20. Show that the normal to the rectangular hyperbola \(x y = c ^ { 2 }\), at the point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\) has equation
$$y = t ^ { 2 } x + \frac { c } { t } - c t ^ { 3 }$$
[*P5 June 2004 Qn 8]
21. Given that \(z = - 2 \sqrt { } 2 + 2 \sqrt { } 2 \mathrm { i }\) and \(w = 1 - \mathrm { i } \sqrt { } 3\), find
- \(\left| \frac { z } { w } \right|\),
- \(\arg \left( \frac { z } { w } \right)\).
- On an Argand diagram, plot points \(A , B , C\) and \(D\) representing the complex numbers \(z\), \(w , \left( \frac { z } { w } \right)\) and 4, respectively.
- Show that \(\angle A O C = \angle D O B\).
- Find the area of triangle \(A O C\).
22. Given that - 2 is a root of the equation \(z ^ { 3 } + 6 z + 20 = 0\), - find the other two roots of the equation,
- show, on a single Argand diagram, the three points representing the roots of the equation,
- prove that these three points are the vertices of a right-angled triangle.
23.
$$f ( x ) = 1 - e ^ { x } + 3 \sin 2 x$$
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.0 < x < 1.4\).
Starting with the interval (1.0, 1.4), use interval bisection three times to find the value of \(\alpha\) to one decimal place.
24.
$$z = - 4 + 6 i$$ - Calculate arg \(z\), giving your answer in radians to 3 decimal places.
(2)
The complex number \(w\) is given by \(w = \frac { A } { 2 - \mathrm { i } }\), where \(A\) is a positive constant. Given that \(| w | = \sqrt { } 20\), - find \(w\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are constants,
(4) - calculate arg \(\frac { W } { Z }\).
(3)
[0pt]
[FP1/P4 June 2005 Qn 5]
25. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(M\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. - Show that an equation of the tangent to \(M\) at \(P\) is
$$p y = x + a p ^ { 2 }$$
The point \(Q \left( 16 a p ^ { 2 } , 8 a p \right)\) also lies on \(M\).
- Write down an equation of the tangent to \(M\) at \(Q\).
[0pt]
[*FP2/P5 June 2005 Qn 5]
26. (a) Express \(\frac { 6 x + 10 } { x + 3 }\) in the form \(p + \frac { q } { x + 3 }\), where \(p\) and \(q\) are integers to be found.
The sequence of real numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 5.2\) and \(u _ { n + 1 } = \frac { 6 u _ { n } + 10 } { u _ { n } + 3 }\). - Prove by induction that \(u _ { n } > 5\), for \(n \in \mathbb { Z } ^ { + }\).
[0pt]
[FP3/P6 June 2005 Qn 1]
27. Prove that \(\sum _ { r = 1 } ^ { n } ( r - 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n - 1 ) n ( n + 4 )\).
28. Given that \(\frac { z + 2 \mathrm { i } } { z - \lambda \mathrm { i } } = \mathrm { i }\), where \(\lambda\) is a positive, real constant, - show that \(z = \left( \frac { \lambda } { 2 } + 1 \right) + \mathrm { i } \left( \frac { \lambda } { 2 } - 1 \right)\).
Given also that \(\arg z = \arctan \frac { 1 } { 2 }\), calculate
- the value of \(\lambda\),
- the value of \(| z | ^ { 2 }\).
29. The temperature \(\theta ^ { \circ } \mathrm { C }\) of a room \(t\) hours after a heating system has been turned on is given by
$$\theta = t + 26 - 20 \mathrm { e } ^ { - 0.5 t } , \quad t \geq 0 .$$
The heating system switches off when \(\theta = 20\). The time \(t = \alpha\), when the heating system switches off, is the solution of the equation \(\theta - 20 = 0\), where \(\alpha\) lies in the interval [1.8, 2].
[0pt] - Using the end points of the interval [1.8, 2], find, by linear interpolation, an approximation to \(\alpha\). Give your answer to 2 decimal places.
- Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on.
30. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. - Show that an equation for the normal to \(C\) at the point \(P \left( a p ^ { 2 } , 2 a p \right)\) is
$$y + p x = 2 a p + a p ^ { 3 }$$
The normals to \(C\) at the points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right) , p \neq q\), meet at the point \(R\).
- Find, in terms of \(a , p\) and \(q\), the coordinates of \(R\).
31. A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix
$$\mathbf { A } = \left( \begin{array} { r r }
- 4 & 2
2 & - 1
\end{array} \right) , \text { where } k \text { is a constant. }$$
Find the image under \(T\) of the line with equation \(y = 2 x + 1\).
[0pt]
[*FP3/P6 January 2006 Qn 3]
32. Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 \left\{ 1 + ( n - 1 ) 2 ^ { n } \right\}\).
[0pt]
[*FP3/P6 January 2006 Qn 5]
33. The complex numbers \(z\) and \(w\) satisfy the simultaneous equations
$$\begin{aligned}
2 z + \mathrm { i } w & = - 1
z - w & = 3 + 3 \mathrm { i }
\end{aligned}$$ - Use algebra to find \(z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
- Calculate arg \(z\), giving your answer in radians to 2 decimal places.
34.
$$f ( x ) = 0.25 x - 2 + 4 \sin \sqrt { } x$$ - Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 0.24\) and \(x = 0.28\).
[0pt] - Starting with the interval [0.24, 0.28], use interval bisection three times to find an interval of width 0.005 which contains \(\alpha\).
[0pt]
[*FP1 June 2006 Qn 6]
35. (a) Find the roots of the equation
$$z ^ { 2 } + 2 z + 17 = 0$$
giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are integers. - Show these roots on an Argand diagram.
[0pt]
[FP1 January 2007 Qn 1]
36. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$\begin{aligned}
& z _ { 1 } = 5 + 3 i
& z _ { 1 } = 1 + p i
\end{aligned}$$
where \(p\) is an integer. - Find \(\frac { z _ { 2 } } { z _ { 1 } }\), in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are expressed in terms of \(p\).
Given that \(\arg \left( \frac { z _ { 2 } } { z _ { 1 } } \right) = \frac { \pi } { 4 }\),
- find the value of \(p\).
37.
$$f ( x ) = x ^ { 3 } + 8 x - 19$$ - Show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
- Show that the real root of \(\mathrm { f } ( x ) = 0\) lies between 1 and 2 .
- Obtain an approximation to the real root of \(\mathrm { f } ( x ) = 0\) by performing two applications of the Newton-Raphson procedure to \(\mathrm { f } ( x )\), using \(x = 2\) as the first approximation. Give your answer to 3 decimal places.
- By considering the change of sign of \(\mathrm { f } ( x )\) over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places.
38.
$$z = \sqrt { 3 } - i$$
\(z ^ { * }\) is the complex conjugate of \(z\). - Show that \(\frac { z } { z * } = \frac { 1 } { 2 } - \frac { \sqrt { } 3 } { 2 } \mathrm { i }\).
- Find the value of \(\left| \frac { z } { z * } \right|\).
- Verify, for \(z = \sqrt { } 3 - \mathrm { i }\), that \(\arg \frac { z } { z ^ { * } } = \arg z - \arg z ^ { * }\).
- Display \(z , z ^ { * }\) and \(\frac { Z } { Z ^ { * } }\) on a single Argand diagram.
- Find a quadratic equation with roots \(z\) and \(z ^ { * }\) in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are real constants to be found.
39. The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right) , p \neq q\), lie on the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. - Show that an equation for the chord \(P Q\) is \(( p + q ) y = 2 ( x + a p q )\).
The normals to \(C\) at \(P\) and \(Q\) meet at the point \(R\).
- Show that the coordinates of \(R\) are \(\left( a \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - a p q ( p + q ) \right)\).
40. Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n ( 2 n - 1 ) ( 2 n + 1 )\).
[0pt]
[FP3 June 2007 Qn 5]
41. Given that
$$f ( n ) = 3 ^ { 4 n } + 2 ^ { 4 n + 2 } ,$$ - show that, for \(k \in \mathbb { Z } ^ { + } , \mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\) is divisible by 15 ,
- prove that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is divisible by 5 ,
[0pt]
[*FP3 June 2007 Qn 6]
42. Given that \(x = - \frac { 1 } { 2 }\) is the real solution of the equation
$$2 x ^ { 3 } - 11 x ^ { 2 } + 14 x + 10 = 0$$
find the two complex solutions of this equation.
43.
$$f ( x ) = 3 x ^ { 2 } + x - \tan \left( \frac { x } { 2 } \right) - 2 , \quad - \pi < x < \pi$$
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 0.7,0.8 ]\).
Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to \(\alpha\). Give your answer to 3 decimal places.
44.
$$z = - 2 + \mathrm { i }$$ - Express in the form \(a + \mathrm { i } b\)
- \(\frac { 1 } { z }\)
- \(z ^ { 2 }\).
- Show that \(\left| z ^ { 2 } - z \right| = 5 \sqrt { } 2\).
- Find arg \(\left( z ^ { 2 } - z \right)\).
- Display \(z\) and \(z ^ { 2 } - z\) on a single Argand diagram.
45. (a) Write down the value of the real root of the equation
$$x ^ { 3 } - 64 = 0 .$$ - Find the complex roots of \(x ^ { 3 } - 64 = 0\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
- Show the three roots of \(x ^ { 3 } - 64 = 0\) on an Argand diagram.
46. The complex number \(z\) is defined by
$$z = \frac { a + 2 \mathrm { i } } { a - \mathrm { i } } , \quad a \in \mathbb { R } , \quad a > 0$$
Given that the real part of \(z\) is \(\frac { 1 } { 2 }\), find - the value of \(a\),
- the argument of \(z\), giving your answer in radians to 2 decimal places.
47.
$$\mathbf { A } = \left( \begin{array} { c r }
k & - 2
1 - k & k
\end{array} \right) , \text { where } k \text { is constant. }$$
A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\). - Find the value of \(k\) for which the line \(y = 2 x\) is mapped onto itself under \(T\).
- Show that \(\mathbf { A }\) is non-singular for all values of \(k\).
- Find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
A point \(P\) is mapped onto a point \(Q\) under \(T\).
The point \(Q\) has position vector \(\binom { 4 } { - 3 }\) relative to an origin \(O\).
Given that \(k = 3\),
- find the position vector of \(P\).