Questions FP1 (1385 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 2012 January Q8
8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).
  1. On the diagram, draw:
    1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
    2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0
      0 & 2 \end{array} \right]$$
  2. Find the matrix of:
    1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
    2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).
      \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}
AQA FP1 2012 January Q9
9 A curve has equation $$y = \frac { x } { x - 1 }$$
  1. Find the equations of the asymptotes of this curve.
  2. Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
  3. It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
    1. Find the two possible values of \(c\).
      (No credit will be given for methods involving differentiation.)
    2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.
AQA FP1 2013 January Q1
1 A curve passes through the point (1,3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { 1 + x ^ { 3 } }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 1.2\). Give your answer to four decimal places.
AQA FP1 2013 January Q2
2
  1. Solve the equation \(w ^ { 2 } + 6 w + 34 = 0\), giving your answers in the form \(p + q \mathrm { i }\), where \(p\) and \(q\) are integers.
  2. It is given that \(z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )\).
    1. Express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are integers.
    2. Find integers \(m\) and \(n\) such that \(z + m z ^ { * } = n \mathrm { i }\).
AQA FP1 2013 January Q3
3
  1. Find the general solution of the equation $$\sin \left( 2 x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
  2. Use your general solution to find the exact value of the greatest solution of this equation which is less than \(6 \pi\).
AQA FP1 2013 January Q4
4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.
AQA FP1 2013 January Q5
5 The roots of the quadratic equation $$x ^ { 2 } + 2 x - 5 = 0$$ are \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Calculate the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(\alpha ^ { 3 } \beta + 1\) and \(\alpha \beta ^ { 3 } + 1\).
AQA FP1 2013 January Q6
6
  1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2
    3 & 0 \end{array} \right]\).
    1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2
      3 & 6 \end{array} \right]\), find the value of \(m\).
    2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0
    0 & - 1 \end{array} \right]\).
    1. Describe the geometrical transformation represented by \(\mathbf { A }\).
    2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1
      1 & 1 \end{array} \right]\), where \(k\) is a surd.
    3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\).
      \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).
AQA FP1 2013 January Q8
8
  1. Show that $$\sum _ { r = 1 } ^ { n } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right) = n ( n + p ) ( n + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
  2. Hence find the value of $$\sum _ { r = 11 } ^ { 20 } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right)$$ (2 marks)
AQA FP1 2013 January Q9
9 An ellipse is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$
  1. Find the \(x\)-coordinates of \(A\) and \(B\).
  2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
    2. Hence find the exact value of \(m\).
    3. Find the coordinates of \(P\).
AQA FP1 2007 June Q1
1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 1
3 & 8 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } 1 & 2
3 & 4 \end{array} \right]$$ The matrix \(\mathbf { M } = \mathbf { A } - 2 \mathbf { B }\).
  1. Show that \(\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1
    - 1 & 0 \end{array} \right]\), where \(n\) is a positive integer.
    (2 marks)
  2. The matrix \(\mathbf { M }\) represents a combination of an enlargement of scale factor \(p\) and a reflection in a line \(L\). State the value of \(p\) and write down the equation of \(L\).
  3. Show that $$\mathbf { M } ^ { 2 } = q \mathbf { I }$$ where \(q\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
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 \(( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }\).