1.02f Solve quadratic equations: including in a function of unknown

The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}

1. State the value of \(x\) which is not in the domain of f. [1 mark]
2. Explain how you know that the function f is many-to-one. [2 marks]
3. 1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
   2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
5. Using set notation, state the range of f. [2 marks]

\includegraphics{figure_1} A stone is thrown over level ground from the top of a tower, \(X\). The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$h(t) = 7 + 21t - 4.9t^2, \quad t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1. Using the model,

1. give a physical interpretation of the meaning of the constant term 7 in the model. [1]
2. find the time taken after the stone is thrown for it to reach ground level. [3]
3. Rearrange \(h(t)\) into the form \(A - B(t - C)^2\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
4. Using your answer to part c or otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached. [2]

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

1. Use the factor theorem to show that \((3 - 2x)\) is a factor of \(f(x)\). [2]
2. Hence show that \(f(x)\) can be written in the form \(f(x) = (3 - 2x)(x + a)^2\) where \(a\) is an integer to be found. [4]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve with equation \(y = f(x)\).

1. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(f(x) \leq 0\)
   2. \(f'(\frac{x}{2}) = 0\)
   [3]

Fig. 12 shows \(x\)- and \(y\)- coordinate axes with origin O and the trajectory of a particle projected from O with speed 28 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x\) m and \(y\) m. Air resistance should be neglected. \includegraphics{figure_12}

1. Show that the equation of the trajectory is given by $$\tan^2\alpha - \frac{160}{x}\tan\alpha + \frac{160y}{x^2} + 1 = 0.$$ (*) [5]
2. 1. Show that if (*) is treated as an equation with \(\tan\alpha\) as a variable and with \(x\) and \(y\) as constants, then (*) has two distinct real roots for \(\tan\alpha\) when \(y < 40 - \frac{x^2}{160}\). [3]
   2. Show the inequality in part (ii)(A) as a locus on the graph of \(y = 40 - \frac{x^2}{160}\) in the Printed Answer Booklet and label it R. [1]

S is the locus of points \((x, y)\) where (*) has one real root for \(\tan\alpha\). T is the locus of points \((x, y)\) where (*) has no real roots for \(\tan\alpha\).

1. Indicate S and T on the graph in the Printed Answer Booklet. [2]
2. State the significance of R, S and T for the possible trajectories of the particle. [3]

A machine can fire a tennis ball from ground level with a maximum speed of 28 m s\(^{-1}\).

1. State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m. [1]

Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]

Water is being emptied out of a sink. The depth of water, \(y\)cm, at time \(t\) seconds, may be modelled by $$y = t^2 - 14t + 49 \quad\quad 0 \leqslant t \leqslant 7.$$

1. Find the value of \(t\) when the depth of water is 25cm. [3]
2. Find the rate of decrease of the depth of water when \(t = 3\). [3]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3x - 2\sqrt{x}\), \(x \geqslant 0\) and the line \(l\) with equation \(y = 8x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\). [5]
2. \includegraphics{figure_4} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\). [3]

1. Sketch the curve with equation $$y = k - \frac{1}{2x}$$ where \(k\) is a positive constant State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. [3]

The straight line \(l\) has equation \(y = 2x + 3\) Given that \(l\) cuts the curve in two distinct places,

1. find the range of values of \(k\), writing your answer in set notation. [6]

Given that \(k\) is a positive constant and \(\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4\)

1. show that \(3k + 5\sqrt{k} - 12 = 0\) [4]
2. Hence, using algebra, find any values of \(k\) such that $$\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$$ [4]

A circle has centre \(C(3, -8)\) and radius \(10\).

1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
3. The line with equation \(y = 2x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]

In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Using algebra, find all solutions of the equation $$3x^3 - 17x^2 - 6x = 0$$ [3]
2. Hence find all real solutions of $$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]

In this question you must show detailed reasoning. Solve the following equations.

1. \(y^6 + 7y^3 - 8 = 0\) [3]
2. \(9^{x+1} + 3^x = 8\) [4]

It is given that $$y = \frac{1}{x+1} + \frac{1}{x-1},$$ where \(x\) and \(y\) are real and positive, and \(i^2 = -1\).

1. Show that $$x = \frac{1 \pm \sqrt{1-y^2}}{y} \quad \text{and} \quad y \leqslant 1.$$ [4]
2. Deduce that $$xy < 2.$$ [2]
3. Indicate the region in the \(x\)-\(y\) plane defined by $$y \leqslant 1 \quad \text{and} \quad xy < 2.$$ [3]

It is given that the trajectory of a projectile which is launched with speed \(V\) at an angle \(\alpha\) above the horizontal has equation $$y = x\tan\alpha - \frac{gx^2}{2V^2}(1 + \tan^2\alpha),$$ where the point of projection is the origin, and the \(x\)- and \(y\)-axes are horizontal and vertically upwards respectively.

1. Express the above equation as a quadratic equation in \(\tan\alpha\) and deduce that the boundary of all accessible points for this projectile has equation $$y = \frac{1}{2gV^2}(V^4 - g^2x^2).$$ [4]
2. A stone is thrown with speed \(\sqrt{gh}\) from the top of a vertical tower, of height \(h\), which stands on horizontal ground. Find
   1. the maximum distance, from the foot of the tower, at which the stone can land, [3]
   2. the angle of elevation at which the stone must be thrown to achieve this maximum distance. [3]