Discriminant and conditions for roots

Find the set of values of \(k\) for which the equation \(2x^2 + kx + 2 = 0\) has no real roots. [4]

1. Given that the equation \(k x ^ { 2 } + 6 k x + 5 = 0 \quad\) where \(k\) is a non zero constant has no real roots, find the range of possible values for \(k\).

1. The equation

$$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
Find the range of possible values of \(c\).

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 \(k\).

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.

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.

1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
2. Hence, using algebra, find the range of possible values of \(p\)

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

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 \(y = 3 x ^ { 2 } + 12 x + 13\).

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.

Find the values of \(k\) for which the equation \((2k - 3)x^2 - kx + (k - 1) = 0\) has equal roots. [4 marks]

The quadratic equation $$4x^2 + bx + 9 = 0$$ has one repeated real root. Find \(b\) Circle your answer. [1 mark] \(b = 0\) \quad \(b = \pm 12\) \quad \(b = \pm 13\) \quad \(b = \pm 36\)

10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).

1 In this question you must show detailed reasoning. Show that the equation \(x = 7 + 2 x ^ { 2 }\) has no real roots.

4

1. 1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
   1. Write down the coordinates of the minimum point of the curve.
   2. Sketch the curve, showing 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 \(y = x ^ { 2 } + 2 x + 5\).

7 \begin{figure}[h] \end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).

1. Find the exact coordinates of the points of intersection of this curve with the axes.
2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).

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 \(C\).

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 \(C\).

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,

• has gradient \(\frac { 1 } { 2 }\)
• intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
• find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
• show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
• Hence, or otherwise, find the value of \(b\)

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

6 The equation of a curve is \(y = ( 2 k - 3 ) x ^ { 2 } - k x - ( k - 2 )\), where \(k\) is a constant. The line \(y = 3 x - 4\) is a tangent to the curve. Find the value of \(k\).

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 \(y = x ^ { 2 } - 4 x + 2\).

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]

The equation \(kx^2 + 4kx + 3 = 0\), where \(k\) is a constant, has no real roots. Prove that $$0 \leqslant k < \frac{3}{4}$$ [4]

10

1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
2. Find the coordinates of \(Q\).
3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).

4

1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).

1. The curve \(C _ { 1 }\) has equation

$$y = x \left( 4 - x ^ { 2 } \right)$$

1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.