SPS SPS SM (SPS SM) 2023 October

In this question you must show detailed reasoning. Find the smallest positive integers \(m\) and \(n\) such that \(\left(\frac{64}{49}\right)^{-\frac{3}{2}} = \frac{m}{n}\) [3]

In this question you must show detailed reasoning. Express \(\frac{8 + \sqrt{7}}{2 + \sqrt{7}}\) in the form \(a + b\sqrt{7}\), where \(a\) and \(b\) are integers. [4]

\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]

A sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 8 \quad \text{and} \quad u_{n+1} = u_n + 3.$$

1. Show that \(u_5 = 20\). [1]
2. The \(n\)th term of the sequence can be written in the form \(u_n = pn + q\). State the values of \(p\) and \(q\). [2]
3. State what type of sequence it is. [1]
4. Find the value of \(N\) such that \(\sum_{n=1}^{2N} u_n - \sum_{n=1}^{N} u_n = 1256\). [5]

In part (ii) of this question you must show detailed reasoning.

1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]

1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]

In this question you must show detailed reasoning. The curve \(C_1\) has equation \(y = 8 - 10x + 6x^2 - x^3\) The curve \(C_2\) has equation \(y = x^2 - 12x + 14\)

1. Verify that when \(x = 1\) the curves \(C_1\) and \(C_2\) intersect. [2]

The curves also intersect when \(x = k\). Given that \(k < 0\)

1. use algebra to find the exact value of \(k\). [5]

The first term of a geometric progression is \(10\) and the common ratio is \(0.8\).

1. Find the fourth term. [2]
2. Find the sum of the first \(20\) terms, giving your answer correct to \(3\) significant figures. [2]
3. The sum of the first \(N\) terms is denoted by \(S_N\), and the sum to infinity is denoted by \(S_\infty\). Show that the inequality \(S_\infty - S_N < 0.01\) can be written as $$0.8^N < 0.0002,$$ and use logarithms to find the smallest possible value of \(N\). [6]

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]