AQA C1 (Core Mathematics 1) 2010 June

1 The trapezium \(A B C D\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-2_298_591_557_737} The line \(A B\) has equation \(2 x + 3 y = 14\) and \(D C\) is parallel to \(A B\).

1. Find the gradient of \(A B\).
2. The point \(D\) has coordinates \(( 3,7 )\).
   1. Find an equation of the line \(D C\).
   2. The angle \(B A D\) is a right angle. Find an equation of the line \(A D\), giving your answer in the form \(m x + n y + p = 0\), where \(m , n\) and \(p\) are integers.
3. The line \(B C\) has equation \(5 y - x = 6\). Find the coordinates of \(B\).

2

1. Express \(( 3 - \sqrt { 5 } ) ^ { 2 }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
2. Hence express \(\frac { ( 3 - \sqrt { 5 } ) ^ { 2 } } { 1 + \sqrt { 5 } }\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
   (4 marks)

3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$$

1. 1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
3. 1. Verify that \(\mathrm { p } ( - 1 ) < \mathrm { p } ( 0 )\).
   2. Sketch the curve with equation \(y = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15\), indicating the values where the curve crosses the coordinate axes.

4 The curve with equation \(y = x ^ { 4 } - 8 x + 9\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721} The point \(( 2,9 )\) lies on the curve.

1. 1. Find \(\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(y = 9\).
2. The point \(A ( 1,2 )\) lies on the curve with equation \(y = x ^ { 4 } - 8 x + 9\).
   1. Find the gradient of the curve at the point \(A\).
   2. Hence find an equation of the tangent to the curve at the point \(A\).

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

6 The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end faces are right-angled triangles with sides of lengths \(3 x \mathrm {~cm}\), \(4 x \mathrm {~cm}\) and \(5 x \mathrm {~cm}\), and the length of the prism is \(y \mathrm {~cm}\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-7_394_825_459_548} The total surface area of the five faces is \(144 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(x y + x ^ { 2 } = 12\).
   2. Hence show that the volume of the block, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = 72 x - 6 x ^ { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that \(V\) has a stationary value when \(x = 2\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 2\).
   (2 marks)

7

1. 1. Express \(2 x ^ { 2 } - 20 x + 53\) in the form \(2 ( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Use your result from part (a)(i) to explain why the equation \(2 x ^ { 2 } - 20 x + 53 = 0\) has no real roots.
2. The quadratic equation \(( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0\) has real roots.
   1. Show that \(7 k ^ { 2 } - 6 k - 1 \leqslant 0\).
   2. Hence find the possible values of \(k\).