SPS SPS FM (SPS FM) 2019

In the question you must show detailed reasoning Solve the equation below, giving your answer in the simplest form $$x\sqrt{32} - \sqrt{24} = (3\sqrt{3} - 5)(\sqrt{6} + x\sqrt{2})$$ [3]

Find the coefficient of the \(x^4\) term in \((2 - 3x)^6\). [3]

A sequence \(u_1, u_2, u_3, ...\) is defined by \(u_n = 3n - 1\), for \(n \geq 1\).

1. Find the values of \(u_1, u_2, u_3\). [1]
2. Find $$\sum_{n=1}^{40} u_n$$ [3]

Show that $$\log_a(x^{10}) - 2\log_a\left(\frac{x^3}{4}\right) = 4\log_a(2x)$$ [3]

Solve the following inequalities giving your answer in set notation:

1. \(|4x + 3| < |x - 8|\) [3]
2. \(\frac{x}{x^2+1} < \frac{1}{2}\) [3]

If \(a\) and \(b\) are odd integers such that 4 is a factor of \((a - b)\), prove by contradiction that 4 cannot be a factor of \((a + b)\). [3]

\includegraphics{figure_7} The diagram shows a circle which passes through the points \(A(2, 9)\) and \(B(10, 3)\). \(AB\) is a diameter of the circle.

1. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the exact coordinates of \(C\). [4]
2. Find the exact area of the triangle formed by \(B\), \(C\) and the centre of the circle [3]

Sketch the curve \(y = 2^{2x+3}\), stating the coordinates of any points of intersection with the axes. [2] The point \(P\) on the curve \(y = 3^{3x+2}\) has \(y\)-coordinate equal to 180. Use logarithms to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. [2] The curves \(y = 2^{2x+3}\) and \(y = 3^{3x+2}\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as $$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$ [3]

1. Given that \(u_{n+1} = 5u_n + 4\), \(u_1 = 4\), prove by induction that \(u_n = 5^n - 1\). [4]
2. For all positive integers, \(n \geq 2\), prove by induction that $$\sum_{r=2}^{n} r^2(r-1) = \frac{1}{12}n(n-1)(n+1)(3n+2)$$ [5]

Show that, for any value of the real constant \(b\), the equation \(x^3 - (b + 1)x + b = 0\) has \(x = 1\) as a solution. Find all values of \(b\) for which this equation has exactly two real solutions [5]

In the question you must show detailed reasoning Given that the coefficients of \(x\), \(x^2\) and \(x^4\) in the expansion of \((1 + kx)^n\) are the consecutive terms of a geometric series, where \(n \geq 4\) and \(k\) is a positive constant

1. Show that $$k = \frac{6(n-1)}{(n-2)(n-3)}$$ [4]
2. For the case when \(k = \frac{7}{2}\), find the value of \(n\). [2]
3. Given that \(k = \frac{7}{5}\), \(n\) is a positive integer, and that the first term of the geometric series is the coefficient of \(x\), find the number of terms required for the sum of the geometric series to exceed \(1.12 \times 10^{12}\). [4]

In the question you must show detailed reasoning Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\), $$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$ is $$\frac{1}{\ln(2\sqrt{2})}$$ [5] [Total marks: 65]