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 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.
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 \(5 \pi\).