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AQA FP1 2010 June Q9
13 marks Standard +0.3
9 A parabola \(P\) has equation \(y ^ { 2 } = x - 2\).
    1. Sketch the parabola \(P\).
    2. On your sketch, draw the two tangents to \(P\) which pass through the point \(( - 2,0 )\).
    1. Show that, if the line \(y = m ( x + 2 )\) intersects \(P\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$m ^ { 2 } x ^ { 2 } + \left( 4 m ^ { 2 } - 1 \right) x + \left( 4 m ^ { 2 } + 2 \right) = 0$$
    2. Show that, if this equation has equal roots, then $$16 m ^ { 2 } = 1$$
    3. Hence find the coordinates of the points at which the tangents to \(P\) from the point \(( - 2,0 )\) touch the parabola \(P\).
AQA FP1 2011 June Q1
5 marks Moderate -0.5
1 A curve passes through the point \(( 2,3 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 + x } }$$ Starting at the point \(( 2,3 )\), use a step-by-step method with a step length of 0.5 to estimate the value of \(y\) at \(x = 3\). Give your answer to four decimal places.
AQA FP1 2011 June Q2
9 marks Standard +0.8
2 The equation $$4 x ^ { 2 } + 6 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = \frac { 3 } { 4 }\).
  3. Find an equation, with integer coefficients, which has roots $$3 \alpha - \beta \text { and } 3 \beta - \alpha$$
AQA FP1 2011 June Q3
7 marks Standard +0.3
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right)$$
  2. Given that $$( z - \mathrm { i } ) \left( z ^ { * } - \mathrm { i } \right) = 24 - 8 \mathrm { i }$$ find the two possible values of \(z\).
AQA FP1 2011 June Q4
10 marks Moderate -0.3
4 The variables \(x\) and \(Y\), where \(Y = \log _ { 10 } y\), are related by the equation $$Y = m x + c$$ where \(m\) and \(c\) are constants.
  1. Given that \(y = a b ^ { x }\), express \(a\) in terms of \(c\), and \(b\) in terms of \(m\).
  2. It is given that \(y = 12\) when \(x = 1\) and that \(y = 27\) when \(x = 5\). On the diagram below, draw a linear graph relating \(x\) and \(Y\).
  3. Use your graph to estimate, to two significant figures:
    1. the value of \(y\) when \(x = 3\);
    2. the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-3_976_1173_1110_484}
AQA FP1 2011 June Q5
7 marks Moderate -0.3
5
  1. Find the general solution of the equation $$\cos \left( 3 x - \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
  2. Use your general solution to find the smallest solution of this equation which is greater than \(5 \pi\).
AQA FP1 2011 June Q6
7 marks Moderate -0.8
6
  1. Expand \(( 5 + h ) ^ { 3 }\).
  2. A curve has equation \(y = x ^ { 3 } - x ^ { 2 }\).
    1. Find the gradient of the line passing through the point \(( 5,100 )\) and the point on the curve for which \(x = 5 + h\). Give your answer in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
    2. Show how the answer to part (b)(i) can be used to find the gradient of the curve at the point \(( 5,100 )\). State the value of this gradient.
AQA FP1 2011 June Q7
9 marks Moderate -0.3
7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$
    1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
    2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
    1. \(\mathrm { A } ^ { 3 }\);
    2. A.
AQA FP1 2011 June Q8
10 marks Standard +0.3
8 A curve has equation \(y = \frac { 1 } { x ^ { 2 } - 4 }\).
    1. Write down the equations of the three asymptotes of the curve.
    2. Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
  2. Hence, or otherwise, solve the inequality $$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$
AQA FP1 2011 June Q9
11 marks Challenging +1.2
9 The diagram shows a parabola \(P\) which has equation \(y = \frac { 1 } { 8 } x ^ { 2 }\), and another parabola \(Q\) which is the image of \(P\) under a reflection in the line \(y = x\). The parabolas \(P\) and \(Q\) intersect at the origin and again at a point \(A\).
The line \(L\) is a tangent to both \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-5_1015_1089_623_479}
    1. Find the coordinates of the point \(A\).
    2. Write down an equation for \(Q\).
    3. Give a reason why the gradient of \(L\) must be - 1 .
    1. Given that the line \(y = - x + c\) intersects the parabola \(P\) at two distinct points, show that $$c > - 2$$
    2. Find the coordinates of the points at which the line \(L\) touches the parabolas \(P\) and \(Q\).
      (No credit will be given for solutions based on differentiation.)
AQA FP1 2012 June Q1
10 marks Standard +0.3
1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)
AQA FP1 2012 June Q2
7 marks Standard +0.3
2 A curve has equation \(y = x ^ { 4 } + x\).
  1. Find the gradient of the line passing through the point \(( - 2,14 )\) and the point on the curve for which \(x = - 2 + h\). Give your answer in the form $$p + q h + r h ^ { 2 } + h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
  2. Show how the answer to part (a) can be used to find the gradient of the curve at the point ( \(- 2,14\) ). State the value of this gradient.
AQA FP1 2012 June Q3
6 marks Moderate -0.8
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
  2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$
AQA FP1 2012 June Q4
6 marks Moderate -0.3
4 Find the general solution, in degrees, of the equation $$\sin \left( 70 ^ { \circ } - \frac { 2 } { 3 } x \right) = \cos 20 ^ { \circ }$$
AQA FP1 2012 June Q5
11 marks Standard +0.3
5 The curve \(C\) has equation \(y = \frac { x } { ( x + 1 ) ( x - 2 ) }\).
The line \(L\) has equation \(y = - \frac { 1 } { 2 }\).
  1. Write down the equations of the asymptotes of \(C\).
  2. The line \(L\) intersects the curve \(C\) at two points. Find the \(x\)-coordinates of these two points.
  3. Sketch \(C\) and \(L\) on the same axes.
    (You are given that the curve \(C\) has no stationary points.)
  4. Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$
AQA FP1 2012 June Q6
11 marks Standard +0.3
6
  1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
  2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1 \\ 1 & - 1 \end{array} \right]\).
    1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
    2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
    3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).
AQA FP1 2012 June Q7
9 marks Moderate -0.3
7 The equation $$24 x ^ { 3 } + 36 x ^ { 2 } + 18 x - 5 = 0$$ has one real root, \(\alpha\).
  1. Show that \(\alpha\) lies in the interval \(0.1 < x < 0.2\).
  2. Starting from the interval \(0.1 < x < 0.2\), use interval bisection twice to obtain an interval of width 0.025 within which \(\alpha\) must lie.
  3. Taking \(x _ { 1 } = 0.2\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four decimal places.
    (4 marks)
AQA FP1 2012 June Q8
15 marks Challenging +1.2
8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}
  1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
  2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p \\ 0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
  3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
  4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
    (No credit will be given for solutions based on differentiation.)
AQA FP1 2013 June Q1
3 marks Moderate -0.8
1 The equation $$x ^ { 3 } - x ^ { 2 } + 4 x - 900 = 0$$ has exactly one real root, \(\alpha\). Taking \(x _ { 1 } = 10\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four significant figures.
(3 marks)
\includegraphics[max width=\textwidth, alt={}]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-02_1659_1709_1048_153}
AQA FP1 2013 June Q2
7 marks Moderate -0.8
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } p & 2 \\ 4 & p \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right]$$
  1. Find, in terms of \(p\), the matrices:
    1. \(\mathbf { A } - \mathbf { B }\);
    2. AB .
  2. Show that there is a value of \(p\) for which \(\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, and state the corresponding value of \(k\).
AQA FP1 2013 June Q3
8 marks Standard +0.3
3
  1. Find the general solution, in degrees, of the equation $$\cos \left( 5 x + 40 ^ { \circ } \right) = \cos 65 ^ { \circ }$$
  2. Given that $$\sin \frac { \pi } { 12 } = \frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$$ express \(\sin \frac { \pi } { 12 }\) in the form \(\left( \cos \frac { \pi } { 4 } \right) ( \cos ( a \pi ) + \cos ( b \pi ) )\), where \(a\) and \(b\) are rational.
    (3 marks)
AQA FP1 2013 June Q4
7 marks Standard +0.3
4
  1. It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
    1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
    2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
  2. It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q\) i is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).
AQA FP1 2013 June Q5
8 marks Standard +0.3
5
  1. A curve has equation \(y = 2 x ^ { 2 } - 5 x\).
    The point \(P\) on the curve has coordinates \(( 1 , - 3 )\).
    The point \(Q\) on the curve has \(x\)-coordinate \(1 + h\).
    1. Show that the gradient of the line \(P Q\) is \(2 h - 1\).
    2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point \(P\) is parallel to the line \(x + y = 0\).
  2. For the improper integral \(\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x\), either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.
AQA FP1 2013 June Q6
11 marks Standard +0.8
6 The equation $$2 x ^ { 2 } + 3 x - 6 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Hence show that \(\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }\).
  3. Find a quadratic equation, with integer coefficients, whose roots are \(\alpha + \frac { \alpha } { \beta ^ { 2 } }\) and \(\beta + \frac { \beta } { \alpha ^ { 2 } }\).
AQA FP1 2013 June Q7
11 marks Standard +0.3
7
  1. Show that the equation \(4 x ^ { 3 } - x - 540000 = 0\) has a root, \(\alpha\), in the interval \(51 < \alpha < 52\).
  2. It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }\).
    1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)\), where \(k\) is an integer to be found.
    2. Hence show that \(6 S _ { n }\) can be written as the product of three consecutive integers.
  3. Find the smallest value of \(N\) for which the sum of the squares of the first \(N\) odd numbers is greater than 180000 .