Questions FP1 (1385 questions)

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OCR FP1 2013 June Q9
9
  1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\).
  2. Hence show that \(\sum _ { r = 1 } ^ { 2 n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { n } { 2 ( 3 n + 1 ) }\).
OCR FP1 2013 June Q10
10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 1
1 & 3 & 2
4 & 1 & 1 \end{array} \right)\).
  1. Find the value of \(a\) for which \(\mathbf { A }\) is singular.
  2. Given that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the equations $$\begin{aligned} a x + 2 y + z & = 1
    x + 3 y + 2 z & = 2
    4 x + y + z & = 3 \end{aligned}$$
OCR FP1 Specimen Q1
1 Use formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$
OCR FP1 Specimen Q2
2 The cubic equation \(x ^ { 3 } - 6 x ^ { 2 } + k x + 10 = 0\) has roots \(p - q , p\) and \(p + q\), where \(q\) is positive.
  1. By considering the sum of the roots, find \(p\).
  2. Hence, by considering the product of the roots, find \(q\).
  3. Find the value of \(k\).
OCR FP1 Specimen Q3
3 The complex number \(2 + \mathrm { i }\) is denoted by \(z\), and the complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Express \(z ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
  2. Show that \(4 z - z ^ { 2 }\) simplifies to a real number, and verify that this real number is equal to \(z z ^ { * }\).
  3. Express \(\frac { z + 1 } { z - 1 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
OCR FP1 Specimen Q4
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 3 ^ { 2 n } - 1$$
  1. Write down the value of \(u _ { 1 }\).
  2. Show that \(u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }\).
  3. Hence prove by induction that each term of the sequence is a multiple of 8 .
OCR FP1 Specimen Q5
5
  1. Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
  2. Hence find an expression in terms of \(n\) for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
  3. State the value of
    (a) \(\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\),
    (b) \(\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\).
OCR FP1 Specimen Q6
6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).
  1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
  2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
  3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
    (a) the real part of \(z\) is zero,
    (b) \(\quad \arg z = \arg a\).
OCR FP1 Specimen Q7
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & - 2
2 & 1 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { A }\).
  2. The value of \(\operatorname { det } \mathbf { A }\) is 5 . Show clearly how this value relates to your diagram in part (i). A represents a sequence of two elementary geometrical transformations, one of which is a rotation \(R\).
  3. Determine the angle of \(R\), and describe the other transformation fully.
  4. State the matrix that represents \(R\), giving the elements in an exact form.
OCR FP1 Specimen Q8
8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1
2 & 3 & - 1
2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  1. Show that the determinant of \(\mathbf { M }\) is \(2 a\).
  2. Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
  3. Hence or otherwise solve the simultaneous equations $$\begin{array} { r } x + 2 y - z = 1
    2 x + 3 y - z = 2
    2 x - y + z = 0 \end{array}$$
  4. Find the value of \(k\) for which the simultaneous equations $$\begin{array} { r } 2 y - z = k
    2 x + 3 y - z = 2
    2 x - y + z = 0 \end{array}$$ have solutions.
  5. Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.
OCR MEI FP1 2005 January Q1
1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3
- 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?
OCR MEI FP1 2005 January Q2
2
  1. Show that \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
  2. Hence use the method of differences to find the sum of the series $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
OCR MEI FP1 2005 January Q3
3
  1. Solve the equation \(\frac { 1 } { x + 2 } = 3 x + 4\).
  2. Solve the inequality \(\frac { 1 } { x + 2 } \leqslant 3 x + 4\).
OCR MEI FP1 2005 January Q4
4 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 )\), giving your answer in a factorised form.
OCR MEI FP1 2005 January Q5
5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.
OCR MEI FP1 2005 January Q6
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\).
OCR MEI FP1 2005 January Q7
7 A curve has equation \(y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }\).
  1. Write down the values of \(x\) for which \(y = 0\).
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or from below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
  5. Solve the inequality \(\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2\).
OCR MEI FP1 2005 January Q8
8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).
  1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
  2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
  3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
  4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).
OCR MEI FP1 2005 January Q9
9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6
0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
  3. By investigating the invariant points of the reflection, find the equation of the mirror line.
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6
    0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
OCR MEI FP1 2006 January Q1
1 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3
1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & - 3
1 & 4 \end{array} \right) , \mathbf { C } = \left( \begin{array} { r r } 1 & - 1
0 & 2
0 & 1 \end{array} \right)\).
  1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
  2. Show that matrix multiplication is not commutative.
OCR MEI FP1 2006 January Q2
2
  1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
  2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).
OCR MEI FP1 2006 January Q3
3 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 )\), expressing your answer in a fully factorised form.
OCR MEI FP1 2006 January Q4
4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2
- 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).
  1. Write down the two equations.
  2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2
    - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?
OCR MEI FP1 2006 January Q5
5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
  2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.
OCR MEI FP1 2006 January Q6
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).