Questions FP1 (1491 questions)

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OCR MEI FP1 2005 June Q3
3 marks Moderate -0.3
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).
OCR MEI FP1 2005 June Q4
5 marks Standard +0.3
4 The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(2 \alpha\) and \(2 \beta\).
OCR MEI FP1 2005 June Q5
5 marks Standard +0.3
5
  1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
  2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
  3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$
OCR MEI FP1 2005 June Q6
7 marks Moderate -0.5
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
OCR MEI FP1 2005 June Q7
6 marks Moderate -0.5
7 Find \(\sum _ { r = 1 } ^ { n } 3 r ( r - 1 )\), expressing your answer in a fully factorised form.
OCR MEI FP1 2005 June Q8
14 marks Challenging +1.2
8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } }\).
  1. Find the equations of the asymptotes.
  2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
  3. Sketch the curve.
  4. Solve the inequality \(\frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } } \geqslant - 1\).
OCR MEI FP1 2005 June Q9
10 marks Standard +0.3
9 The quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers, has roots \(2 + \mathrm { j }\) and - 2 j .
  1. Write down the other roots of the equation.
  2. Find the values of \(A , B , C\) and \(D\).
OCR MEI FP1 2005 June Q10
12 marks Standard +0.3
10
  1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
  2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$
OCR MEI FP1 2008 June Q1
4 marks Easy -1.2
1
  1. Write down the matrix for reflection in the \(y\)-axis.
  2. Write down the matrix for enlargement, scale factor 3, centred on the origin.
  3. Find the matrix for reflection in the \(y\)-axis, followed by enlargement, scale factor 3 , centred on the origin.
OCR MEI FP1 2008 June Q2
7 marks Standard +0.3
2 Indicate on a single Argand diagram
  1. the set of points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\),
  2. the set of points for which \(\arg ( z - 2 \mathrm { j } ) = \pi\),
  3. the two points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\) and \(\arg ( z - 2 \mathrm { j } ) = \pi\).
OCR MEI FP1 2008 June Q3
3 marks Moderate -0.3
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1 \\ 2 & 2 \end{array} \right)\).
OCR MEI FP1 2008 June Q4
5 marks Moderate -0.8
4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).
OCR MEI FP1 2008 June Q5
5 marks Moderate -0.5
5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4 \\ 3 & 2 & 5 \\ 4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2 \\ 14 & - 14 & 7 \\ - 5 & 7 & - 4 \end{array} \right)\).
  1. Calculate AB.
  2. Write down \(\mathbf { A } ^ { - 1 }\).
OCR MEI FP1 2008 June Q6
5 marks Moderate -0.3
6 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 3 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(2 \alpha , 2 \beta\) and \(2 \gamma\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2008 June Q7
7 marks Standard +0.3
7
  1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) for all integers \(r\).
  2. Hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\). Section B (36 marks)
OCR MEI FP1 2008 June Q8
12 marks Standard +0.3
8 A curve has equation \(y = \frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) }\).
  1. Write down the equations of the three asymptotes.
  2. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0\).
OCR MEI FP1 2008 June Q9
11 marks Moderate -0.3
9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\). \(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.
  1. Write down the other two roots.
  2. Represent these four roots on an Argand diagram.
  3. Find the values of \(A , B , C\) and \(D\).
OCR MEI FP1 2008 June Q10
13 marks Standard +0.3
10
  1. Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$
  2. Prove the same result by mathematical induction.
OCR FP1 2009 January Q1
4 marks Easy -1.2
1 Express \(\frac { 2 + 3 \mathrm { i } } { 5 - \mathrm { i } }\) in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answer.
OCR FP1 2009 January Q2
4 marks Moderate -0.8
2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ a & 5 \end{array} \right)\). Find
  1. \(\mathbf { A } ^ { - 1 }\),
  2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right)\).
OCR FP1 2009 January Q3
6 marks Moderate -0.8
3 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)\), expressing your answer in a fully factorised form.
OCR FP1 2009 January Q4
4 marks Standard +0.3
4 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are \(2 \times 2\) non-singular matrices and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, simplify $$\mathbf { B } ( \mathbf { A B } ) ^ { - 1 } \mathbf { A } - \mathbf { I } .$$
OCR FP1 2009 January Q5
5 marks Standard +0.3
5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations $$\begin{aligned} 2 x - y + z & = 7 \\ 3 y + z & = 4 \\ x + k y + k z & = 5 \end{aligned}$$ do not have a unique solution for \(x , y\) and \(z\).
OCR FP1 2009 January Q6
9 marks Moderate -0.8
6
  1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
  2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
  3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
  4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).
OCR FP1 2009 January Q7
7 marks Standard +0.3
7 It is given that \(u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }\), where \(n\) is a positive integer.
  1. Show that \(u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }\).
  2. Prove by induction that \(u _ { n }\) is a multiple of 7 .