Questions — AQA FP1 (176 questions)

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AQA FP1 2006 January Q1
1
  1. Show that the equation $$x ^ { 3 } + 2 x - 2 = 0$$ has a root between 0.5 and 1 .
  2. Use linear interpolation once to find an estimate of this root. Give your answer to two decimal places.
AQA FP1 2006 January Q2
2
  1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
    1. \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
    2. \(\int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
  2. Explain briefly why the integrals in part (a) are improper integrals.
AQA FP1 2006 January Q3
3 Find the general solution, in degrees, for the equation $$\sin \left( 4 x + 10 ^ { \circ } \right) = \sin 50 ^ { \circ }$$
AQA FP1 2006 January Q4
4 A curve has equation $$y = \frac { 6 x } { x - 1 }$$
  1. Write down the equations of the two asymptotes to the curve.
  2. Sketch the curve and the two asymptotes.
  3. Solve the inequality $$\frac { 6 x } { x - 1 } < 3$$
AQA FP1 2006 January Q5
5
    1. Calculate \(( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )\).
    2. Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation $$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$ where \(z ^ { * }\) is the conjugate of \(z\).
  2. The quadratic equation $$x ^ { 2 } + p x + q = 0$$ in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
    1. Write down the other root of the equation.
    2. Find the sum and product of the two roots of the equation.
    3. Hence state the values of \(p\) and \(q\).
AQA FP1 2006 January Q6
6 [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 $$y = k x ^ { n }$$ where \(k\) and \(n\) are constants.
Experimental evidence has provided the following approximate values:
\(x\)417150300
\(y\)1.85.03050
  1. Complete the table in Figure 1, showing values of \(X\) and \(Y\), where $$X = \log _ { 10 } x \quad \text { and } \quad Y = \log _ { 10 } y$$ Give each value to two decimal places.
  2. Show that if \(y = k x ^ { n }\), then \(X\) and \(Y\) must satisfy an equation of the form $$Y = a X + b$$
  3. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
  4. Find an estimate for the value of \(n\).
AQA FP1 2006 January Q7
7
  1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1
    - 1 & 0 \end{array} \right]$$
    1. Describe the transformation T geometrically.
    2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
    3. Explain briefly why the transformation T followed by T is the identity transformation.
  2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1
    0 & 1 \end{array} \right]$$
    1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
    2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).
AQA FP1 2006 January Q8
8 A curve has equation \(y ^ { 2 } = 12 x\).
  1. Sketch the curve.
    1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
    2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
    1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
    2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
    3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
    4. In the case where \(c = 4\), determine whether the line intersects the curve or not.
AQA FP1 2007 January Q1
1
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 16 = 0\);
    2. \(x ^ { 2 } - 2 x + 17 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Hence, or otherwise, verify that \(x = 1 + \mathrm { i }\) satisfies the equation $$x ^ { 3 } + 2 x - 4 \mathrm { i } = 0$$
AQA FP1 2007 January Q2
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 }
\frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$
  1. Calculate:
    1. \(\mathbf { A } + \mathbf { B }\);
    2. \(\mathbf { B A }\).
  2. Describe fully the geometrical transformation represented by each of the following matrices:
    1. \(\mathbf { A }\);
    2. \(\mathbf { B }\);
    3. \(\mathbf { B A }\).
AQA FP1 2007 January Q3
3 The quadratic equation $$2 x ^ { 2 } + 4 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 1\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 }\).
AQA FP1 2007 January Q4
4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { b }$$ where \(a\) and \(b\) are constants.
  1. Using logarithms to base 10 , reduce the relation \(y = a x ^ { b }\) to a linear law connecting \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
  2. The diagram shows the linear graph that results from plotting \(\log _ { 10 } y\) against \(\log _ { 10 } x\).
    \includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411} Find the values of \(a\) and \(b\).
AQA FP1 2007 January Q5
5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$
  1. Write down the equations of the three asymptotes to the curve.
  2. Sketch the curve.
    (You are given that the curve has no stationary points.)
  3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$
AQA FP1 2007 January Q6
6
    1. Expand \(( 2 r - 1 ) ^ { 2 }\).
    2. Hence show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
  2. Hence find the sum of the squares of the odd numbers between 100 and 200 .
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.