1.02h Express solutions: using 'and', 'or', set and interval notation

A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)

1. Sketch a graph to show the region R. Shade the region R.
2. Find the area of R [6 marks]

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]

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]

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]

In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$ is convergent.

1. Find the set of possible values of \(x\), giving your answer in set notation. [5]
2. Given that the sum to infinity of the series is \(\frac{2}{3}\), find the value of \(x\). [3]

The circle \(C\) has equation $$x^2 + y^2 + 10x - 4y + 1 = 0$$

1. Find
   1. the coordinates of the centre of \(C\)
   2. the exact radius of \(C\) [2]
   The line with equation \(y = k\), where \(k\) is a constant, cuts \(C\) at two distinct points.
2. Find the range of values for \(k\), giving your answer in set notation. [2]
3. The line with equation \(y = mx + 4\) is a tangent to \(C\). Find possible exact values of \(m\). [5]

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]