Questions — AQA (3620 questions)

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AQA C3 2007 June Q1
7 marks Moderate -0.3
1
  1. Differentiate \(\ln x\) with respect to \(x\).
  2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).
AQA C3 2007 June Q2
9 marks Moderate -0.3
2
  1. Differentiate \(( x - 1 ) ^ { 4 }\) with respect to \(x\).
  2. The diagram shows the curve with equation \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\) for \(x \geqslant 1\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-02_789_1180_1190_431} The shaded region \(R\) is bounded by the curve \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\), the lines \(x = 2\) and \(x = 4\), and the \(x\)-axis. Find the exact value of the volume of the solid formed when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \sqrt { x ^ { 3 } }\) onto the graph of \(y = 2 \sqrt { ( x - 1 ) ^ { 3 } }\).
AQA C3 2007 June Q3
7 marks Moderate -0.8
3
  1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
    (2 marks)
  2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
    1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
    2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
    (2 marks)
AQA C3 2007 June Q4
12 marks Moderate -0.3
4 [Figure 1, printed on the insert, is provided for use in this question.]
  1. Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to \(\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
  2. The curve \(y = 3 ^ { x }\) intersects the line \(y = x + 3\) at the point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 0.5 and 1.5.
    2. Show that the equation \(3 ^ { x } = x + 3\) can be rearranged into the form $$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
    3. Use the iteration \(x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\) to two significant figures.
    4. The sketch on Figure 1 shows part of the graphs of \(y = \frac { \ln ( x + 3 ) } { \ln 3 }\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
AQA C3 2007 June Q5
9 marks Moderate -0.3
5 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { x - 2 } \text { for } x \geqslant 2 \\ & \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for real values of } x , x \neq 0 \end{aligned}$$
  1. State the range of f .
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 1\).
  3. The inverse of f is \(\mathrm { f } ^ { - 1 }\). Find \(\mathrm { f } ^ { - 1 } ( x )\).
AQA C3 2007 June Q6
9 marks Moderate -0.3
6
  1. Use integration by parts to find \(\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x\).
    1. Use the substitution \(u = \sqrt { x }\) to show that $$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
    2. Find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x\).
AQA C3 2007 June Q7
10 marks Standard +0.3
7
  1. A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { x }\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    1. Find the \(x\)-coordinate of each of the stationary points of the curve.
    2. Using your answer to part (a)(ii), determine the nature of each of the stationary points.
AQA C3 2007 June Q8
12 marks Standard +0.3
8
  1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
  2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
  4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.
AQA C3 2015 June Q1
8 marks Standard +0.3
1
  1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x\), giving your answer to three decimal places.
    [0pt] [4 marks]
  2. Find the exact value of the gradient of the curve \(y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )\) at the point on the curve where \(x = 2\).
    [0pt] [4 marks]
AQA C3 2015 June Q2
13 marks Moderate -0.3
2
  1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
  2. Solve the equation \(x = 4 - | 2 x + 1 |\).
  3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
  4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
    [0pt] [4 marks] \section*{Answer space for question 2}
    1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}
AQA C3 2015 June Q3
14 marks Standard +0.3
3
  1. It is given that the curves with equations \(y = 6 \ln x\) and \(y = 8 x - x ^ { 2 } - 3\) intersect at a single point where \(x = \alpha\).
    1. Show that \(\alpha\) lies between 5 and 6 .
    2. Show that the equation \(x = 4 + \sqrt { 13 - 6 \ln x }\) can be rearranged into the form $$6 \ln x + x ^ { 2 } - 8 x + 3 = 0$$
    3. Use the iterative formula $$x _ { n + 1 } = 4 + \sqrt { 13 - 6 \ln x _ { n } }$$ with \(x _ { 1 } = 5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
  2. A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = 6 \ln x + x ^ { 2 } - 8 x + 3\).
    1. Find the exact values of the coordinates of the stationary points of the curve.
    2. Hence, or otherwise, find the exact values of the coordinates of the stationary points of the curve with equation $$y = 2 \mathrm { f } ( x - 4 )$$
AQA C3 2015 June Q4
9 marks Moderate -0.3
4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$
  1. Find the range of f.
  2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 0\).
  3. Find an expression for \(\operatorname { gg } ( x )\), giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
    [0pt] [3 marks]
AQA C3 2015 June Q5
9 marks Standard +0.3
5
  1. By writing \(\tan x\) as \(\frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x\).
    [0pt] [2 marks]
  2. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
    [0pt] [4 marks]
  3. The region bounded by the curve \(y = ( 5 \sqrt { x } ) \sec x\), the \(x\)-axis from 0 to 1 and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
    [0pt] [3 marks]
AQA C3 2015 June Q6
5 marks Moderate -0.3
6
  1. Sketch, on the axes below, the curve with equation \(y = \sin ^ { - 1 } ( 3 x )\), where \(y\) is in radians. State the exact values of the coordinates of the end points of the graph.
  2. Given that \(x = \frac { 1 } { 3 } \sin y\), write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) and hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). \section*{Answer space for question 6}
    1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}
AQA C3 2015 June Q7
7 marks Standard +0.3
7 Use the substitution \(u = 6 - x ^ { 2 }\) to find the value of \(\int _ { 1 } ^ { 2 } \frac { x ^ { 3 } } { \sqrt { 6 - x ^ { 2 } } } \mathrm {~d} x\), giving your answer in the form \(p \sqrt { 5 } + q \sqrt { 2 }\), where \(p\) and \(q\) are rational numbers.
[0pt] [7 marks]
AQA C3 2015 June Q8
10 marks Standard +0.3
8
  1. Show that the equation \(4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k\), where \(k \neq 4\), can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
  2. Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
    [0pt] [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-18_72_113_1055_159}
    \includegraphics[max width=\textwidth, alt={}]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-20_2288_1707_221_153}
AQA C4 Q5
10 marks Standard +0.3
5
    1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
    2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
  2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
  3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
  4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
AQA C4 2006 January Q1
8 marks Moderate -0.8
1
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
    1. Find f(1).
    2. Show that \(\mathrm { f } ( - 2 ) = 0\).
    3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).
AQA C4 2006 January Q2
11 marks Moderate -0.3
2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$
AQA C4 2006 January Q3
6 marks Moderate -0.3
3 It is given that \(3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  1. Find the value of \(R\).
  2. Show that \(\alpha \approx 33.7 ^ { \circ }\).
  3. Hence write down the maximum value of \(3 \cos \theta - 2 \sin \theta\) and find a positive value of \(\theta\) at which this maximum value occurs.
AQA C4 2006 January Q4
9 marks Moderate -0.8
4 On 1 January 1900, a sculpture was valued at \(\pounds 80\).
When the sculpture was sold on 1 January 1956, its value was \(\pounds 5000\).
The value, \(\pounds V\), of the sculpture is modelled by the formula \(V = A k ^ { t }\), where \(t\) is the time in years since 1 January 1900 and \(A\) and \(k\) are constants.
  1. Write down the value of \(A\).
  2. Show that \(k \approx 1.07664\).
  3. Use this model to:
    1. show that the value of the sculpture on 1 January 2006 will be greater than £200 000;
    2. find the year in which the value of the sculpture will first exceed \(\pounds 800000\).
AQA C4 2006 January Q5
15 marks Standard +0.3
5
    1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
    2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
  2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
  3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
  4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
AQA C4 2006 January Q6
7 marks Moderate -0.8
6
  1. Express \(\cos 2 x\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
  2. Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }\), where \(a\) is an integer.
AQA C4 2006 January Q7
10 marks Moderate -0.3
7 The quadrilateral \(A B C D\) has vertices \(A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )\) and \(D ( 2 , - 3 , - 1 )\).
The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ 1 \\ - 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right]\).
    1. Find the vector \(\overrightarrow { A B }\).
    2. Show that the line \(A B\) is parallel to \(l _ { 1 }\).
    3. Verify that \(D\) lies on \(l _ { 1 }\).
  2. The line \(l _ { 2 }\) passes through \(D ( 2 , - 3 , - 1 )\) and \(M ( 4,1,1 )\).
    1. Find the vector equation of \(l _ { 2 }\).
    2. Find the angle between \(l _ { 2 }\) and \(A C\).
AQA C4 2006 January Q8
9 marks Moderate -0.3
8
  1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$ to find \(t\) in terms of \(x\), given that \(x = 70\) when \(t = 0\).
  2. Liquid fuel is stored in a tank. At time \(t\) minutes, the depth of fuel in the tank is \(x \mathrm {~cm}\). Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
    1. Explain what happens when \(x = 6\).
    2. Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm .