1.02n Sketch curves: simple equations including polynomials

\includegraphics{figure_1} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \((-5, 0)\), \((-2, 0)\), \((1.5, 0)\) and \((0, -30)\).

1. Use the intersections with both axes to express the equation of the curve in a factorised form. [2]
2. Hence show that the equation of the curve may be written as \(y = 2x^3 + 11x^2 - x - 30\). [2]
3. Draw the line \(y = 5x + 10\) accurately on the graph. The curve and this line intersect at \((-2, 0)\); find graphically the \(x\)-coordinates of the other points of intersection. [3]
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2x^2 + 7x - 20 = 0.$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. [5]

Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).

1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]

\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). The values of \(x\) where it crosses the \(x\)-axis are \(-5\), \(-2\) and \(2\), and it crosses the \(y\)-axis at \((0, -20)\).

1. Express f(\(x\)) in factorised form. [2]
2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4. Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place. [6]
4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]

The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).

1. Sketch the curve. [2]
2. Use calculus to find the equation of the tangent to the curve at A. [4]
3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]

The function f is even and has domain \(\mathbb{R}\). For \(x \geq 0\), f(x) = \(x^2 - 4ax\), where \(a\) is a positive constant.

1. In the space below, sketch the curve with equation \(y = \text{f}(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
2. Find, in terms of \(a\), the value of f(2a) and the value of f(-2a). [2]

Given that \(a = 3\),

1. use algebra to find the values of \(x\) for which f(x) = 45. [4]

The functions f and g are defined by $$\text{f}: x \alpha |x - a| + a, \quad x \in \mathbb{R},$$ $$\text{g}: x \alpha 4x + a, \quad x \in \mathbb{R}.$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of any points at which your graphs meet the axes. [5]
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect. [3]
3. Find an expression for fg(x). [2]
4. Solve, for \(x\) in terms of \(a\), the equation $$\text{fg}(x) = 3a.$$ [3]

The functions f and g are defined by \(\text{f: } x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, 0 \leq x \leq 4,\) \(\text{g: } x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.\)

1. Find the range of f. [3]
2. Given that gf(2) = 16, find the value of \(\lambda\). [3]

1. Sketch, on the same set of axes, the graphs of $$y = 2 - e^{-x} \text{ and } y = \sqrt{x}.$$ [3] [It is not necessary to find the coordinates of any points of intersection with the axes.] Given that f(x) = \(e^{-x} + \sqrt{x} - 2\), \(x \geq 0\),
2. explain how your graphs show that the equation f(x) = 0 has only one solution, [1]
3. show that the solution of f(x) = 0 lies between \(x = 3\) and \(x = 4\). [2]

The iterative formula \(x_{n+1} = (2 - e^{-x_n})^2\) is used to solve the equation f(x) = 0.

1. Taking \(x_0 = 4\), write down the values of \(x_1\), \(x_2\), \(x_3\) and \(x_4\), and hence find an approximation to the solution of f(x) = 0, giving your answer to 3 decimal places. [4]

\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]

The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through 360° about the \(y\)-axis to form a solid shape \(S\).

1. Find the volume of \(S\), giving your answer in the form \(\pi(a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]

The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,

1. show that the volume of the tower is approximately 15500 m\(^3\). [2]

Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.

1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
2. Find the time for which the ball is in flight. [4 marks]
3. Find the horizontal distance travelled by the ball. [3 marks]
4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]

A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after 3.3 seconds.

1. Calculate the height above ground from which the lead was dropped. [2 marks]

According to a simple model, the ball hits the ground at the same time as the lead.

1. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. [2 marks]

The piece of lead is now thrown again from \(P\), with speed 7 ms\(^{-1}\) at an angle of 30° to the horizontal, as shown. \includegraphics{figure_6}

1. Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown. [4 marks]
2. Deduce that \(y = \frac{\sqrt{3}}{3}x - \frac{2}{15}x^2\). [3 marks]
3. Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\). [5 marks]

A particle is projected with speed 49 m s\(^{-1}\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac{x^2(1 + \tan^2 \theta)}{490}.$$ [4]

\includegraphics{figure_8} The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta_1\) and \(\theta_2\), and the corresponding points where the particle returns to the plane are \(A_1\) and \(A_2\) respectively (see diagram).

1. Find \(\theta_1\) and \(\theta_2\). [4]
2. Calculate the distance between \(A_1\) and \(A_2\). [5]

A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).

1. Write down the equations of all the asymptotes of \(C\). [2 marks]
2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
   1. Find the \(y\)-coordinate of the stationary point. [1 mark]
   2. Sketch the curve \(C\). [2 marks]
3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]

A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).

1. Write down the equations of the asymptotes of \(C\). [2 marks]
2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]

A curve has equation \(y = \frac{x^2}{(x-2)(x+1)}\).

1. Write down the equations of the three asymptotes. [3]
2. Determine whether the curve approaches the horizontal asymptote from above or from below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
3. Sketch the curve. [4]
4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]

A curve has equation \(y = \frac{x^2 - 4}{(x-3)(x+1)(x-1)}\).

1. Write down the coordinates of the points where the curve crosses the axes. [3]
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
3. Determine whether the curve approaches the horizontal asymptote from above or below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
4. Sketch the curve. [4]

\includegraphics{figure_5} The diagram shows the curve with equation \(y = f(x)\), where $$f(x) = 2x^3 - 9x^2 + 12x - 4.36.$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha\), \(x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).

1. The Newton-Raphson method is to be used to find the roots of the equation \(f(x) = 0\), with \(x_1 = k\).
   1. To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\)? [2]
   2. What happens if \(1 < k < 2\)? [2]
2. Sketch the curve with equation \(y^2 = f(x)\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points. [4]