SPS SPS SM (SPS SM) 2024 October

A is inversely proportional to B. B is inversely proportional to the square of C. When A is 2, C is 8. Find C when A is 12. [3]

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 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]

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]

\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]

Given that

• lines \(l_1\) and \(l_2\) intersect at the point C
• line \(l_1\) crosses the \(x\)-axis at the point A

1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]

In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
2. Find the time when the mass is 30 grams. [3]
3. Sketch the graph of \(m\) against \(t\). [2]

Express \(\frac{a^{\frac{1}{2}} - a^{\frac{2}{3}}}{a^{\frac{1}{3}} - a}\) in the form \(a^m + \sqrt{a^n}\), where \(m\) and \(n\) are integers and \(a \neq 0\) or 1. [4]

A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:

• The line through the centres of circle C and circle D has gradient 1.
• Circle D touches circle C at exactly one point.
• The centre of circle D lies in the first quadrant.
• Circle D has the same radius as circle C.

Find the coordinates of the centre of circle D. [5]

In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$

1. 1. Show that \((x + 1)\) is a factor of f(x). [1]
   2. Hence find the exact roots of the equation f(x) = 0. [4]
2. 1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
   2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]

The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of k, giving a reason for your answer. [4]
2. Find the value of \(\sum_{r=2}^{\infty} u_r\) [3]