Questions — AQA FP1 (176 questions)

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AQA FP1 2010 June Q1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)