Edexcel C2 (Core Mathematics 2)

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

1. the value of \(p\),
2. the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).

2. (a) Find \(\quad \int \left( 3 + 4 x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) \mathrm { d } x\).
(b) Hence evaluate \(\quad \int _ { 1 } ^ { 2 } \left( 3 + 4 x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) \mathrm { d } x\).

1. Solve

$$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2 , \quad x > 2 .$$

5. 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,
2. the first term,
3. the sum to infinity.
   

   5. continued

1. The circle \(C\), with centre \(A\), has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 12 = 0$$

1. Find the coordinates of \(A\).
2. Show that the radius of \(C\) is 5 . The points \(P , Q\) and \(R\) lie on \(C\). The length of \(P Q\) is 10 and the length of \(P R\) is 3 .
3. Find the length of \(Q R\), giving your answer to 1 decimal place.
   

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1. The first four terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { n }\) are

$$1 + A x + B x ^ { 2 } + B x ^ { 3 } + \ldots$$ 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 \(3 = ( n - 2 ) k\). Given that \(A = 4\),
2. find the value of \(n\) and the value of \(k\).
   

   7. continued

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1. (a) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\cos \left( x - 20 ^ { \circ } \right) = - 0.437\), giving your answers to the nearest degree.
   (b) Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which

$$3 \tan \theta = 2 \cos \theta$$

1. A pencil holder is in the shape of an open circular cylinder of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The surface area of the cylinder (including the base) is \(250 \mathrm {~cm} ^ { 2 }\).
   1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by \(V = 125 r - \frac { \pi r ^ { 3 } } { 2 }\).
   2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value.
   3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\).
   4. Calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the maximum volume of the pencil holder.
      

   9. continued

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10. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2 x - \frac { 2 } { \sqrt { x } } , \quad x > 0$$ 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 \(b = 4\). The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2 .
2. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\).
3. Find the coordinates of the point \(D\). The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(A D\) and the \(x\)-axis.
4. Use integration to find the area of \(R\).
   
   1. continued
   2. continued