1.02d Quadratic functions: graphs and discriminant conditions

7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.

1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
2. 1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
   2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$

8 The quadratic equation \(( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0\) has real roots.

1. Show that \(4 k ^ { 2 } - 9 k - 9 \geqslant 0\).
2. Hence find the possible values of \(k\).

7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).

1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
2. The curve \(C\) and the line \(L\) intersect in two distinct points.
   1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
   2. Hence find the possible values of \(k\).

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).

1. Show that \(a = 2\) and \(b = 4\).
2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
   (No credit will be given for solutions based on differentiation.)

11 A circle \(C\) has centre \(( 0,10 )\) and radius \(\sqrt { 20 }\) A line \(L\) has equation \(y = m x\) 11

1. 1. Show that the \(x\)-coordinate of any point of intersection of \(L\) and \(C\) satisfies the equation $$\left( 1 + m ^ { 2 } \right) x ^ { 2 } - 20 m x + 80 = 0$$ 11
      1. (ii) Find the values of \(m\) for which the equation in part (a)(i) has equal roots.
         11
   2. Two lines are drawn from the origin which are tangents to \(C\). Find the coordinates of the points of contact between the tangents and \(C\).

4 The equation \(9 x ^ { 2 } + 4 x + p ^ { 2 } = 0\) has no real solutions for \(x\). Find the set of possible values of \(p\).
Fully justify your answer.
[0pt] [4 marks]

1 Four possible sketches of \(y = a x ^ { 2 } + b x + c\) are shown below.
Given \(b ^ { 2 } - 4 a c = 0\) and \(a , b\) and \(c\) are non-zero constants, which sketch is the only one that could possibly be correct? Tick ( \(\checkmark\) ) one box. A \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_383_303_995_550} \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1128_1009} B \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1562_1009} C \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_223_300_1868_548} \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_108_109_2001_1009} D \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_388_301_2305_549}
□

13 A vehicle, of total mass 1200 kg , is travelling along a straight, horizontal road at a constant speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This vehicle begins to accelerate at a constant rate.
After 40 metres it reaches a speed of \(17 \mathrm {~ms} ^ { - 1 }\) Find the resultant force acting on the vehicle during the period of acceleration.

3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box. \(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □ \(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □ \(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □ \(b = 0 \Rightarrow \alpha = - \beta\) □

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)

• has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
• cuts the \(x\)-axis at the origin and at \(x = 4\)
• has a minimum turning point at ( \(2 , - 4.8\) )
  1. find \(\mathrm { f } ( x )\)

Given that \(C _ { 2 }\)

The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.

Use algebra to find the coordinates of \(P\).

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

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).

2 Let \(\mathrm { f } ( x ) = x ^ { 2 } + k x + 4\), where \(k\) is a constant.

1. Find an expression for the discriminant of f in terms of \(k\).
2. Hence find the range of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has two distinct real roots.

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

1. Show that the curve meets the line \(y = k\) when \(2 x ^ { 2 } - ( 2 k + 1 ) x + ( 3 k - 1 ) = 0\), and hence show that no part of the curve exists in the interval \(\frac { 1 } { 2 } < y < \frac { 9 } { 2 }\).
2. Deduce the coordinates of the turning points of this curve.

12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

2

1. 1. Find the value of the discriminant of \(x ^ { 2 } + 3 x + 5\).
   2. Use your value from part (i) to determine the number of real roots of the equation \(x ^ { 2 } + 3 x + 5 = 0\).
2. Find the non-zero value of \(k\) for which the equation \(k x ^ { 2 } + 3 x + 5 = 0\) has only one distinct real root.

2

1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

The diagram below shows a sketch of the graph of \(y = f ( x )\), where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.

Show that the curve with equation \(x^2 - 3xy - 40 = 0\) and the line with equation \(3x + y + k = 0\) meet for all values of the constant \(k\). [5]

The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).

1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]

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

1. Find the set of values of \(p\) for which the equation \(\mathrm{f}(x) = p\) has no real roots. [3]

The function g is defined by \(\mathrm{g} : x \mapsto 2x^2 - 6x + 5\) for \(0 \leqslant x \leqslant 4\).

1. Express \(\mathrm{g}(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Find the range of g. [2]

The function h is defined by \(\mathrm{h} : x \mapsto 2x^2 - 6x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which h has an inverse. [1]
2. For this value of \(k\), find an expression for \(\mathrm{h}^{-1}(x)\). [3]

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