AQA C1 (Core Mathematics 1) 2005 January

1 The point \(A\) has coordinates \(( 11,2 )\) and the point \(B\) has coordinates \(( - 1 , - 1 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation $$x - 4 y = 3$$
2. The line with equation \(3 x + 5 y = 26\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).

2 A curve has equation \(y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P\) on the curve has coordinates \(( 2,3 )\).
   1. Show that the gradient of the curve at \(P\) is 5 .
   2. Hence find an equation of the normal to the curve at \(P\), expressing your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
3. Determine whether \(y\) is increasing or decreasing when \(x = 1\).

3 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0\).

1. By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle.
3. The line with equation \(y = x + 4\) intersects the circle at the points \(P\) and \(Q\).
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
   2. Find the coordinates of \(P\) and \(Q\).

4

1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
   1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
   2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
   3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below.
   \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
   1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
   2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
3. 1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
   2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.

5

1. Simplify \(( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )\).
2. Express \(\sqrt { 12 }\) in the form \(m \sqrt { 3 }\), where \(m\) is an integer.
3. Express \(\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm .
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
   3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
   4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence determine whether the stationary value is a maximum or minimum.

7

1. Simplify \(( k + 5 ) ^ { 2 } - 12 k ( k + 2 )\).
2. The quadratic equation \(3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0\) has real roots.
   1. Show that \(( k - 1 ) ( 11 k + 25 ) \leqslant 0\).
   2. Hence find the possible values of \(k\).