1.02e Complete the square: quadratic polynomials and turning points

Prove or disprove each of the following statements:

1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [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]

\(f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}\)

1. Write \(f(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers to be found. [3]
2. Sketch the curve with equation \(y = f(x)\) showing any points of intersection with the coordinate axes and the coordinates of any turning point. [3]
3. 1. Describe fully the transformation that maps the curve with equation \(y = f(x)\) onto the curve with equation \(y = g(x)\) where $$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
   2. Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$ [4]

1. Determine the values of \(p\) and \(q\) for which $$x^2 - 6x + 10 \equiv (x - p)^2 + q.$$ [2]

1. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int_3^4 \frac{1}{x^2 - 6x + 10} \, dx.$$ [3]

1. Determine the value of $$\int_3^4 \frac{x}{x^2 - 6x + 10} \, dx,$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined. [5]