Questions FP1 - A-Level Maths

Edexcel FP1 Q6

Given that $z = 3 + 4 \mathrm { i }$ and $w = - 1 + 7 \mathrm { i }$,
1. find $| w |$.
The complex numbers $z$ and $w$ are represented by the points $A$ and $B$ on an Argand diagram.
Show points $A$ and $B$ on an Argand diagram.
Prove that $\triangle O A B$ is an isosceles right-angled triangle.
Find the exact value of $\arg \left( \frac { z } { w } \right)$.

Edexcel FP1 Q7

7. The point $P \left( 2 p , \frac { 2 } { p } \right)$ and the point $Q \left( 2 q , \frac { 2 } { q } \right)$, where $p \neq - q$, lie on the rectangular hyperbola with equation $x y = 4$. The tangents to the curve at the points $P$ and $Q$ meet at the point $R$.
Show that at the point $R$, $$x = \frac { 4 p q } { p + q } \text { and } y = \frac { 4 } { p + q }$$ [*P5 June 2002 Qn 7]

Edexcel FP1 Q8

8. For $n \in \mathbb { Z } ^ { + }$prove that

$2 ^ { 3 n + 2 } + 5 ^ { n + 1 }$ is divisible by 3 ,
$\left( \begin{array} { r r } - 2 & - 1
9 & 4 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 - 3 n & - n
9 n & 3 n + 1 \end{array} \right)$.

Edexcel FP1 Q9

9. $$f ( x ) = 2 \sin 2 x + x - 2 .$$ The root $\alpha$ of the equation $\mathrm { f } ( x ) = 0$ lies in the interval $[ 2 , \pi ]$.
Using the end points of this interval find, by linear interpolation, an approximation to $\alpha$.
[0pt] [*P4 January 2003 Qn 4]

Edexcel FP1 Q10

10. Given that $z = 3 - 3 \mathrm { i }$ express, in the form $a + \mathrm { i } b$, where $a$ and $b$ are real numbers,

$z ^ { 2 }$,
$\frac { 1 } { z }$.
Find the exact value of each of $| z | , \left| z ^ { 2 } \right|$ and $\left| \frac { 1 } { z } \right|$. The complex numbers $z , z ^ { 2 }$ and $\frac { 1 } { z }$ are represented by the points $A , B$ and $C$ respectively on an Argand diagram. The real number 1 is represented by the point $D$, and $O$ is the origin.
Show the points $A$, $B$, $C$ and $D$ on an Argand diagram.
Prove that $\triangle O A B$ is similar to $\triangle O C D$.

Edexcel FP1 Q11

11. (a) Using that 3 is the real root of the cubic equation $x ^ { 3 } - 27 = 0$, show that the complex roots of the cubic satisfy the quadratic equation $x ^ { 2 } + 3 x + 9 = 0$.
(b) Hence, or otherwise, find the three cube roots of 27, giving your answers in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$.
(c) Show these roots on an Argand diagram.

Edexcel FP1 Q12

12. $$f ( x ) = 3 ^ { x } - x - 6$$

Show that $\mathrm { f } ( x ) = 0$ has a root $\alpha$ between $x = 1$ and $x = 2$.
Starting with the interval (1,2), use interval bisection three times to find an interval of width 0.125 which contains $\alpha$.
[0pt] [*P4 June 2003 Qn 4]

Edexcel FP1 Q13

13. $$z = \frac { a + 3 \mathrm { i } } { 2 + a \mathrm { i } } , \quad a \in \mathbb { R }$$

Given that $a = 4$, find $| z |$.
Show that there is only one value of $a$ for which $\arg z = \frac { \pi } { 4 }$, and find this value.

Edexcel FP1 Q14

14. $$f ( n ) = ( 2 n + 1 ) 7 ^ { n } - 1$$ Prove by induction that, for all positive integers $n , f ( n )$ is divisible by 4 .

Edexcel FP1 Q15

15. Given that $z = 2 - 2 \mathrm { i }$ and $w = - \sqrt { 3 } + \mathrm { i }$,

find the modulus and argument of $w z ^ { 2 }$.
(6)
Show on an Argand diagram the points $A , B$ and $C$ which represent $z , w$ and $w z ^ { 2 }$ respectively, and determine the size of angle $B O C$.

Edexcel FP1 Q16

16. (a) Show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { 1 } { 6 } n ( n + 7 ) ( 2 n + 7 ) .$$ (b) Hence calculate the value of $\quad \sum _ { r = 10 } ^ { 40 } ( r + 1 ) ( r + 5 )$.
[0pt] [P4 June 2004 Qn 1]

Edexcel FP1 Q17

17. $$f ( x ) = 2 ^ { x } + x - 4$$ The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1,2].
Use linear interpolation on the values at the end points of this interval to find an approximation to $\alpha$.
[0pt] [*P4 June 2004 Qn 2]

Edexcel FP1 Q18

18. The complex number $z = a + \mathrm { i } b$, where $a$ and $b$ are real numbers, satisfies the equation $$z ^ { 2 } + 16 - 30 i = 0$$

Show that $a b = 15$.
Write down a second equation in $a$ and $b$ and hence find the roots of $$z ^ { 2 } + 16 - 30 i = 0$$

Edexcel FP1 Q19

Given that $z = 1 + \sqrt { } 3 \mathrm { i }$ and that $\frac { w } { z } = 2 + 2 \mathrm { i }$, find
1. $w$ in the form $a + \mathrm { i } b$, where $a , b \in \mathbb { R }$,
2. the argument of $w$,
3. the exact value for the modulus of $w$.
On an Argand diagram, the point $A$ represents $z$ and the point $B$ represents $w$.
Draw the Argand diagram, showing the points $A$ and $B$.
Find the distance $A B$, giving your answer as a simplified surd.

Edexcel FP1 Q20

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 $a + \mathrm { i } b$
1. $\frac { 1 } { z }$
2. $z ^ { 2 }$.
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 $P$.

AQA FP1 2005 January Q1

1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots $\alpha$ and $\beta$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Find the value of $\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }$.
Find a quadratic equation which has roots $$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$

AQA FP1 2005 January Q2

2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
Calculate the $y$-coordinates of the points of intersection of the curve with the line $x = 1$. Give your answers in the form $p \sqrt { 2 }$, where $p$ is a rational number.
The curve is translated one unit in the positive $x$ direction. Write down the equation of the curve after the translation.

AQA FP1 2005 January Q3

3 It is given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real numbers.

Write down, in terms of $x$ and $y$, an expression for $z ^ { * }$, the complex conjugate of $z$.
Find, in terms of $x$ and $y$, the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
Find the complex number $z$ such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$

AQA FP1 2005 January Q4

4 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

$\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x$;
(3 marks)
$\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x$;
$\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x$.

AQA FP1 2005 January Q5

5

The transformation $T _ { 1 }$ is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
The transformation $T _ { 2 }$ is an anticlockwise rotation about the origin through an angle of $60 ^ { \circ }$. Find the matrix of the transformation $T _ { 2 }$. Use surds in your answer where appropriate.
(3 marks)
Find the matrix of the transformation obtained by carrying out $T _ { 1 }$ followed by $T _ { 2 }$.
(3 marks)

AQA FP1 2005 January Q6

6 The angle $x$ radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$

Find the general solution of this equation, giving the roots as exact values in terms of $\pi$.
Find the number of roots of the equation which lie between 0 and $2 \pi$.

AQA FP1 2005 January Q7

7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables $x$ and $y$ are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where $a$ and $b$ are constants. Experimental evidence has provided the following approximate values:

$x$	1.5	4.0	5.0	6.5	8.0
$y$	5.0	6.3	7.0	8.0	9.0

On Figure 1, draw a linear graph connecting the variables $X$ and $Y$, where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
From your graph, find approximate values for the constants $a$ and $b$.

AQA FP1 2005 January Q8

8 [Figure 2, printed on the insert, is provided for use in this question.]
The diagram shows a part of the graph of $y = \mathrm { f } ( x )$, where $$f ( x ) = x ^ { 3 } - 2 x - 1$$ The point $P$ has coordinates $( 1 , - 2 )$.
\includegraphics[max width=\textwidth, alt={}, center]{a77cc9c3-5ff6-4abc-931e-e811740267f2-05_606_565_717_740}

Taking $x _ { 1 } = 1$ as a first approximation to a root of the equation $\mathrm { f } ( x ) = 0$, use the NewtonRaphson method to find a second approximation, $x _ { 2 }$, to the root.
On Figure 2, draw a straight line to illustrate the Newton-Raphson method as used in part (a). Mark $x _ { 1 }$ and $x _ { 2 }$ on Figure 2
By considering $f ( 2 )$, show that the second approximation found in part (a) is not as good as the first approximation.
Taking $x _ { 1 } = 1.6$ as a first approximation to the root, use the Newton-Raphson method to find a second approximation to the root. Give your answer to three decimal places.
(2 marks)

AQA FP1 2005 January Q9

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$

Write down the equations of the two asymptotes to the curve $y = \mathrm { f } ( x )$.
By considering the expression $x ^ { 2 } + 2 x + 2$ :
1. show that the graph of $y = \mathrm { f } ( x )$ does not intersect the $x$-axis;
2. find the non-real roots of the equation $\mathrm { f } ( x ) = 0$.
1. Show that, if the equation $\mathrm { f } ( x ) = k$ has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
2. Deduce that the graph of $y = \mathrm { f } ( x )$ has exactly one stationary point and find its coordinates.

AQA FP1 2008 January Q1

1 It is given that $z _ { 1 } = 2 + \mathrm { i }$ and that $z _ { 1 } { } ^ { * }$ is the complex conjugate of $z _ { 1 }$.
Find the real numbers $x$ and $y$ such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

Questions FP1 (1385 questions)

Browse by module

Browse by board