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 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 \(\frac { \beta ^ { 2 } } { \alpha }\).
AQA FP1 2010 January Q2
2 The complex number \(z\) is defined by $$z = 1 + \mathrm { i }$$
  1. Find the value of \(z ^ { 2 }\), giving your answer in its simplest form.
  2. Hence show that \(z ^ { 8 } = 16\).
  3. Show that \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).
AQA FP1 2010 January Q3
3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$
AQA FP1 2010 January Q4
4 It is given that $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 4
3 & 1 \end{array} \right]$$ and that \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  1. Show that \(( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }\) for some integer \(k\).
  2. Given further that $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 3
    p & 1 \end{array} \right]$$ find the integer \(p\) such that $$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$
AQA FP1 2010 January Q5
5
  1. Explain why \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\) is an improper integral.
  2. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\);
    2. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).
AQA FP1 2010 January Q6
6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}
  1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0
    0 & 2 \end{array} \right]\).
    1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
    2. On Figure 1, draw the rectangle \(R _ { 2 }\).
  2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
    1. On Figure 1, draw the rectangle \(R _ { 3 }\).
    2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
  3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).
AQA FP1 2010 January Q7
7 A curve \(C\) has equation \(y = \frac { 1 } { ( x - 2 ) ^ { 2 } }\).
    1. Write down the equations of the asymptotes of the curve \(C\).
    2. Sketch the curve \(C\).
  2. The line \(y = x - 3\) intersects the curve \(C\) at a point which has \(x\)-coordinate \(\alpha\).
    1. Show that \(\alpha\) lies within the interval \(3 < x < 4\).
    2. Starting from the interval \(3 < x < 4\), use interval bisection twice to obtain an interval of width 0.25 within which \(\alpha\) must lie.
AQA FP1 2010 January Q8
8
  1. Show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$ can be expressed in the form $$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(k\) is a rational number and \(a , b\) and \(c\) are integers.
  2. Show that there is exactly one positive integer \(n\) for which $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$
AQA FP1 2010 January Q9
9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).
  1. Show that \(a = 2\) and \(b = 4\).
  2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
  3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
  4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
    (No credit will be given for solutions based on differentiation.)
AQA FP1 2005 June Q1
1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 4
4 & 3 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 2
2 & 0 \end{array} \right]$$
  1. Calculate the matrices:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { A B }\).
  2. Show that \(\mathbf { A } + \mathbf { B } - \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
    (2 marks)
AQA FP1 2005 June Q2
2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x$$ where the angle \(2 x\) is measured in radians.
Starting at the point \(( 0.5,1 )\) on the curve, use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 0.7\). Give your answer to three significant figures.
(6 marks)
AQA FP1 2005 June Q3
3
  1. Use the formulae $$\begin{gathered} \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )
    \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 } \end{gathered}$$ and $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$ (4 marks)
  2. Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)
AQA FP1 2005 June Q4
4 The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x$$
  1. Express \(\mathrm { f } ( 2 + h ) - \mathrm { f } ( 2 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
  2. Use your answer to part (a) to find the value of \(\mathrm { f } ^ { \prime } ( 2 )\).
AQA FP1 2005 June Q5
5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :
  1. \(\quad \tan 3 x = \sqrt { 3 }\);
  2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).
AQA FP1 2005 June Q6
6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
    2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
    3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
  2. Find in the form \(p + \mathrm { i } q\) the values of:
    1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
    2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
  3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).
AQA FP1 2005 June Q7
7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}
  1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2
    - 2 & 2 \end{array} \right]$$
    1. Calculate the coordinates of \(D , E\) and \(F\).
    2. Draw the triangle \(D E F\) on Figure 1.
  2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
    1. the scale factor of the enlargement;
    2. the magnitude of the angle of the rotation.
AQA FP1 2005 June Q8
8 The diagram shows a part of the curve $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$$ and a chord \(P Q\) of the curve, where \(P\) lies on the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-05_751_1072_680_459}
  1. Write down the coordinates of \(P\).
  2. The gradient of the chord \(P Q\) is 2 . Find the coordinates of \(Q\).