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AQA C1 2016 June Q6
3 marks
6
  1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
    1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
    2. Sketch the curve, giving the value of the \(y\)-intercept.
  2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
    1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
    2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]
AQA C1 2016 June Q7
3 marks
7 The diagram shows the sketch of a curve and the tangent to the curve at \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0d5b9235-af2b-4fd5-8fcf-b2b45e3c0a3c-14_519_817_356_614} The curve has equation \(y = 4 - x ^ { 2 } - 3 x ^ { 3 }\) and the point \(P ( - 2,24 )\) lies on the curve. The tangent at \(P\) crosses the \(x\)-axis at \(Q\).
    1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
    2. Hence find the \(x\)-coordinate of \(Q\).
    1. Find \(\int _ { - 2 } ^ { 1 } \left( 4 - x ^ { 2 } - 3 x ^ { 3 } \right) \mathrm { d } x\).
    2. The point \(R ( 1,0 )\) lies on the curve. Calculate the area of the shaded region bounded by the curve and the lines \(P Q\) and \(Q R\).
      [0pt] [3 marks]
AQA C1 2016 June Q8
8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
    1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
    2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).
Edexcel C1 Q1
  1. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
    (b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
  2. Solve the simultaneous equations
$$\begin{gathered} x - 3 y + 1 = 0
x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{gathered}$$
Edexcel C1 Q3
  1. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.
    1. Show that \(p = 4\).
    2. Find the value of the 40th term of this series.
    3. \(\mathrm { f } ( x ) = x ^ { 2 } - k x + 9\), where \(k\) is a constant.
    4. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions.
    Given that \(k = 4\),
  2. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,
Edexcel C1 Q5
5. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } }$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
Edexcel C1 Q6
6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$
Edexcel C1 Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{922202a6-3455-433f-ac8f-673daefaa7d2-3_574_574_879_662}
\end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
Edexcel C1 Q1
  1. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
  2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
Edexcel C1 Q2
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q3
3. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
(b) Solve the simultaneous equations $$\begin{aligned} & x = 2 y - 2
& x ^ { 2 } = y ^ { 2 } + 7 \end{aligned}$$
Edexcel C1 Q4
  1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$
  1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
  2. Using integration, find \(\mathrm { f } ( x )\).
Edexcel C1 Q5
5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
  2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
  3. Find the exact coordinates of the mid-point of \(A C\).
Edexcel C1 Q6
6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
Edexcel C1 Q7
7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q8
8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END
Edexcel C1 Q1
1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
Edexcel C1 Q2
2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$
  1. Write down the first two terms of the series.
  2. Find the common difference of the series. Given that \(n = 50\),
  3. find the sum of the series.
Edexcel C1 Q3
3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\).
  2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
Edexcel C1 Q4
4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
Edexcel C1 Q5
5. Find the set of values for \(x\) for which
  1. \(6 x - 7 < 2 x + 3\),
  2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
  3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
Edexcel C1 Q6
6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
Edexcel C1 Q7
7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),
  1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
  2. show that the population of fish dies out during the sixth year.
  3. Find the value of \(d\) which would leave the population each year unchanged.
Edexcel C1 Q8
8. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
  2. Find the \(x\)-coordinate of \(Q\).
  3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
  4. Find the length of \(R S\), giving your answer as a surd.
Edexcel C1 Q9
9. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.
  1. Find the gradient of \(A B\).
  2. Calculate the value of \(k\).
  3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
  4. Find the exact value of the area of \(\triangle A B C\).
  5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.