Edexcel C2 (Core Mathematics 2)

1. Using the factor theorem, show that \((x + 3)\) is a factor of $$x^3 - 3x^2 - 10x + 24.$$ [2]
2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]

\(f(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.

1. Given that f(n) has a remainder of 3 when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
2. Show that f(n) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
3. Hence show that f(n) is divisible by 3 for all positive integer values of \(n\). [2]

Find the values of \(\theta\), to 1 decimal place, in the interval \(-180 \leq \theta < 180\) for which $$2 \sin^2 \theta° - 2 \sin \theta° = \cos^2 \theta°.$$ [8]

Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after \(x\) years, the total value of an initial £1 investment is £\(y\), where $$y = 1.04^x.$$

1. Sketch the graph of \(y = 1.04^x\), \(x \geq 0\). [2]
2. Calculate, to the nearest £, the total value of an initial £800 investment after 10 years. [2]
3. Use logarithms to find the number of years it takes to double the total value of any initial investment. [3]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P\) (ln 2, \(p + 2q\)) on \(C\), the gradient is 5.

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately 53.8 [5]

\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

\includegraphics{figure_1} Figure 1 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac{1}{2}\pi\). [5]

Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),

1. the perimeter of shape \(X\), [2]
2. the perimeter of shape \(Y\). [3]
3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of f(x),

1. prove that \(n = 6k + 2\). Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero, [3]
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\), [4]
3. expand f(x) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]