SPS SPS FM (SPS FM) 2024 October

Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),

1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
2. Find the maximum value of \(f(x)\). [1]
3. Explain why \(f^{-1}(x)\) does not exist. [1]

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]

1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left(2 + \frac{1}{3}kx\right)^6\), where \(k\) is a constant. [3]
2. In the expansion of \((3 - 4x)\left(2 + \frac{1}{3}kx\right)^6\), the constant term is equal to the coefficient of \(x^2\). Determine the exact value of \(k\), given that \(k\) is positive. [3]

The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]

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]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan\alpha\). [3]

Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]