Edexcel C1 (Core Mathematics 1)

1. Given that \(( 2 + \sqrt { } 7 ) ( 4 - \sqrt { 7 } ) = a + b \sqrt { } 7\), where a and \(b\) are integers,
   1. find the value of a and the value of \(b\).
   Given that \(\frac { 2 + \sqrt { 7 } } { 4 + \sqrt { 7 } } = c + d \sqrt { 7 }\) where \(c\) and \(d\) are rational numbers,
2. find the value of \(c\) and the value of \(d\).

2. (a) Prove, by completing the square, that the roots of the equation \(x ^ { 2 } + 2 k x + c = 0\), where \(k\) and \(c\) are constants, are \(- k \pm \sqrt { } \left( k ^ { 2 } - c \right)\). The equation \(x ^ { 2 } + 2 k x \pm 81 = 0\) has equal roots.
(b) Find the possible values of \(k\).

3. \begin{figure}[h] \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.

1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
3. find the value of \(k\),
4. find the coordinates of \(D\).

4. \begin{figure}[h] \end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$

1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
   (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
   (a) Prove that \(k ( 25 k - 8 ) \geq 0\).
   (b) Hence find the set of possible values of \(k\).
   (c) Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
3. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.
   (a) Find the amount that Anne will pay in the 40th year.
   (b) Find the total amount that Anne will pay in over the 40 years.

Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
(c) find the value of \(d\).

8. 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.
   [0pt] [P1 June 2001 Question 8*]

9. Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),

1. find the exact value of \(x\) and the exact value of \(y\),
2. calculate the exact value of \(2 ^ { y - x }\).
   [0pt] [P1 January 2002 Question 1]

10. The straight line \(l _ { 1 }\) has equation \(4 y + x = 0\). The straight line \(l _ { 2 }\) has equation \(y = 2 x - 3\).

1. On the same axes, sketch the graphs of \(l _ { 1 }\) and \(l _ { 2 }\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).
2. Calculate, as exact fractions, the coordinates of \(A\).
3. Find an equation of the line through \(A\) which is perpendicular to \(l _ { 1 }\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

11. 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.

12. 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.

13. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
(b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
[0pt] [P1 June 2002 Question 1]

14. 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 \(y = \mathrm { f } ( x )\).

15. (a) By completing the square, find in terms of \(k\) the roots of the equation $$x ^ { 2 } + 2 k x - 7 = 0$$ (b) Prove that, for all values of \(k\), the roots of \(x ^ { 2 } + 2 k x - 7 = 0\) are real and different.
(c) Given that \(k = \sqrt { } 2\), find the exact roots of the equation.
[0pt] [P1 June 2002 Question 4]

16. \section*{Figure 3} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.

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.

17. (a) Solve the inequality $$3 x - 8 > x + 13$$ (b) Solve the inequality $$x ^ { 2 } - 5 x - 14 > 0$$

1. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is

$$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011.

19. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

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 \(k\).

20. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that \(C\) passes through the point \(P ( 2,6 )\),

1. find \(y\) in terms of \(x\).
2. Verify that \(C\) passes through the point ( 4,0 ).
3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
4. Calculate the exact \(x\)-coordinate of \(Q\).
   21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$
5. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
6. Find \(\int y \mathrm {~d} x\).
   22. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
7. Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2
   x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
   1. The straight line \(l _ { 1 }\) with equation \(y = \frac { 3 } { 2 } x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \(( 5 , - 3 )\).
   The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(Q\).
8. Calculate the coordinates of the mid-point of \(P Q\).
9. Find an equation for \(l _ { 2 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integer constants. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\).
10. Calculate the exact coordinates of \(R\).
    24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
11. Use integration to find \(y\) in terms of \(x\).
12. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
    25. Find the set of values for \(x\) for which
13. \(6 x - 7 < 2 x + 3\),
14. \(2 x ^ { 2 } - 11 x + 5 < 0\),
15. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
    [0pt] [P1 June 2003 Question 2]
    26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \(( 280 + x )\) phones will be sold in the second month, \(( 280 + 2 x )\) in the third month, and so on. Using this model with \(x = 5\), calculate
16. 1. the number of phones sold in the 36th month,
    2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
       Using the same model,
17. find the least value of \(x\) required to achieve this target.
    [0pt] [P1 June 2003 Question 3]
    27. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
18. 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 .
19. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
20. Find the exact coordinates of the mid-point of \(A C\).
    28. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
21. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
22. 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.
    [0pt] [P1 June 2003 Question 8*]
    29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$
23. Write down the first two terms of the series.
24. Find the common difference of the series. Given that \(n = 50\),
25. find the sum of the series.
    30. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where \(c\) is a constant.
26. Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
    31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0
    & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
    1. 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 $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
    \includegraphics[max width=\textwidth, alt={}]{b85f4635-aa93-4c6a-9d1f-2ef5bac1b48c-15_668_748_358_699}
    The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
27. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
28. Find an equation for the line through \(C\) and \(M\). Given that the \(x\)-coordinate of \(C\) is 4 ,
29. find the \(y\)-coordinate of \(C\),
30. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
    34. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.
31. Show that \(p = 4\).
32. Find the value of the 40th term of this series.
33. Prove that the sum of the first \(n\) terms of the series is a perfect square.
    35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$
34. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions. Given that \(k = 4\),
35. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,
    36. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$
36. 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 \(( 4,30 )\).
37. Using integration, find \(\mathrm { f } ( x )\).
    37. \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{b85f4635-aa93-4c6a-9d1f-2ef5bac1b48c-17_687_1074_351_539}
    Figure 2 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\).
    The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).
38. Write down the coordinates of \(A\).
39. Find, using algebra, the coordinates of \(P\) and \(Q\).
40. Show that \(\angle P A Q\) is a right angle.
    38. 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.
41. 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.
42. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
       [0pt] [P2 January 2004 Question 2]
       39. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.
43. Find the coordinates of the mid-point of \(A B\).
44. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
    40. Giving your answers in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers, find
45. \(( 3 - \sqrt { } 8 ) ^ { 2 }\),
46. \(\frac { 1 } { 4 - \sqrt { 8 } }\).
    41. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
47. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
48. form a quadratic inequality in \(x\).
49. by solving your inequalities, find the set of possible values of \(x\).
    42. The curve \(C\) has equation \(y = x ^ { 2 } - 4\) and the straight line \(l\) has equation \(y + 3 x = 0\).
50. In the space below, sketch \(C\) and \(l\) on the same axes.
51. Write down the coordinates of the points at which \(C\) meets the coordinate axes.
52. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\).
    43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
53. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
54. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).