AQA C1 (Core Mathematics 1) 2015 June

1 The line \(A B\) has equation \(3 x + 5 y = 7\).

1. Find the gradient of \(A B\).
2. Find an equation of the line that is perpendicular to the line \(A B\) and which passes through the point \(( - 2 , - 3 )\). Express your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
3. The line \(A C\) has equation \(2 x - 3 y = 30\). Find the coordinates of \(A\).

2 The point \(P\) has coordinates \(( \sqrt { 3 } , 2 \sqrt { 3 } )\) and the point \(Q\) has coordinates \(( \sqrt { 5 } , 4 \sqrt { 5 } )\). Show that the gradient of \(P Q\) can be expressed as \(n + \sqrt { 15 }\), stating the value of the integer \(n\).
[0pt] [5 marks]

3 The diagram shows a sketch of a curve and a line.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-06_520_588_351_742} The curve has equation \(y = x ^ { 4 } + 3 x ^ { 2 } + 2\). The points \(A ( - 1,6 )\) and \(B ( 2,30 )\) lie on the curve.

1. Find an equation of the tangent to the curve at the point \(A\).
2. 1. Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 4 } + 3 x ^ { 2 } + 2 \right) \mathrm { d } x\).
   2. Calculate the area of the shaded region bounded by the curve and the line \(A B\).
      [0pt] [3 marks] \(4 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0\).

1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
2. 1. State the coordinates of \(C\).
   2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
   [0pt] [2 marks]

5

1. Express \(x ^ { 2 } + 3 x + 2\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
2. A curve has equation \(y = x ^ { 2 } + 3 x + 2\).
   1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
3. The curve with equation \(y = x ^ { 2 } + 3 x + 2\) is translated by the vector \(\left[ \begin{array} { l } 2
   4 \end{array} \right]\). Find the equation of the resulting curve in the form \(y = x ^ { 2 } + b x + c\).

6 The diagram shows a cylindrical container of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The container has an open top and a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-12_389_426_404_751} The external surface area of the container's curved surface and base is \(48 \pi \mathrm {~cm} ^ { 2 }\).
When the radius of the base is \(r \mathrm {~cm}\), the volume of the container is \(V \mathrm {~cm} ^ { 3 }\).

1. 1. Find an expression for \(h\) in terms of \(r\).
   2. Show that \(V = 24 \pi r - \frac { \pi } { 2 } r ^ { 3 }\).
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
   2. Find the positive value of \(r\) for which \(V\) is stationary, and determine whether this stationary value is a maximum value or a minimum value.
      [0pt] [4 marks]

7

1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]

8 A curve has equation \(y = x ^ { 2 } + ( 3 k - 4 ) x + 13\) and a line has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that the \(x\)-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
2. Given that the line and the curve do not intersect:
   1. show that \(9 k ^ { 2 } - 32 k - 16 < 0\);
   2. find the possible values of \(k\).
      \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}