Edexcel C2 (Core Mathematics 2)

\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find

1. the value of \(p\), [2]
2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]

1. Find \(\int \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [3]
2. Hence evaluate \(\int_1^2 \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [2]

\includegraphics{figure_1} Figure 1 shows a logo \(ABD\). The logo is formed from triangle \(ABC\). The mid-point of \(AC\) is \(D\) and \(BC = AD = DC = 6\) cm. \(\angle BCA = 0.4\) radians. The curve \(BD\) is an arc of a circle with centre \(C\) and radius 6 cm.

1. Write down the length of the arc \(BD\). [1]
2. Find the length of \(AB\). [3]
3. Write down the perimeter of the logo \(ABD\), giving your answer to 3 significant figures. [1]

Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]

The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find

1. the value of the common ratio, [3]
2. the first term, [2]
3. the sum to infinity. [2]

The circle \(C\), with centre \(A\), has equation $$x^2 + y^2 - 6x + 4y - 12 = 0.$$

1. Find the coordinates of \(A\). [2]
2. Show that the radius of \(C\) is 5. [2]

The points \(P\), \(Q\) and \(R\) lie on \(C\). The length of \(PQ\) is 10 and the length of \(PR\) is 3.

1. Find the length of \(QR\), giving your answer to 1 decimal place. [3]

The first four terms, in ascending powers of \(x\), of the binomial expansion of \((1 + kx)^n\) are $$1 + Ax + Bx^2 + Bx^3 + \ldots,$$ where \(k\) is a positive constant and \(A\), \(B\) and \(n\) are positive integers.

1. By considering the coefficients of \(x^2\) and \(x^3\), show that \(3 = (n - 2) k\). [4]

Given that \(A = 4\),

1. find the value of \(n\) and the value of \(k\). [4]

1. Solve, for \(0 \leq x < 360°\), the equation \(\cos (x - 20°) = -0.437\), giving your answers to the nearest degree. [4]
2. Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$3 \tan \theta = 2 \cos \theta.$$ [6]

A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).

1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$ The point \(A(1, 5)\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B(b, 0)\), where \(b\) is a constant and \(b > 0\).

1. Verify that \(b = 4\). [1]

The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2.

1. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\). [4]
2. Find the coordinates of the point \(D\). [1]

The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(AD\) and the \(x\)-axis.

1. Use integration to find the area of \(R\). [6]