Questions FP1 (1385 questions)

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AQA FP1 2005 June Q9
9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$
    1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
    2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
  2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
  3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).
    SurnameOther Names
    Centre NumberCandidate Number
    Candidate Signature
    General Certificate of Education
    June 2005
    Advanced Subsidiary Examination MATHEMATICS
    MFP1
    Unit Further Pure 1 ASSESSMENT and
    QUALIFICATIONS
    ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
    Fill in the boxes at the top of this page.
    Fasten this insert securely to your answer book.
AQA FP1 2006 June Q1
1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
    1. Expand \(( \alpha + \beta ) ^ { 3 }\).
    2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
  3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
AQA FP1 2006 June Q2
2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$ Starting at the point \(( 2,3 )\) on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 2.4\). Give your answer to three decimal places.
AQA FP1 2006 June Q3
3 Show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r \right) = k n ( n + 1 ) ( n - 1 )$$ where \(k\) is a rational number.
AQA FP1 2006 June Q4
4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).
AQA FP1 2006 June Q5
5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
- \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$
  1. Find the matrix:
    1. \(\mathbf { M } ^ { 2 }\);
    2. \(\mathbf { M } ^ { 4 }\).
  2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
  3. Find the matrix \(\mathbf { M } ^ { 2006 }\).
AQA FP1 2006 June Q6
6 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 $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
  2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).
AQA FP1 2006 June Q7
7
  1. Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
  2. The diagram shows the hyperbola \(H\) with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
    \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
    By completing the square, describe a geometrical transformation by which the hyperbola \(H\) can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
AQA FP1 2006 June Q8
8
  1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
    1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) 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)(i) to find the value of \(f ^ { \prime } ( 1 )\).
  2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
    \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
    The graphs intersect at the point \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
    2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
  3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
    (3 marks)
AQA FP1 2006 June Q9
9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$
    1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
    2. Write down the equations of all the asymptotes of \(C\).
    1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
    2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve \(C\).