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 \(P ( 2,6 )\),
- find \(y\) in terms of \(x\).
- Verify that \(C\) passes through the point ( 4,0 ).
- Find an equation of the tangent to \(C\) at \(P\).
The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
- Calculate the exact \(x\)-coordinate of \(Q\).
21.
$$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$ - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Find \(\int y \mathrm {~d} x\).
22. (a) Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\). - Solve the simultaneous equations
$$\begin{gathered}
x = 2 y - 2
x ^ { 2 } = y ^ { 2 } + 7
\end{gathered}$$
- 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 \(Q\). - Calculate the coordinates of the mid-point of \(P Q\).
- 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 \(R\).
- Calculate the exact coordinates of \(R\).
24.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$ - Use integration to find \(y\) in terms of \(x\).
- 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 - \(6 x - 7 < 2 x + 3\),
- \(2 x ^ { 2 } - 11 x + 5 < 0\),
- 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 - the number of phones sold in the 36th month,
- 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,
- 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 \(B\). - 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 .
- Write down an equation for \(l _ { 2 }\).
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
- Find the exact coordinates of the mid-point of \(A C\).
28. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\), - find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
- 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
$$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$ - Write down the first two terms of the series.
- Find the common difference of the series.
Given that \(n = 50\),
- 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. - 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
$$\begin{aligned}
& x - 3 y + 1 = 0
& x ^ { 2 } - 3 x y + y ^ { 2 } = 11
\end{aligned}$$
- 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={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_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. - Find the gradient of \(A B\).
The point \(M\) is the mid-point of \(A B\).
- Find an equation for the line through \(C\) and \(M\).
Given that the \(x\)-coordinate of \(C\) is 4 ,
- find the \(y\)-coordinate of \(C\),
- 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. - Show that \(p = 4\).
- Find the value of the 40th term of this series.
- Prove that the sum of the first \(n\) terms of the series is a perfect square.
35.
$$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$ - Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions.
Given that \(k = 4\),
- 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
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$ - 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 \(( 4,30 )\).
- Using integration, find \(\mathrm { f } ( x )\).
37.
\section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_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 \(Q\). - Write down the coordinates of \(A\).
- Find, using algebra, the coordinates of \(P\) and \(Q\).
- 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. - 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.
- Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
- calculate the value of \(a\),
- 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.
- Find the coordinates of the mid-point of \(A B\).
- 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 - \(( 3 - \sqrt { } 8 ) ^ { 2 }\),
- \(\frac { 1 } { 4 - \sqrt { 8 } }\).
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 , - form a linear inequality in \(x\).
Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
- form a quadratic inequality in \(x\).
- 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 \(y + 3 x = 0\). - In the space below, sketch \(C\) and \(l\) on the same axes.
- Write down the coordinates of the points at which \(C\) meets the coordinate axes.
- Using algebra, find the coordinates of the points at which \(l\) intersects \(C\).
43.
$$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$ - Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
- Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).