Edexcel C2 (Core Mathematics 2)

Expand \((3 - 2x)^4\) in ascending powers of \(x\) and simplify each coefficient. [4]

\includegraphics{figure_1} Figure 1 shows triangle \(PQR\) in which \(PQ = x\), \(PR = 7 - x\), \(QR = x + 1\) and \(\angle PQR = 60°\). Using the cosine rule, find the value of \(x\). [4]

Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [6]

Find all values of \(x\) in the interval \(0 \leq x < 360°\) for which $$2\sin^2 x - 2\cos x - \cos^2 x = 1.$$ [8]

1. Sketch the curve \(y = 5^{x-1}\), showing the coordinates of any points of intersection with the coordinate axes. [2]
2. Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5^{x-1}\) intersects
   1. the straight line \(y = 10\),
   2. the curve \(y = 2^x\). [6]

\(f(x) = 2x^3 + 3x^2 - 6x + 1\).

1. Find the remainder when \(f(x)\) is divided by \((2x - 1)\). [2]
2. 1. Find the remainder when \(f(x)\) is divided by \((x + 2)\).
   2. Hence, or otherwise, solve the equation $$2x^3 + 3x^2 - 6x - 8 = 0,$$ giving your answers to 2 decimal places where appropriate. [7]

1. Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac{a(1-r^n)}{1-r}.$$ [4]
2. Evaluate \(\sum_{r=1}^{12} (5 \times 2^r)\). [5]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y = 5 + x - x^2\) and the normal to the curve at the point \(P(1, 5)\).

1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again. [2]
3. Show that the area of the shaded region bounded by the curve and the straight line \(PQ\) is \(\frac{4}{3}\). [6]

\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$

1. Find the coordinates of the centre and the radius of \(C\). [3]

\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).

1. Find the coordinates of \(P\) and \(Q\). [3]

The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).

1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]