1.02d Quadratic functions: graphs and discriminant conditions

The equation of a curve is \(y = x^2 - 6x + k\), 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]
2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]

Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]

\(x^2 + 2x + 3 \equiv (x + a)^2 + b\).

1. Find the values of the constants \(a\) and \(b\). [2]
2. Sketch the graph of \(y = x^2 + 2x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes. [3]
3. Find the value of the discriminant of \(x^2 + 2x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). [2]

The equation \(x^2 + kx + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Find the set of possible values of \(k\), giving your answer in surd form. [4]

The equation \(x^2 + 2px + (3p + 4) = 0\), where \(p\) is a positive constant, has equal roots.

1. Find the value of \(p\). [4]
2. For this value of \(p\), solve the equation \(x^2 + 2px + (3p + 4) = 0\). [2]

The equation \(2x^2 - 3x - (k + 1) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\). [4]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

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

1. Prove that \(k(25k - 8) \geq 0\). [2]
2. Hence find the set of possible values of \(k\). [4]
3. Write down the values of \(k\) for which the equation \(x^2 + 5kx + 2k = 0\) has equal roots. [1]

1. Solve the equation \(4x^2 + 12x = 0\). [3]

\(f(x) = 4x^2 + 12x + c\), where \(c\) is a constant.

1. Given that \(f(x) = 0\) has equal roots, find the value of \(c\) and hence solve \(f(x) = 0\). [4]

\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]

Given that \(k = 4\),

1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]

Given that $$x^2 + 10x + 36 = (x + a)^2 + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\). [3]
2. Hence show that the equation \(x^2 + 10x + 36 = 0\) has no real roots. [2]

The equation \(x^2 + 10x + k = 0\) has equal roots.

1. Find the value of \(k\). [2]
2. For this value of \(k\), sketch the graph of \(y = x^2 + 10x + k\), showing the coordinates of any points at which the graph meets the coordinate axes. [4]

The circle \(C\) has centre \(X(3, 5)\) and radius \(r\) The line \(l\) has equation \(y = 2x + k\), where \(k\) is a constant.

1. Show that \(l\) and \(C\) intersect when $$5x^2 + (4k - 26)x + k^2 - 10k + 34 - r^2 = 0$$ [3]

Given that \(l\) is a tangent to \(C\),

1. show that \(5r^2 = (k + p)^2\), where \(p\) is a constant to be found. [3]

\includegraphics{figure_2} The line \(l\)

• cuts the \(y\)-axis at the point \(A\)
• touches the circle \(C\) at the point \(B\)

as shown in Figure 2. Given that \(AB = 2r\)

1. find the value of \(k\) [6]

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),

1. Find the value of \(c\), [2]
2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]

Given that \(f(x) = 15 - 7x - 2x^2\),

1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Calculate the coordinates of the stationary point of \(f(x)\). [3]

$$f(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find the range of \(f\). [1]
2. Write down the domain and range of \(f^{-1}\). [2]
3. Sketch the graph of \(f^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]

Given that \(g(x) = |x - 4|, x \in \mathbb{R}\),

1. find an expression for \(gf(x)\). [2]
2. Solve \(gf(x) = 8\). [5]

The function \(f\) is even and has domain \(\mathbb{R}\). For \(x \geq 0\), \(f(x) = x^2 - 4ax\), where \(a\) is a positive constant.

1. In the space below, sketch the curve with equation \(y = f(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
2. Find, in terms of \(a\), the value of \(f(2a)\) and the value of \(f(-2a)\). [2]

Given that \(a = 3\),

1. use algebra to find the values of \(x\) for which \(f(x) = 45\). [4]

The equation \(2x^2 + 2kx + (k + 2) = 0\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies $$k^2 - 2k - 4 > 0$$ [3]
2. Find the set of possible values of \(k\). [4]

\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]

Given that \(k = 4\),

1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{(k^2 - c)}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]