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AQA FP1 2009 June Q5
9 marks Standard +0.3
5
  1. Find the general solution of the equation $$\cos ( 3 x - \pi ) = \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
  2. From your general solution, find all the solutions of the equation which lie between \(10 \pi\) and \(11 \pi\).
AQA FP1 2009 June Q6
11 marks Standard +0.3
6 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 1$$
  1. Sketch the ellipse \(E\), showing the coordinates of the points of intersection of the ellipse with the coordinate axes.
  2. The ellipse \(E\) is stretched with scale factor 2 parallel to the \(y\)-axis. Find and simplify the equation of the curve after the stretch.
  3. The original ellipse, \(E\), is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\). The equation of the translated ellipse is $$4 x ^ { 2 } + 3 y ^ { 2 } - 8 x + 6 y = 5$$ Find the values of \(a\) and \(b\).
AQA FP1 2009 June Q7
11 marks Standard +0.3
7
  1. Using surd forms where appropriate, find the matrix which represents:
    1. a rotation about the origin through \(30 ^ { \circ }\) anticlockwise;
    2. a reflection in the line \(y = \frac { 1 } { \sqrt { 3 } } x\).
  2. The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { c c } 1 & \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$ represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
  3. The transformation represented by \(\mathbf { A }\) is followed by the transformation represented by \(\mathbf { B }\), where $$\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 3 } & - 1 \\ 1 & \sqrt { 3 } \end{array} \right]$$ Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.
AQA FP1 2009 June Q8
15 marks Standard +0.8
8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$
  1. Write down the equations of the three asymptotes to the curve.
  2. Show that the curve has no point of intersection with the line \(y = - 1\).
    1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
    2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
  4. Hence find the coordinates of the two stationary points on the curve.
AQA FP1 2010 June Q1
6 marks Moderate -0.5
1 A curve passes through the point ( 1,3 ) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + x ^ { 3 }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the \(y\)-coordinate of the point on the curve for which \(x = 1.3\). Give your answer to three decimal places.
(No credit will be given for methods involving integration.)
AQA FP1 2010 June Q2
6 marks Moderate -0.3
2 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 $$( 1 - 2 i ) z - z ^ { * }$$
  2. Hence find the complex number \(z\) such that $$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$
    PARTREFERENCE
    \(\_\_\_\_\)\(\_\_\_\_\)
AQA FP1 2010 June Q3
5 marks Moderate -0.3
3 Find the general solution, in degrees, of the equation $$\cos \left( 5 x - 20 ^ { \circ } \right) = \cos 40 ^ { \circ }$$
\includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-04_2228_1705_475_155}
AQA FP1 2010 June Q4
8 marks Moderate -0.8
4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants.
The following approximate values of \(x\) and \(y\) have been found.
\(\boldsymbol { x }\)2468
\(\boldsymbol { y }\)6.010.518.028.2
  1. Complete the table below, showing values of \(X\), where \(X = x ^ { 2 }\).
  2. On the diagram below, draw a linear graph relating \(X\) and \(y\).
  3. Use your graph to find estimates, to two significant figures, for:
    1. the value of \(x\) when \(y = 15\);
    2. the values of \(a\) and \(b\).
  4. \(\boldsymbol { x }\)2468
    \(\boldsymbol { X }\)
    \(\boldsymbol { y }\)6.010.518.028.2
  5. \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-05_771_1586_1772_274}
AQA FP1 2010 June Q5
6 marks Moderate -0.8
5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).
  1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
  2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
    \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}
AQA FP1 2010 June Q6
11 marks Moderate -0.3
6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:
  1. A ;
  2. B ;
  3. \(\quad \mathbf { A } ^ { 2 }\);
  4. \(\quad \mathbf { B } ^ { 2 }\);
  5. AB.
    □
AQA FP1 2010 June Q7
10 marks Moderate -0.8
7
    1. Write down the equations of the two asymptotes of the curve \(y = \frac { 1 } { x - 3 }\).
    2. Sketch the curve \(y = \frac { 1 } { x - 3 }\), showing the coordinates of any points of intersection with the coordinate axes.
    3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line \(y = 2 x - 5\).
    1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
    2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}
AQA FP1 2010 June Q8
10 marks Standard +0.8
8 The quadratic equation $$x ^ { 2 } - 4 x + 10 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 2 } { 5 }\).
  3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha + \frac { 2 } { \beta }\) and \(\beta + \frac { 2 } { \alpha }\).
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 $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$