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 2016 June Q4
2 marks
4
  1. Given that \(\sin \frac { \pi } { 3 } = \cos \frac { \pi } { k }\), state the value of the integer \(k\).
  2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }$$ giving your answer, in its simplest form, in terms of \(\pi\).
  3. Hence, given that \(\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }\), show that there is only one finite value for \(\tan x\) and state its exact value.
    [0pt] [2 marks]
AQA FP1 2016 June Q5
4 marks
5
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(\sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 } = 3 n \left( 4 n ^ { 2 } - 1 \right)\).
  2. Hence express \(\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 }\) as a product of four linear factors in terms of \(n\).
    [0pt] [4 marks]
AQA FP1 2016 June Q6
6 marks
6 A parabola with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant, is translated by the vector \(\left[ \begin{array} { l } 2
3 \end{array} \right]\) to give the curve \(C\). The curve \(C\) passes through the point (4, 7).
  1. Show that \(a = 2\).
  2. Find the values of \(k\) for which the line \(k y = x\) does not meet the curve \(C\).
    [0pt] [6 marks]
AQA FP1 2016 June Q7
11 marks
7
  1. Solve the equation \(x ^ { 2 } + 4 x + 20 = 0\), giving your answers in the form \(c + d \mathrm { i }\), where \(c\) and \(d\) are integers.
  2. The roots of the quadratic equation $$z ^ { 2 } + ( 4 + i + q i ) z + 20 = 0$$ are \(w\) and \(w ^ { * }\).
    1. In the case where \(q\) is real, explain why \(q\) must be - 1 .
    2. In the case where \(w = p + 2 \mathrm { i }\), where \(p\) is real, find the possible values of \(q\).
      [0pt] [5 marks] \(8 \quad\) The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { l l } 2 & 0
      0 & 1 \end{array} \right]\).
    1. Find the matrix \(\mathbf { A } ^ { 2 }\).
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
  4. Given that the matrix \(\mathbf { B }\) represents a reflection in the line \(x + \sqrt { 3 } y = 0\), find the matrix \(\mathbf { B }\), giving the exact values of any trigonometric expressions.
  5. Hence find the coordinates of the point \(P\) which is mapped onto \(( 0 , - 4 )\) under the transformation represented by \(\mathbf { A } ^ { 2 }\) followed by a reflection in the line \(x + \sqrt { 3 } y = 0\).
    [0pt] [6 marks] \(9 \quad\) A curve \(C\) has equation \(y = \frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) }\).
    The line \(L\) has equation \(y = \frac { 1 } { 2 } ( x - 1 )\).
  6. Write down the equations of the asymptotes of \(C\).
  7. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
  8. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes.
  9. Hence solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )\).
AQA FP1 2013 January Q7
  1. Show that there is a linear relationship between \(Y\) and \(X\).
  2. The graph of \(Y\) against \(X\) is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of \(n\) and the value of \(a\).
AQA FP1 2015 June Q6
  1. Sketch the curve \(C _ { 1 }\), stating the values of its intercepts with the coordinate axes.
  2. The curve \(C _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } k
    0 \end{array} \right]\), where \(k < 0\), to give a curve \(C _ { 2 }\). Given that \(C _ { 2 }\) passes through the origin \(( 0,0 )\), find the equations of the asymptotes of \(C _ { 2 }\).
    [0pt] [3 marks]
AQA FP1 2005 January Q1
1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots $$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$
AQA FP1 2005 January Q2
2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
  1. Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
  2. Calculate the \(y\)-coordinates of the points of intersection of the curve with the line \(x = 1\). Give your answers in the form \(p \sqrt { 2 }\), where \(p\) is a rational number.
  3. The curve is translated one unit in the positive \(x\) direction. Write down the equation of the curve after the translation.
AQA FP1 2005 January Q3
3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
  1. Write down, in terms of \(x\) and \(y\), an expression for \(z ^ { * }\), the complex conjugate of \(z\).
  2. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
  3. Find the complex number \(z\) such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$
AQA FP1 2005 January Q4
4 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
  1. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x\);
    (3 marks)
  2. \(\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x\);
  3. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).
AQA FP1 2005 January Q5
5
  1. The transformation \(T _ { 1 }\) is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1
    1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
  2. The transformation \(T _ { 2 }\) is an anticlockwise rotation about the origin through an angle of \(60 ^ { \circ }\). Find the matrix of the transformation \(T _ { 2 }\). Use surds in your answer where appropriate.
    (3 marks)
  3. Find the matrix of the transformation obtained by carrying out \(T _ { 1 }\) followed by \(T _ { 2 }\).
    (3 marks)
AQA FP1 2005 January Q6
6 The angle \(x\) radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$
  1. Find the general solution of this equation, giving the roots as exact values in terms of \(\pi\).
  2. Find the number of roots of the equation which lie between 0 and \(2 \pi\).
AQA FP1 2005 January Q7
7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants. Experimental evidence has provided the following approximate values:
\(x\)1.54.05.06.58.0
\(y\)5.06.37.08.09.0
  1. On Figure 1, draw a linear graph connecting the variables \(X\) and \(Y\), where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
  2. From your graph, find approximate values for the constants \(a\) and \(b\).
AQA FP1 2005 January Q8
8 [Figure 2, printed on the insert, is provided for use in this question.]
The diagram shows a part of the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } - 2 x - 1$$ The point \(P\) has coordinates \(( 1 , - 2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{a77cc9c3-5ff6-4abc-931e-e811740267f2-05_606_565_717_740}
  1. Taking \(x _ { 1 } = 1\) as a first approximation to a root of the equation \(\mathrm { f } ( x ) = 0\), use the NewtonRaphson method to find a second approximation, \(x _ { 2 }\), to the root.
  2. On Figure 2, draw a straight line to illustrate the Newton-Raphson method as used in part (a). Mark \(x _ { 1 }\) and \(x _ { 2 }\) on Figure 2
  3. By considering \(f ( 2 )\), show that the second approximation found in part (a) is not as good as the first approximation.
  4. Taking \(x _ { 1 } = 1.6\) as a first approximation to the root, use the Newton-Raphson method to find a second approximation to the root. Give your answer to three decimal places.
    (2 marks)
AQA FP1 2005 January Q9
9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$
  1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
  2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
    1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
    2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
    1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
    2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.
AQA FP1 2008 January Q1
1 It is given that \(z _ { 1 } = 2 + \mathrm { i }\) and that \(z _ { 1 } { } ^ { * }\) is the complex conjugate of \(z _ { 1 }\).
Find the real numbers \(x\) and \(y\) such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$
AQA FP1 2008 January Q2
2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x }$$ Starting at the point \(( 1,4 )\) on the curve, use a step-by-step method with a step length of 0.01 to estimate the value of \(y\) at \(x = 1.02\). Give your answer to six significant figures.
AQA FP1 2008 January Q3
3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).
AQA FP1 2008 January Q4
4
  1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
  2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .
AQA FP1 2008 January Q5
5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}
  1. Write down the equations of the two asymptotes.
  2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
  3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
    1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
    2. the equations of the hyperbola and the asymptotes after the translation.
AQA FP1 2008 January Q6
6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3
3 & - \sqrt { 3 } \end{array} \right]$$
    1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ }
      \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
  2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
    1. the scale factor of the enlargement;
    2. the equation of the mirror line of the reflection.
  3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).
AQA FP1 2008 January Q7
7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}
    1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
    2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
    3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
  2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
    1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
    2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.
AQA FP1 2008 January Q8
8
    1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
    2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
  2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
  3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$
AQA FP1 2008 January Q9
9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$
  1. Write down the equations of the three asymptotes of \(C\).
  2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
    (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve \(C\).
AQA FP1 2010 January Q1
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
  3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).