1.02g Inequalities: linear and quadratic in single variable

The equation $$(k + 3)x^2 + 6x + k = 5$$, where \(k\) is a constant, has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k^2 - 2k - 24 < 0$$ [4]
2. Hence find the set of possible values of \(k\). [2]

You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).

1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
2. State the range of \(gf(x)\). [1]
3. Solve the inequality \(|3x - 1| > 1\). [3]

The functions f and g are defined for all real values of \(x\) by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

\includegraphics{figure_3} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\). Given that • \(C\) has equation \(y = f(x)\) where \(f(x)\) is a quadratic expression in \(x\) • \(C\) cuts the \(x\)-axis at \(0\) and \(6\) • \(l\) cuts the \(y\)-axis at \(60\) and intersects \(C\) at the point \((10, 80)\) use inequalities to define the region \(R\) shown shaded in Figure 3. [5]

In this question you must show detailed reasoning. A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form using set notation. [6]

The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]

1. Write \(3x^2 + 24x + 5\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]

The finite region R is enclosed by the curve \(y = 3x^2 + 24x + 5\) and the \(x\)-axis.

1. State the inequalities that define R, including its boundaries. [2]

The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]

Solve the inequalities

1. \(3 - 8x > 4\), [2]
2. \((2x - 4)(x - 3) < 12\). [5]

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

The line \(l\) passes through the points \(A(-3, 0)\) and \(B\left(\frac{5}{3}, 22\right)\)

1. Find the equation of \(l\) giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are constants. [3]

\includegraphics{figure_2} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\). Given that

• \(C\) has equation \(y = 2x^2 + 5x - 3\)
• the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)

1. use inequalities to define \(R\). [2]

In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the cuboid \(ABCDEFGH\) where \(AD = 3\) cm, \(AF = (2x + 1)\) cm and \(DC = (x - 2)\) cm. The volume of the cuboid is at most 9 cm³. Find the range of possible values of \(x\). Give your answer in interval notation. [5]

Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities are satisfied. $$3y \geqslant 4x \qquad y - x \leqslant 1 \qquad y \geqslant (x-1)^2$$ You should indicate the region for which the inequalities hold by labelling the region R. [4]

In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]

Solve the inequality $$\log_3(2x^2 - x) - \log_3(2x^2 - 3x + 1) > 1.$$ [5]

It is given that $$y = \frac{1}{x+1} + \frac{1}{x-1},$$ where \(x\) and \(y\) are real and positive, and \(i^2 = -1\).

1. Show that $$x = \frac{1 \pm \sqrt{1-y^2}}{y} \quad \text{and} \quad y \leqslant 1.$$ [4]
2. Deduce that $$xy < 2.$$ [2]
3. Indicate the region in the \(x\)-\(y\) plane defined by $$y \leqslant 1 \quad \text{and} \quad xy < 2.$$ [3]