Questions FP1 (1385 questions)

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AQA FP1 2007 January Q7
7 The function f is defined for all real numbers by $$f ( x ) = \sin \left( x + \frac { \pi } { 6 } \right)$$
  1. Find the general solution of the equation \(\mathrm { f } ( x ) = 0\).
  2. The quadratic function g is defined for all real numbers by $$\mathrm { g } ( x ) = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 4 } x ^ { 2 }$$ It can be shown that \(\mathrm { g } ( x )\) gives a good approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
    1. Show that \(\mathrm { g } ( 0.05 )\) and \(\mathrm { f } ( 0.05 )\) are identical when rounded to four decimal places.
    2. A chord joins the points on the curve \(y = \mathrm { g } ( x )\) for which \(x = 0\) and \(x = h\). Find an expression in terms of \(h\) for the gradient of this chord.
    3. Using your answer to part (b)(ii), find the value of \(\mathrm { g } ^ { \prime } ( 0 )\).
AQA FP1 2007 January Q8
8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$
  1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
  3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
    1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
    2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).
AQA FP1 2009 January Q1
1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.
AQA FP1 2009 January Q2
2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.
  1. Write down the other root of this equation.
  2. Find the values of \(b\) and \(c\).
AQA FP1 2009 January Q3
3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$
AQA FP1 2009 January Q4
4 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right)$$
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = n ^ { 3 }\).
  2. Hence show that \(\sum _ { r = n + 1 } ^ { 2 n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = k n ^ { 3 }\) for some integer \(k\).
AQA FP1 2009 January Q5
5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k
k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k
k & k \end{array} \right]$$ where \(k\) is a constant.
  1. Find, in terms of \(k\) :
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { A } ^ { 2 }\).
  2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
  3. It is now given that \(k = 1\).
    1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
    2. The matrix \(\mathbf { A }\) 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.
AQA FP1 2009 January Q6
6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$
    1. Write down the equations of the three asymptotes of this curve.
    2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
    3. Sketch the curve.
      (You are given that the curve has no stationary points.)
  2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$
AQA FP1 2009 January Q7
7 The points \(P ( a , c )\) and \(Q ( b , d )\) lie on the curve with equation \(y = \mathrm { f } ( x )\). The straight line \(P Q\) intersects the \(x\)-axis at the point \(R ( r , 0 )\). The curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis at the point \(S ( \beta , 0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{38c2a2c8-84cc-4bd2-b3ad-f9dee59763ba-4_951_971_470_539}
  1. Show that $$r = a + c \left( \frac { b - a } { c - d } \right)$$
  2. Given that $$a = 2 , b = 3 \text { and } \mathrm { f } ( x ) = 20 x - x ^ { 4 }$$
    1. find the value of \(r\);
    2. show that \(\beta - r \approx 0.18\).
AQA FP1 2009 January Q8
8 For each of the following improper integrals, find the value of the integral or explain why it does not have a value:
  1. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x\);
  2. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\);
  3. \(\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x\).
AQA FP1 2009 January Q9
9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$
  1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
  2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
    1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
    2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
    3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.
AQA FP1 2011 January Q1
1 The quadratic equation \(x ^ { 2 } - 6 x + 18 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find a quadratic equation, with integer coefficients, which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
  3. Hence find the values of \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
AQA FP1 2011 January Q2
2
  1. Find, in terms of \(p\) and \(q\), the value of the integral \(\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
  2. Show that only one of the following improper integrals has a finite value, and find that value:
    1. \(\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\);
    2. \(\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
AQA FP1 2011 January Q3
3
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a rotation about the origin through \(90 ^ { \circ }\) clockwise;
    2. a rotation about the origin through \(180 ^ { \circ }\).
  2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { r r } 2 & 4
    - 1 & - 3 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } - 2 & 1
    - 4 & 3 \end{array} \right]$$
    1. Calculate the matrix \(\mathbf { A B }\).
    2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix, for some integer \(k\).
  3. Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(( \mathbf { A } + \mathbf { B } ) ^ { 2 }\);
    3. \(( \mathbf { A } + \mathbf { B } ) ^ { 4 }\).
AQA FP1 2011 January Q4
4 Find the general solution of the equation $$\sin \left( 4 x - \frac { 2 \pi } { 3 } \right) = - \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
(6 marks)
AQA FP1 2011 January Q5
5
  1. It is given that \(z _ { 1 } = \frac { 1 } { 2 } - \mathrm { i }\).
    1. Calculate the value of \(z _ { 1 } ^ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
    2. Hence verify that \(z _ { 1 }\) is a root of the equation $$z ^ { 2 } + z ^ { * } + \frac { 1 } { 4 } = 0$$
  2. Show that \(z _ { 2 } = \frac { 1 } { 2 } + \mathrm { i }\) also satisfies the equation in part (a)(ii).
  3. Show that the equation in part (a)(ii) has two equal real roots.
AQA FP1 2011 January Q6
6 The diagram shows a circle \(C\) and a line \(L\), which is the tangent to \(C\) at the point \(( 1,1 )\). The equations of \(C\) and \(L\) are $$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$ respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395} The circle \(C\) is now transformed by a stretch with scale factor 2 parallel to the \(x\)-axis. The image of \(C\) under this stretch is an ellipse \(E\).
  1. On the diagram below, sketch the ellipse \(E\), indicating the coordinates of the points where it intersects the coordinate axes.
  2. Find equations of:
    1. the ellipse \(E\);
    2. the tangent to \(E\) at the point \(( 2,1 )\).
      \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}
AQA FP1 2011 January Q7
7 A graph has equation $$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$
  1. Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
  2. Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation $$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
  3. Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
  4. Hence find the coordinates of the two stationary points on the curve.
    (No credit will be given for methods involving differentiation.)
AQA FP1 2011 January Q8
8
  1. The equation $$x ^ { 3 } + 2 x ^ { 2 } + x - 100000 = 0$$ has one real root. Taking \(x _ { 1 } = 50\) as a first approximation to this root, use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to the root.
    1. Given that \(S _ { n } = \sum _ { r = 1 } ^ { n } r ( 3 r + 1 )\), use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$S _ { n } = n ( n + 1 ) ^ { 2 }$$
    2. The lowest integer \(n\) for which \(S _ { n } > 100000\) is denoted by \(N\). Show that $$N ^ { 3 } + 2 N ^ { 2 } + N - 100000 > 0$$
  3. Find the value of \(N\), justifying your answer.
AQA FP1 2012 January Q2
2 Show that only one of the following improper integrals has a finite value, and find that value:
  1. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x\);
  2. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x\).
AQA FP1 2012 January Q3
3
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 9 = 0\);
    2. \(( x + 2 ) ^ { 2 } + 9 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + 2 \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Given that \(z = 1 + 2 \mathrm { i }\), find the value of $$z ^ { * } - z ^ { 3 }$$
AQA FP1 2012 January Q4
4
  1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 4 r - 3 ) = k n ( n + 1 ) \left( 2 n ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
  2. Hence evaluate $$\sum _ { r = 20 } ^ { 40 } r ^ { 2 } ( 4 r - 3 )$$ (2 marks)
AQA FP1 2012 January Q5
5 The diagram below (not to scale) shows a part of a curve \(y = \mathrm { f } ( x )\) which passes through the points \(A ( 2 , - 10 )\) and \(B ( 5,22 )\).
    1. On the diagram, draw a line which illustrates the method of linear interpolation for solving the equation \(\mathrm { f } ( x ) = 0\). The point of intersection of this line with the \(x\)-axis should be labelled \(P\).
    2. Calculate the \(x\)-coordinate of \(P\). Give your answer to one decimal place.
    1. On the same diagram, draw a line which illustrates the Newton-Raphson method for solving the equation \(\mathrm { f } ( x ) = 0\), with initial value \(x _ { 1 } = 2\). The point of intersection of this line with the \(x\)-axis should be labelled \(Q\).
    2. Given that the gradient of the curve at \(A\) is 8 , calculate the \(x\)-coordinate of \(Q\). Give your answer as an exact decimal.
      \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-3_876_1063_1779_523}
AQA FP1 2012 January Q6
6 Find the general solution of each of the following equations:
  1. \(\quad \tan \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { \sqrt { 3 } }\);
  2. \(\quad \tan ^ { 2 } \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { 3 }\).
AQA FP1 2012 January Q7
7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$
  1. Find the equations of the asymptotes of \(H\).
  2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
  3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3
    0 \end{array} \right]\).
    1. Write down the equation of the translated curve.
    2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
  4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\).
    \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}