graphs to compare the central tendency and spread of the marks in Mathematics and English.

Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data. [2]
2. Use your graph to estimate the median of the data. [1]
3. For 80 of the drivers, the time taken was at least \(T\) minutes. Use your graph to estimate the value of \(T\). [2]
4. Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park. [4]

The daily rainfall, \(x\) mm, in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data. [2]
2. On 100 of the days, the rainfall was \(k\) mm or more. Use your graph to estimate the value of \(k\). [2]
3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village. [6]

There are 900 students in a certain year-group. An identical puzzle is given to each student and the time taken, \(t\) minutes, to complete the puzzle is recorded. These times are summarised in the following frequency table.

On the grid, draw a cumulative frequency graph to represent the data. Use your graph to estimate the median time taken by these students to complete the puzzle.

The circumferences, \(c\) cm, of some trees in a wood were measured. The results are summarised in the table.

(i) On the grid, draw a cumulative frequency graph to represent the information.

(ii) Estimate the percentage of trees which have a circumference larger than 75 cm.

The time taken by a car to accelerate from 0 to 30 metres per second was measured correct to the nearest second. The results from 48 cars are summarised in the following table.

(i) On the grid, draw a cumulative frequency graph to represent this information. [3]

(ii) 35 of these cars accelerated from 0 to 30 metres per second in a time more than \(t\) seconds. Estimate the value of \(t\). [2]

The following histogram represents the lengths of worms in a garden.

(i) Calculate the frequencies represented by each of the four histogram columns.

(ii) On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden.

(iii) Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.

(iv) Calculate an estimate of the mean length of worms in the garden.

Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below.

(i) Estimate the median and the interquartile range of the lengths.

(ii) Estimate how many caterpillars had a length of between 2 and 3.5 cm.

(iii) 6% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).

The times taken by 120 children to complete a particular puzzle are represented in the cumulative frequency graph.

(a) Use the graph to estimate the interquartile range of the data.

35% of the children took longer than \(T\) seconds to complete the puzzle.

(b) Use the graph to estimate the value of \(T\).

In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.

(i) On graph paper draw a box-and-whisker plot to summarise this information.An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.

(ii) Show that there are no outliers.

On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.

1. Draw a cumulative frequency graph to illustrate the data.
2. 28% of these daffodils are of height h cm or more. Estimate h.
3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils.

The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.

1. On graph paper, draw a cumulative frequency graph to represent these results.
2. 64 people weigh more than \(c\) kg. Use your graph to find the value of \(c\).
3. Calculate estimates of the mean and standard deviation of the weights.

The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.

1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
2. Comment on the salaries of the people in this sample.
3. An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
   1. How high must a salary be in order to be classified as an outlier?
   2. Show that none of the salaries is low enough to be classified as an outlier.

A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.

1. Draw a cumulative frequency graph on graph paper to illustrate this information.
2. Estimate the number of days when over 30 rooms were occupied.
3. On 75% of the days at most n rooms were occupied. Estimate the value of n.

There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.

1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
2. 80% of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
3. Find how many schools have between 201 and 250 (inclusive) pupils.
4. Calculate an estimate of the mean number of pupils per school.

Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

(a) Draw a cumulative frequency graph to illustrate the data.

(b) Use your graph to estimate the 70th percentile of the data.

The birth weights of random samples of 900 babies born in country A and 900 babies born in country B are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies.

During January the numbers of people entering a store during the first hour after opening were as follows.

1. Find the values of a and b.
2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
3. Use your graph to estimate the median time after opening that people entered the store.
4. Calculate estimates of the mean, m minutes, and standard deviation, s minutes, of the time after opening that people entered the store.
5. Use your graph to estimate the number of people entering the store between \(m - \frac{1}{2}s\) and \(m + \frac{1}{2}s\) minutes after opening.

The arrival times of 204 trains were noted and the number of minutes, t, that each train was late was recorded. The results are summarised in the table.

1. Explain what -2 ≤ t < 0 means about the arrival times of trains.
2. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.

In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.

1. On graph paper, show these results using a fully labelled cumulative frequency graph.
2. Use your graph to estimate how many people watched more than 50 hours of television each week.

The times, t minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.

(a) Draw a cumulative frequency graph to illustrate the data.

(b) Use your graph to estimate the 60th percentile of the data.

It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.

(c) Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.

The distances, x m, travelled to school by 140 children were recorded. The results are summarised in the table below.

(a) On the grid, draw a cumulative frequency graph to represent these results.

(b) Use your graph to estimate the interquartile range of the distances.

(c) Calculate estimates of the mean and standard deviation of the distances.

The heights in cm of 160 sunflower plants were measured. The results are summarised on the following cumulative frequency curve.

(a) Use the graph to estimate the number of plants with heights less than 100 cm.

(b) Use the graph to estimate the 65th percentile of the distribution.

(c) Use the graph to estimate the interquartile range of the heights of these plants.

A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km, are summarised in the following table.

(a) Draw a cumulative frequency graph to illustrate the data.

(b) For 30% of these journeys the distance travelled is \(d\) km or more. Use your graph to estimate the value of \(d\).

(c) Calculate an estimate of the mean distance travelled for the 150 journeys.

The times, t minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.

(a) Draw a cumulative frequency graph to illustrate the data.

(b) 24% of the students take k minutes or longer to complete the challenge. Use your graph to estimate the value of k.

(c) Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.

A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.

1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
2. State two comparisons of the times taken by girls and boys.

A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.

23, 19, 32, 14, 25, 22, 26, 36, 45, 42, 47, 28, 17, 38, 15, 46, 18, 26, 22, 41, 19, 21, 28, 24, 30

On graph paper draw a box-and-whisker plot to represent the data.

The marks of the pupils in a certain class in a History examination are as follows.

28, 33, 55, 38, 42, 39, 27, 48, 51, 37, 57, 49, 33

The marks of the pupils in a Physics examination are summarised as follows.

Lower quartile: 28, Median: 39, Upper quartile: 67. The lowest mark was 17 and the highest mark was 74.

1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
2. State one difference, which can be seen from the diagram, between the marks for History and Physics.

The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.

(ii) Find the median and the quartiles.

(iii) On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.

In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes.

1. Find the lower quartile.
2. Represent the information by a box-and-whisker plot, using a scale of 2 cm to represent 60 minutes.

The weights, in kg, of 15 rugby players in the Rebels club and 15 soccer players in the Sharks club are shown below.

(a) Represent the data by drawing a back-to-back stem-and-leaf diagram with Rebels on the left-hand side of the diagram.

(b) Find the median and the interquartile range for the Rebels.

A box-and-whisker plot for the Sharks is shown below.

(c) On the same diagram, draw a box-and-whisker plot for the Rebels.

(d) Make one comparison between the weights of the players in the Rebels club and the weights of the players in the Sharks club.

The number of Olympic medals won in the 2012 Olympic Games by the top \(27\) countries is shown below.

Find the median and quartiles of the data and draw a box-and-whisker plot on the grid.

The weights, x kg, of 120 students in a sports college are recorded. The results are summarised in the following table.

Calculate estimates for the mean and standard deviation of the weights of the 120 students.

The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y^2 = 42\,850\), where \(y\) is the age of a Senior member in years.

(i) Find the mean age of all 32 members of the club.

(ii) Find the standard deviation of the ages of all 32 members of the club.

Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

(i) Calculate the mean price of all 51 holidays.

(ii) The prices of individual holidays with Farfield Travel are denoted by $x_F$ and the prices of individual holidays with Lacket Travel are denoted by $x_L$. By first finding $\sum x_F^2$ and $\sum x_L^2$, find the standard deviation of the prices of all 51 holidays.

Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows.

37, 40, 45, 38, 36, 38, 42, 38, 40, 39

1. Find the mode.
2. Find the median and the interquartile range.

The ages of a group of 12 people at an Art class have mean 48.7 years and standard deviation 7.65 years. The ages of a group of 7 people at another Art class have mean 38.1 years and standard deviation 4.2 years.\n\n(i) Find the mean age of all 19 people.\n\n(ii) The individual ages in years of people in the first Art class are denoted by \(x\) and those in the second Art class by \(y\). By first finding \(\Sigma x^2\) and \(\Sigma y^2\), find the standard deviation of the ages of all 19 people.

Rani and Diksha go shopping for clothes.

(i) Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs $5.50 and the coat costs $90. The mean cost of Rani’s 8 items is $29. Find the cost of a sweater.

(ii) Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha’s 5 items is $26 and the standard deviation is $0. Explain how you can tell that Diksha spends $104 on shirts.

120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.

Calculate estimates of the mean and standard deviation of the reading times.

The table shows the mean and standard deviation of the weights of some turkeys and geese.

(i) Find the mean weight of the 27 birds.

(ii) The weights of individual turkeys are denoted by \(x_t\) kg and the weights of individual geese by \(x_g\) kg. By first finding \(\Sigma x_t^2\) and \(\Sigma x_g^2\), find the standard deviation of the weights of all 27 birds.

Find the mean and variance of the following data. 5 \(-2\) 12 7 \(-3\) 2 \(-6\) 4 0 8

The heights, \(x\) cm, of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm. A person whose height was 161.8 cm left the group. (i) Find the mean height of the remaining group of 27 people. (ii) Find \(\Sigma x^2\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.

Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g. For the lemons, the mean weight is 118 g and the standard deviation is 12 g.

(i) Find the mean weight of the 45 fruits.

(ii) The individual weights of the oranges in grams are denoted by \(x_o\), and the individual weights of the lemons in grams are denoted by \(x_l\). By first finding \(\Sigma x_o^2\) and \(\Sigma x_l^2\), find the variance of the weights of the 45 fruits.

The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

Calculate estimates for the mean and standard deviation of the times taken by the athletes.

Ashfaq and Kuljit have done a school statistics project on the prices of a particular model of headphones for MP3 players. Ashfaq collected prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.

1. Name a suitable statistical diagram for Ashfaq to represent his data, together with a reason for choosing this particular diagram. [2]
2. Name a suitable statistical diagram for Kuljit to represent her data, together with a reason for choosing this particular diagram. [2]

The following are the times, in minutes, taken by 11 runners to complete a 10 km run.

48.3, 55.2, 59.9, 67.7, 60.5, 75.6, 62.5, 57.4, 53.4, 49.2, 64.1

Find the mean and standard deviation of these times.

Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, \(\Sigma x = 64\) and \(\Sigma x^2 = 352\), where \(x\) is the age of a car in years.

(i) Find the mean age of all 24 cars.

(ii) Find the standard deviation of the ages of all 24 cars.

The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below.

15, 10, 48, 10, 19, 14, 16

1. Find the mean and standard deviation of these times.
2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case.
3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate.

The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

(i) The mean cost of Fei’s rides is $2.50 and the standard deviation of the costs of Fei’s rides is $0. Explain how you can tell that the roller coaster and the water slide each cost $2.50 per ride. [2]

(ii) The mean cost of Graeme’s rides is $3.76. Find the standard deviation of the costs of Graeme’s rides. [5]

Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below.

32, 35, 45, 37, 38, 44, 33, 39, 36, 45

Find the mean and standard deviation of the lengths of these leaves.

32 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.

1. How many teams play in only 1 match?
2. How many teams play in exactly 2 matches?
3. Draw up a frequency table for the numbers of matches which the teams play.
4. Calculate the mean and variance of the numbers of matches which the teams play.

The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be:

40, 42, 45, 41, 352, 40, 50, 48, 51, 49, 47.

Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.

A group of 10 married couples and 3 single men found that the mean age \(\bar{x}_w\) of the 10 women was 41.2 years and the standard deviation of the women’s ages was 15.1 years. For the 13 men, the mean age \(\bar{x}_m\) was 46.3 years and the standard deviation was 12.7 years.

(i) Find the mean age of the whole group of 23 people.

(ii) The individual women’s ages are denoted by \(x_w\) and the individual men’s ages by \(x_m\). By first finding \(\Sigma x_w^2\) and \(\Sigma x_m^2\), find the standard deviation for the whole group.

A study of the ages of car drivers in a certain country produced the results shown in the table.

Percentage of drivers in each age group

Illustrate these results diagrammatically.

Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years.

The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms.

The mean time was calculated to be 27.5 minutes.

1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26. [4]
2. Find the standard deviation of these times. [2]

The ages, \(x\) years, of 18 people attending an evening class are summarised by the following totals: \(\Sigma x = 745, \Sigma x^2 = 33951\).

(i) Calculate the mean and standard deviation of the ages of this group of people. [3]

(ii) One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find the age of the person who left and the standard deviation of the ages of the remaining 17 people. [4]

Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.

1. Find the mean and standard deviation of the scores for team A. [2]
2. State with a reason which team has the more consistent scores. [2]

The mean and standard deviation for team B are 130.75 and 29.63 respectively.

A computer can generate random numbers which are either 0 or 2. On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

Twenty children were asked to estimate the height of a particular tree. Their estimates, in metres, were as follows.

4.1, 4.2, 4.4, 4.5, 4.6, 4.8, 5.0, 5.2, 5.3, 5.4, 5.5, 5.8, 6.0, 6.2, 6.3, 6.4, 6.6, 6.8, 6.9, 19.4

(a) Find the mean of the estimated heights.

(b) Find the median of the estimated heights.

(c) Give a reason why the median is likely to be more suitable than the mean as a measure of the central tendency for this information.

A sports club has a volleyball team and a hockey team. The heights of the 6 members of the volleyball team are summarised by \(\Sigma x = 1050\) and \(\Sigma x^2 = 193700\), where \(x\) is the height of a member in cm. The heights of the 11 members of the hockey team are summarised by \(\Sigma y = 1991\) and \(\Sigma y^2 = 366400\), where \(y\) is the height of a member in cm.

(a) Find the mean height of all 17 members of the club.

(b) Find the standard deviation of the heights of all 17 members of the club.

Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows.

50, 45, 62, 30, 40, 55, 110, 38, 52, 60, 55, 40

1. Find the median and the interquartile range for the data.
2. Give a disadvantage of using the mean as a measure of the central tendency in this case.

The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x^2\).

Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40 500.

2. Find the mean and standard deviation of all these 30 values of \(x\).

The times in minutes taken to run a marathon were recorded for a group of 13 marathon runners and were found to be as follows.

180, 275, 235, 242, 311, 194, 246, 229, 238, 768, 332, 227, 228

State which of the mean, mode or median is most suitable as a measure of central tendency for these times. Explain why the other measures are less suitable.

The heights of the 11 members of the Anvils are denoted by \(x\) cm. It is given that \(\Sigma x = 1923\) and \(\Sigma x^2 = 337221\). The Anvils are joined by 3 new members whose heights are 166 cm, 172 cm and 182 cm. Find the standard deviation of the heights of all 14 members of the Anvils.

(a) Sketch the graph of \(y = |4x - 2|\).

(b) Solve the inequality \(1 + 3x < |4x - 2|\).

Solve the inequality: \(|3x - a| > 2|x + 2a|\), where \(a\) is a positive constant.

Solve the inequality: \(|2x - 1| < 3|x + 1|\)

Solve the inequality: \(2|3x - 1| < |x + 1|\)

Solve the inequality: \(2 - 5x > 2|x - 3|\)

Solve the inequality: \(|2x - 1| > 3|x + 2|\)

Solve the inequality: \(|x - 2| < 3x - 4\)

Sketch the graph of \(y = |x - 2|\).

Solve the inequality: \(|2x - 3| > 4|x + 1|\)

Solve the inequality: \(3|2x - 1| > |x + 4|\)

Find the set of values of x satisfying the inequality:

2|2x - a| < |x + 3a|, Where a is a positive constant.

Solve the inequality \(|5x - 3| < 2|3x - 7|\).

Solve the inequality: \(|x - 3| < 3x - 4\)

Solve the inequality: \(|2x + 1| < 3|x - 2|\)

Solve the inequality: \(|x - 4| < 2|3x + 1|\)

Solve the inequality: \(2|x - 2| > |3x + 1|\)

Solve the equation: \(2|x - 1| = 3|x|\)

Solve the inequality: \(|2x - 5| > 3|2x + 1|\)

Solve the inequality: \(|x - 2| > 2x - 3\)

Solve the inequality: \(|3x - 1| < |2x + 5|\)

Find the set of values of x satisfying the inequality: \(|x + 2a| > 3|x - a|\), where a is a positive constant.

Solve the inequality: \(|4x + 3| > |x|\)

(a) Sketch the graph of \(y = |2x + 3|\).

(b) Solve the inequality \(3x + 8 > |2x + 3|\).

Solve the equation: \(|x - 2| = \left|\frac{1}{3}x\right|\)

Solve the equation: \(|4x - 1| = |x - 3|\)

Find the set of values of x satisfying the inequality:

\(3|x - 1| < |2x + 1|\)

Solve the inequality: \(|x| < |5 + 2x|\)

Solve the inequality: \(2|x - 3| > |3x + 1|\)

Solve the inequality: \(|x - 3| > 2|x + 1|\)

Solve the inequality: \(|x + 3a| > 2|x - 2a|\), where \(a\) is a positive constant.

Solve the inequality: \(2 - 3x < |x - 3|\)

Solve the inequality: \(|x - 2| > 3|2x + 1|\)

Solve the inequality: \(2x > |x - 1|\)

Solve the inequality: \(3x + 5 < |2x + 1|\)

Solve the inequality: \(|x - 3a| > |x - a|\), where \(a\) is a positive constant.

Solve the inequality: \(|2x + 1| < |x|\)

Solve the inequality: \(|x - 2| < 3 - 2x\)

Solve the inequality: \(|9 - 2x| < 1\)

Sketch the graph of \(y = |2x + 1|\)

Find, in terms of a, the set of values of x satisfying the inequality:

2|3x + a| < |2x + 3a|, where a is a positive constant.

Solve the inequality: \(|2x + 3| > 3|x + 2|\)

Solve the inequality: \(|2x - 3| < 3x + 2\)

Sketch the graph of \(y = |2x - 3|\).

The polynomial \(2x^3 + ax^2 + bx + 6\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \((x + 2)\) the remainder is \(-38\) and when \(p(x)\) is divided by \((2x - 1)\) the remainder is \(\frac{19}{2}\).

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

The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x + 2)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 1)\) the remainder is 2.

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

Find the quotient and remainder when \(6x^4 + x^3 - x^2 + 5x - 6\) is divided by \(2x^2 - x + 1\).

The polynomial \(6x^3 + ax^2 + bx - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is \(-24\). Find the values of \(a\) and \(b\).

The polynomial \(x^4 + 3x^3 + ax + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \(x^2 + x - 1\) the remainder is \(2x + 3\). Find the values of \(a\) and \(b\).

The polynomial \(x^4 + 2x^3 + ax + b\), where \(a\) and \(b\) are constants, is divisible by \(x^2 - x + 1\). Find the values of \(a\) and \(b\).

Find the quotient and remainder when \(x^4\) is divided by \(x^2 + 2x - 1\).

The polynomial \(4x^4 + ax^2 + 11x + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(x^2 - x + 2\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the real roots of the equation \(p(x) = 0\).

The polynomial \(4x^3 + ax + 2\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\).

1. Find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(p(x)\),
   2. solve the inequality \(p(x) > 0\), justifying your answer.

Show that \((x + 1)\) is a factor of \(4x^3 - x^2 - 11x - 6\).

The polynomial \(8x^3 + ax^2 + bx - 1\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((2x + 1)\) the remainder is 1.

(i) Find the values of \(a\) and \(b\).

(ii) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

The polynomial \(2x^3 + ax^2 - 11x + b\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \((2x - 1)\) and that when \(p(x)\) is divided by \((x + 1)\) the remainder is 12.

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

The polynomial \(4x^3 + ax^2 + bx - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x+1)\) and \((x+2)\) are factors of \(p(x)\).

(i) Find the values of \(a\) and \(b\).

(ii) When \(a\) and \(b\) have these values, find the remainder when \(p(x)\) is divided by \((x^2 + 1)\).

The polynomial \(ax^3 + bx^2 + x + 3\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((3x + 1)\) is a factor of \(p(x)\), and that when \(p(x)\) is divided by \((x - 2)\) the remainder is 21. Find the values of \(a\) and \(b\).

The polynomial \(f(x)\) is defined by

\(f(x) = x^3 + ax^2 - ax + 14\),

where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(f(x)\).

(i) Find the value of \(a\).

(ii) Show that, when \(a\) has this value, the equation \(f(x) = 0\) has only one real root.

The polynomial \(8x^3 + ax^2 + bx + 3\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((2x - 1)\) the remainder is 1.

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(p(x)\) is divided by \(2x^2 - 1\).

The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((3x + 1)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, factorise \(p(x)\) completely.

Find the quotient and remainder when \(2x^2\) is divided by \(x + 2\).

The polynomial \(p(x)\) is defined by

\(p(x) = x^3 - 3ax + 4a\),

where \(a\) is a constant.

(i) Given that \((x - 2)\) is a factor of \(p(x)\), find the value of \(a\).

(ii) When \(a\) has this value,

(a) factorise \(p(x)\) completely,

(b) find all the roots of the equation \(p(x^2) = 0\).

The polynomial \(p(x)\) is defined by

\(p(x) = ax^3 - x^2 + 4x - a\),

where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(p(x)\).

Find the value of \(a\) and hence factorise \(p(x)\).

The polynomial \(x^4 + 3x^3 + ax + 3\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(x^2 - x + 1\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the real roots of the equation \(p(x) = 0\).

The polynomial \(ax^3 + bx^2 + 5x - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x - 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x - 2)\) the remainder is 12.

(i) Find the values of \(a\) and \(b\).

(ii) When \(a\) and \(b\) have these values, find the quadratic factor of \(p(x)\).

Find the quotient and remainder when \(2x^4 - 27\) is divided by \(x^2 + x + 3\).

The polynomial \(f(x)\) is defined by

\(f(x) = 12x^3 + 25x^2 - 4x - 12\).

(i) Show that \(f(-2) = 0\) and factorise \(f(x)\) completely.

(ii) Given that

\(12 \times 27^y + 25 \times 9^y - 4 \times 3^y - 12 = 0\),

state the value of \(3^y\) and hence find \(y\) correct to 3 significant figures.

The polynomial \(p(z)\) is defined by

\(p(z) = z^3 + mz^2 + 24z + 32\),

where \(m\) is a constant. It is given that \((z + 2)\) is a factor of \(p(z)\).

1. Find the value of \(m\).
2. Hence, showing all your working, find
   1. the three roots of the equation \(p(z) = 0\),
   2. the six roots of the equation \(p(z^2) = 0\).

The polynomial \(2x^3 + 5x^2 + ax + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is 9.

(i) Find the values of \(a\) and \(b\).

(ii) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

The polynomial \(4x^3 - 4x^2 + 3x + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(2x^2 - 3x + 3\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, solve the inequality \(p(x) < 0\), justifying your answer.

The polynomial \(x^4 + 3x^2 + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \(x^2 + x + 2\) is a factor of \(p(x)\). Find the value of \(a\) and the other quadratic factor of \(p(x)\).

The polynomial \(x^3 - 2x + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((x + 2)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, find the quadratic factor of \(p(x)\).

The polynomial \(x^4 + 5x + a\) is denoted by \(p(x)\). It is given that \(x^2 - x + 3\) is a factor of \(p(x)\).

(i) Find the value of \(a\) and factorise \(p(x)\) completely.

(ii) Hence state the number of real roots of the equation \(p(x) = 0\), justifying your answer.

The polynomial \(2x^3 + ax^2 - 4\) is denoted by \(p(x)\). It is given that \((x - 2)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

When \(a\) has this value,

(ii) factorise \(p(x)\),

(iii) solve the inequality \(p(x) > 0\), justifying your answer.

The polynomial \(x^4 - 2x^3 - 2x^2 + a\) is denoted by \(f(x)\). It is given that \(f(x)\) is divisible by \(x^2 - 4x + 4\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, show that \(f(x)\) is never negative.

The polynomial \(x^4 + 4x^2 + x + a\) is denoted by \(p(x)\). It is given that \((x^2 + x + 2)\) is a factor of \(p(x)\).

Find the value of \(a\) and the other quadratic factor of \(p(x)\).

The polynomial \(2x^4 + ax^3 + bx - 1\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \(x^2 - x + 1\) the remainder is \(3x + 2\).

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

The polynomial \(ax^3 + x^2 + bx + 3\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \((2x - 1)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is 5.

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

The polynomial \(ax^3 - 10x^2 + bx + 8\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x-2)\) is a factor of both \(p(x)\) and \(p'(x)\).

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

(b) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

Find the quotient and remainder when \(8x^3 + 4x^2 + 2x + 7\) is divided by \(4x^2 + 1\).

Find the quotient and remainder when \(2x^4 + 1\) is divided by \(x^2 - x + 2\).