1.02d Quadratic functions: graphs and discriminant conditions

8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.

1. Find the value of \(p\) and the value of \(q\).
2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}

1. Given the simultaneous equations

$$\begin{aligned} & 2 x + y = 1 \\ & x ^ { 2 } - 4 k y + 5 k = 0 \end{aligned}$$ where \(k\) is a non zero constant,

1. show that $$x ^ { 2 } + 8 k x + k = 0$$ Given that \(x ^ { 2 } + 8 k x + k = 0\) has equal roots,
2. find the value of \(k\).
3. For this value of \(k\), find the solution of the simultaneous equations.

9. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 2 } + 8\) The line \(L\) has equation \(y = 3 x + k\), where \(k\) is a positive constant.

1. Sketch \(C\) and \(L\) on separate diagrams, showing the coordinates of the points at which \(C\) and \(L\) cut the axes. Given that line \(L\) is a tangent to \(C\),
2. find the value of \(k\).

1. The equation

$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
2. Hence find the set of possible values of \(p\).

8. The straight line with equation \(y = 3 x - 7\) does not cross or touch the curve with equation \(y = 2 p x ^ { 2 } - 6 p x + 4 p\), where \(p\) is a constant.

1. Show that \(4 p ^ { 2 } - 20 p + 9 < 0\)
2. Hence find the set of possible values of \(p\).
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1. The function f is defined by

$$\mathrm { f } : x \mapsto | x - 2 | - 3 , x \in \mathbb { R }$$

1. Solve the equation \(\mathrm { f } ( x ) = 1\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 4 x + 11 , x \geq 0$$
2. Find the range of g .
3. Find \(g f ( - 1 )\).

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants.

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)

1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
5. Find the minimum area of triangle \(O A B\).

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6

1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).

8

1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

7

1. Calculate the discriminant of each of the following:
   1. \(x ^ { 2 } + 6 x + 9\),
   2. \(x ^ { 2 } - 10 x + 12\),
   3. \(x ^ { 2 } - 2 x + 5\).
   4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_391_446_628_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_449_625_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_389_442_630_1384} \captionsetup{labelformat=empty} \caption{Fig. 3}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_446_1119_644} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_396_447_1119_1137} \captionsetup{labelformat=empty} \caption{Fig. 5}
      \end{figure} State with reasons which of the diagrams corresponds to the curve
      (a) \(y = x ^ { 2 } + 6 x + 9\),
      (b) \(y = x ^ { 2 } - 10 x + 12\),
      (c) \(y = x ^ { 2 } - 2 x + 5\).

4

1. Find the discriminant of \(k x ^ { 2 } - 4 x + k\) in terms of \(k\).
2. The quadratic equation \(k x ^ { 2 } - 4 x + k = 0\) has equal roots. Find the possible values of \(k\)

10

1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).

3 The quadratic equation \(x ^ { 2 } + k x + k = 0\) has no real roots for \(x\).

1. Write down the discriminant of \(x ^ { 2 } + k x + k\) in terms of \(k\).
2. Hence find the set of values that \(k\) can take.

11

1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

8 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.

12

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.

13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).

3

1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.

11 \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.

1. Write down the radius of the circle and the coordinates of C .
2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .

12

1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).

11

1. Multiply out \(( x - p ) ( x - q )\).
2. You are given that \(p = 2 + \sqrt { 3 }\) and \(q = 2 - \sqrt { 3 }\) are the roots of a quadratic equation. Find \(p + q\) and \(p q\) and hence find the quadratic equation with roots \(x = p\) and \(x = q\).
3. Solve the quadratic equation \(x ^ { 2 } + 5 x - 7 = 0\) giving the roots exactly.
4. Show that \(x = 1\) is the only root of the equation \(x ^ { 3 } + 2 x - 3 = 0\).
5. A quadratic equation \(x ^ { 2 } + r x + s = 0\), where \(r\) and \(s\) are integers, has two roots. One root is \(x = 3 + \sqrt { 5 }\). Without finding \(r\) or \(s\), write down the other root.

10

1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
   Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.

11

1. Show algebraically that the equation \(x ^ { 2 } - 6 x + 10 = 0\) has no real roots.
2. Solve algebraically the simultaneous equations \(y = x ^ { 2 } - 6 x + 10\) and \(y + 2 x = 7\).
3. Plot the graph of the function \(y = x ^ { 2 } - 6 x + 10\) on graph paper, taking \(1 \mathrm {~cm} = 1\) unit on each axis, with the \(x\) axis from 0 to 6 and the \(y\) axis from - 2 to 10 .
   On the same axes plot the line with equation \(y + 2 x = 7\) showing clearly where the line cuts the quadratic curve.
4. Explain why these \(x\) coordinates satisfy the equation \(x ^ { 2 } - 4 x + 3 = 0\). Plot a graph of the function \(y = x ^ { 2 } - 4 x + 3\) on the same axes to illustrate your answer.