1.02d Quadratic functions: graphs and discriminant conditions

1 Find the set of values of \(m\) for which the line with equation \(y = m x + 1\) and the curve with equation \(y = 3 x ^ { 2 } + 2 x + 4\) intersect at two distinct points.

1

1. Express \(16 x ^ { 2 } - 24 x + 10\) in the form \(( 4 x + a ) ^ { 2 } + b\).
2. It is given that the equation \(16 x ^ { 2 } - 24 x + 10 = k\), where \(k\) is a constant, has exactly one root. Find the value of this root.

5 The equation of a curve is \(y = 4 x ^ { 2 } - k x + \frac { 1 } { 2 } k ^ { 2 }\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

1. Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac { 3 } { 4 }\), find the values of \(k\) and \(a\).
2. Given instead that \(a = - \frac { 7 } { 2 }\), find the values of \(k\) for which the line is a tangent to the curve. [5]

11 The function f is given by \(\mathrm { f } ( x ) = 4 \cos ^ { 4 } x + \cos ^ { 2 } x - k\) for \(0 \leqslant x \leqslant 2 \pi\), where \(k\) is a constant.

1. Given that \(k = 3\), find the exact solutions of the equation \(\mathrm { f } ( x ) = 0\).
2. Use the quadratic formula to show that, when \(k > 5\), the equation \(\mathrm { f } ( x ) = 0\) has no solutions.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A line has equation \(y = 3 x + k\) and a curve has equation \(y = x ^ { 2 } + k x + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.

2 A curve has equation \(y = x ^ { 2 } + 2 c x + 4\) and a straight line has equation \(y = 4 x + c\), where \(c\) is a constant. Find the set of values of \(c\) for which the curve and line intersect at two distinct points.

1 A line has equation \(y = 3 x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant. Show that the line and the curve meet for all values of \(k\).

7 The straight line \(\mathrm { y } = \mathrm { x } + 5\) meets the curve \(2 \mathrm { x } ^ { 2 } + 3 \mathrm { y } ^ { 2 } = \mathrm { k }\) at a single point \(P\).

1. Find the value of the constant \(k\).
2. Find the coordinates of \(P\).

3 The equation of a curve is \(y = 2 x ^ { 2 } + m ( 2 x + 1 )\), where \(m\) is a constant, and the equation of a line is \(y = 6 x + 4\). Show that, for all values of \(m\), the line intersects the curve at two distinct points.

4 A curve has equation \(y = 3 x ^ { 2 } - 4 x + 4\) and a straight line has equation \(y = m x + m - 1\), where \(m\) is a constant. Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.

2 A curve has equation \(y = k x ^ { 2 } + 2 x - k\) and a line has equation \(y = k x - 2\), where \(k\) is a constant. Find the set of values of \(k\) for which the curve and line do not intersect.

3

1. Find the set of values of \(k\) for which the equation \(8 x ^ { 2 } + k x + 2 = 0\) has no real roots.
2. Solve the equation \(8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).

11 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
2. It is given instead that \(a = 5\) and that the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots then \(a ^ { 2 } < 4 ( b - a )\).

10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).

1. Show that the whole of the curve lies above the \(x\)-axis.
2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
4. Find the value of \(k\) for which the line is a tangent to the curve.

4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).

1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
2. Show that one of these points is also the stationary point of \(C\).

2 Find the set of values of \(k\) for which the line \(y = k x - 4\) intersects the curve \(y = x ^ { 2 } - 2 x\) at two distinct points.

2 Find the set of values of \(m\) for which the line \(y = m x + 4\) intersects the curve \(y = 3 x ^ { 2 } - 4 x + 7\) at two distinct points.

10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x + k , x \in \mathbb { R }\), where \(k\) is a constant.

1. In the case where \(k = 3\), solve the equation \(\mathrm { ff } ( x ) = 25\). The function g is defined by \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x + 8 , x \in \mathbb { R }\).
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has no real solutions. The function \(h\) is defined by \(h : x \mapsto x ^ { 2 } - 6 x + 8 , x > 3\).
3. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 3 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto x ^ { 2 } + 4 x , \quad x \in \mathbb { R } . \end{aligned}$$

1. Solve the equation \(\mathrm { ff } ( x ) = 11\).
2. Find the range of g .
3. Find the set of values of \(x\) for which \(\mathrm { g } ( x ) > 12\).
4. Find the value of the constant \(p\) for which the equation \(\mathrm { gf } ( x ) = p\) has two equal roots. Function h is defined by \(\mathrm { h } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant k\), and it is given that h has an inverse.
5. State the smallest possible value of \(k\).
6. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

8

1. Express \(2 x ^ { 2 } - 10 x + 8\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and use your answer to state the minimum value of \(2 x ^ { 2 } - 10 x + 8\).
2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } - 10 x + 8 = k x\) has no real roots.

2 The equation of a curve is \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis.
2. Find the value of \(k\) for which the line \(y + 2 x = 7\) is a tangent to the curve.

9 The curve \(C _ { 1 }\) has equation \(y = x ^ { 2 } - 4 x + 7\). The curve \(C _ { 2 }\) has equation \(y ^ { 2 } = 4 x + k\), where \(k\) is a constant. The tangent to \(C _ { 1 }\) at the point where \(x = 3\) is also the tangent to \(C _ { 2 }\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).

1 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 k x + k = 0\) has distinct real roots.

1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).