Questions — AQA FP1 (176 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA FP1 2007 June Q2
2
  1. Show that the equation $$x ^ { 3 } + x - 7 = 0$$ has a root between 1.6 and 1.8.
  2. Use interval bisection twice, starting with the interval in part (a), to give this root to one decimal place.
AQA FP1 2007 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 $$z - 3 \mathbf { i } z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
  2. Find the complex number \(z\) such that $$z - 3 \mathrm { i } z ^ { * } = 16$$
AQA FP1 2007 June Q4
4 The quadratic equation $$2 x ^ { 2 } - x + 4 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }\).
  3. Find a quadratic equation with integer coefficients such that the roots of the equation are $$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$ (3 marks)
AQA FP1 2007 June Q5
5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.] The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants. The following approximate values of \(x\) and \(y\) have been found.
\(x\)1234
\(y\)3.846.149.8215.7
  1. Complete the table in Figure 1, showing values of \(x\) and \(Y\), where \(Y = \log _ { 10 } y\). Give each value of \(Y\) to three decimal places.
  2. Show that, if \(y = a b ^ { x }\), then \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
  3. Draw on Figure 2 a linear graph relating \(x\) and \(Y\).
  4. Hence find estimates for the values of \(a\) and \(b\).
AQA FP1 2007 June Q6
6 Find the general solution of the equation $$\sin \left( 2 x - \frac { \pi } { 2 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
AQA FP1 2007 June Q7
7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$
  1. Write down the equations of the two asymptotes to the curve.
  2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
  3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$
AQA FP1 2007 June Q8
8 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
  2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).
AQA FP1 2007 June Q9
9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}
  1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
  2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
  3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
  4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
  5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
    \end{figure}
AQA FP1 2008 June Q1
1 The equation $$x ^ { 2 } + x + 5 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }\).
  4. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
AQA FP1 2008 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 $$3 \mathrm { i } z + 2 z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
  2. Find the complex number \(z\) such that $$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$
AQA FP1 2008 June Q3
3 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
  2. \(\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
AQA FP1 2008 June Q4
4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are related by an equation of the form $$y = a x + \frac { b } { x + 2 }$$ where \(a\) and \(b\) are constants.
  1. The variables \(X\) and \(Y\) are defined by \(X = x ( x + 2 ) , Y = y ( x + 2 )\). Show that \(Y = a X + b\).
  2. The following approximate values of \(x\) and \(y\) have been found:
    \(x\)1234
    \(y\)0.401.432.403.35
    1. Complete the table in Figure 1, showing values of \(X\) and \(Y\).
    2. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
    3. Estimate the values of \(a\) and \(b\).
AQA FP1 2008 June Q5
5
  1. Find, in radians, the general solution of the equation $$\cos \left( \frac { x } { 2 } + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answer in terms of \(\pi\).
  2. Hence find the smallest positive value of \(x\) which satisfies this equation.
AQA FP1 2008 June Q6
6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 0 & 2
2 & 0 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { r r } 2 & 0
0 & - 2 \end{array} \right]$$
  1. Calculate the matrix \(\mathbf { A B }\).
  2. Show that \(\mathbf { A } ^ { 2 }\) is of the form \(k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  3. Show that \(( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }\).
AQA FP1 2008 June Q7
7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$
  1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
    1. Write down the equations of the two asymptotes of \(C\).
    2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
  3. Sketch the curve \(C\) and its two asymptotes.
AQA FP1 2008 June Q8
8 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-04_866_883_1318_550}
  1. Find the matrix of the stretch which maps \(T _ { 1 }\) to \(T _ { 2 }\).
  2. The triangle \(T _ { 2 }\) is reflected in the line \(y = x\) to give a third triangle, \(T _ { 3 }\). On Figure 3, draw the triangle \(T _ { 3 }\).
  3. Find the matrix of the transformation which maps \(T _ { 1 }\) to \(T _ { 3 }\).
AQA FP1 2008 June Q9
9 The diagram shows the parabola \(y ^ { 2 } = 4 x\) and the point \(A\) with coordinates \(( 3,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-05_732_657_370_689}
  1. Find an equation of the straight line having gradient \(m\) and passing through the point \(A ( 3,4 )\).
  2. Show that, if this straight line intersects the parabola, then the \(y\)-coordinates of the points of intersection satisfy the equation $$m y ^ { 2 } - 4 y + ( 16 - 12 m ) = 0$$
  3. By considering the discriminant of the equation in part (b), find the equations of the two tangents to the parabola which pass through \(A\).
    (No credit will be given for solutions based on differentiation.)
  4. Find the coordinates of the points at which these tangents touch the parabola.
AQA FP1 2009 June Q1
1 The equation $$2 x ^ { 2 } + x - 8 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(4 \alpha ^ { 2 }\) and \(4 \beta ^ { 2 }\). Give your answer in the form \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are integers.
AQA FP1 2009 June Q2
2 A curve has equation $$y = x ^ { 2 } - 6 x + 5$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates 2 and \(2 + h\) respectively.
  1. Find, in terms of \(h\), the gradient of the line \(A B\), giving your answer in its simplest form.
  2. Explain how the result of part (a) can be used to find the gradient of the curve at \(A\). State the value of this gradient.
AQA FP1 2009 June Q3
3 The complex number \(z\) is defined by $$z = x + 2 \mathrm { i }$$ where \(x\) is real.
  1. Find, in terms of \(x\), the real and imaginary parts of:
    1. \(z ^ { 2 }\);
    2. \(z ^ { 2 } + 2 z ^ { * }\).
  2. Show that there is exactly one value of \(x\) for which \(z ^ { 2 } + 2 z ^ { * }\) is real.
AQA FP1 2009 June Q4
4 The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants.
  1. Given that \(Y = \log _ { 10 } y\), show that \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
  2. The diagram shows the linear graph which has equation \(Y = m x + c\).
    \includegraphics[max width=\textwidth, alt={}, center]{932d4c7e-6514-4543-b1d1-753fca5a08fd-5_744_720_833_699} Use this graph to calculate:
    1. an approximate value of \(y\) when \(x = 2.3\), giving your answer to one decimal place;
    2. an approximate value of \(x\) when \(y = 80\), giving your answer to one decimal place.
      (You are not required to find the values of \(m\) and \(c\).)
AQA FP1 2009 June Q5
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
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
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
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.