1.02d Quadratic functions: graphs and discriminant conditions

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0\).

1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(D\) has coordinates (7, -2).
   1. Verify that the point \(D\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(D\), giving your answer in the form \(m x + n y = p\), where \(m , n\) and \(p\) are integers.
4. 1. A line has equation \(y = k x\). Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
   2. Find the values of \(k\) for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (d)(ii).

5

1. 1. Sketch the curve with equation \(y = x ( x - 2 ) ^ { 2 }\).
   2. Show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   2. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x - 3 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
3. Hence show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) has only one real root and state the value of this root.

7

1. 1. Express \(4 - 10 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = 4 - 10 x - x ^ { 2 }\).
2. The curve \(C\) has equation \(y = 4 - 10 x - x ^ { 2 }\) and the line \(L\) has equation \(y = k ( 4 x - 13 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
   2. Given that the curve \(C\) and the line \(L\) intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
   3. Solve the inequality \(4 k ^ { 2 } + 33 k + 29 > 0\).

2

1. Factorise \(x ^ { 2 } - 4 x - 12\).
2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
3. 1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
   2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden. \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
   1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
   2. Write down:
      1. the coordinates of \(C\);
      2. the radius of the circle.
      3. Sketch the circle.
      4. A line has equation \(y = k x + 6\), where \(k\) is a constant.
         1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
         2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
         3. Hence find the values of \(k\) for which the line is a tangent to the circle.

4

1. 1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
   1. Find the coordinates of the vertex of the curve.
   2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
   3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).

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

1. Show that the \(x\)-coordinate of any point of intersection of the curve and the line satisfies the equation $$2 x ^ { 2 } - ( 2 k + 1 ) x + 3 k - 1 = 0$$
2. The curve and the line intersect at two distinct points.
   1. Show that \(4 k ^ { 2 } - 20 k + 9 > 0\).
   2. Find the possible values of \(k\).

6 The cubic polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = ( x - 2 ) \left( x ^ { 2 } + x + 3 \right)\).

1. Show that \(\mathrm { p } ( x )\) can be written in the form \(x ^ { 3 } + a x ^ { 2 } + b x - 6\), where \(a\) and \(b\) are constants whose values are to be found.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   (2 marks)
3. Prove that the equation \(( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0\) has only one real root and state its value.
   (3 marks)

8 A line has equation \(y = m x - 1\), where \(m\) is a constant.
A curve has equation \(y = x ^ { 2 } - 5 x + 3\).

1. Show that the \(x\)-coordinate of any point of intersection of the line and the curve satisfies the equation $$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
2. Find the values of \(m\) for which the equation \(x ^ { 2 } - ( 5 + m ) x + 4 = 0\) has equal roots.
   (4 marks)
3. Describe geometrically the situation when \(m\) takes either of the values found in part (b).
   (1 mark)

2

1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).

7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).

1. Find:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
2. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
3. A line has equation \(y = 2 k - x\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
   2. Find the values of \(k\) for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).

7

1. 1. Express \(2 x ^ { 2 } - 20 x + 53\) in the form \(2 ( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Use your result from part (a)(i) to explain why the equation \(2 x ^ { 2 } - 20 x + 53 = 0\) has no real roots.
2. The quadratic equation \(( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0\) has real roots.
   1. Show that \(7 k ^ { 2 } - 6 k - 1 \leqslant 0\).
   2. Hence find the possible values of \(k\).

4

1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   (3 marks)
2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
   1. Find the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
   3. Sketch the curve, stating the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 5 x + 7\).

5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
2. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
3. 1. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   2. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

7 The quadratic equation $$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$ has real roots.

1. Show that \(7 k ^ { 2 } - 48 k + 80 \leqslant 0\).
2. Find the possible values of \(k\).

8 A curve has equation \(y = x ^ { 2 } + ( 3 k - 4 ) x + 13\) and a line has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
2. Given that the line and the curve do not intersect:
   1. show that \(9 k ^ { 2 } - 32 k - 16 < 0\);
   2. find the possible values of \(k\).
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6

1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
   1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
   2. Sketch the curve, giving the value of the \(y\)-intercept.
2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
   2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]

2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).

5. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.

1. Prove that \(k ( 25 k - 8 ) \geq 0\).
2. Hence find the set of possible values of \(k\).
3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.

1. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\).

You are given that \(\mathrm { f } ( x ) = 4 x ^ { 2 } + 12 x + c\), where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).

2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).

4. (a) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(b) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).

7. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,

1. show that $$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
2. find the set of possible values of \(k\),
3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.

3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
   [0pt] [P1 June 2002 Question 3]