Questions — OCR (4619 questions)

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OCR FP1 2008 June Q7
7 marks Moderate -0.8
7 Describe fully the geometrical transformation represented by each of the following matrices:
  1. \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\),
  2. \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\),
  3. \(\left( \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right)\),
  4. \(\left( \begin{array} { r r } 0.8 & 0.6 \\ - 0.6 & 0.8 \end{array} \right)\).
OCR FP1 2008 June Q8
7 marks Standard +0.3
8 The quadratic equation \(x ^ { 2 } + k x + 2 k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
OCR FP1 2008 June Q9
11 marks Standard +0.8
9
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
  2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
  3. Hence solve the quartic equation \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).
OCR FP1 2008 June Q10
11 marks Standard +0.8
10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & 8 & 10 \\ 2 & 1 & 2 \\ 4 & 3 & 6 \end{array} \right)\). The matrix \(\mathbf { B }\) is such that \(\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1 \\ 1 & 1 & 0 \\ 1 & 3 & 0 \end{array} \right)\).
  1. Show that \(\mathbf { A B }\) is non-singular.
  2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
  3. Find \(\mathbf { B } ^ { - 1 }\).
OCR FP1 2013 June Q1
6 marks Moderate -0.5
1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).
OCR FP1 2013 June Q2
7 marks Moderate -0.8
2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 5 & 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 2 & - 5 \end{array} \right)\) and \(\mathbf { C } = \binom { 3 } { 2 }\).
  1. Find \(3 \mathbf { A } - 4 \mathbf { B }\).
  2. Find CB. Determine whether \(\mathbf { C B }\) is singular or non-singular, giving a reason for your answer.
OCR FP1 2013 June Q3
6 marks Standard +0.3
3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
OCR FP1 2013 June Q4
6 marks Standard +0.3
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2 \\ 0 & 1 \end{array} \right) .$$
OCR FP1 2013 June Q5
6 marks Moderate -0.8
5 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right)\), giving your answer in a fully factorised form.
OCR FP1 2013 June Q6
7 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{2ba2e0bf-d20a-41ab-a77c-86a08e700b40-2_885_803_1425_630} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.
  1. Write down, in complex number form, the equations of \(l\) and \(C\).
    [0pt]
  2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]
OCR FP1 2013 June Q7
8 marks Moderate -0.8
7
  1. Find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin.
  2. Find the matrix that represents a reflection in the \(x\)-axis.
  3. Hence find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin, followed by a reflection in the \(x\)-axis.
  4. Describe a single transformation that is represented by your answer to part (iii).
OCR FP1 2013 June Q8
6 marks Standard +0.8
8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).
OCR FP1 2013 June Q9
8 marks Standard +0.3
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
12 marks Standard +0.3
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
5 marks Moderate -0.5
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
8 marks Standard +0.3
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
8 marks Moderate -0.3
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
8 marks Standard +0.3
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
8 marks Standard +0.8
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
10 marks Standard +0.3
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
11 marks Standard +0.8
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
14 marks Standard +0.8
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 C4 Q1
4 marks Moderate -0.3
  1. Express
$$\frac { 2 x } { 2 x ^ { 2 } + 3 x - 5 } \div \frac { x ^ { 3 } } { x ^ { 2 } - x }$$ as a single fraction in its simplest form.
OCR C4 Q2
7 marks Standard +0.3
2. A curve has the equation $$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q3
8 marks Standard +0.3
3. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).
  1. Find the values of \(a\) and \(n\).
  2. Show that \(k = - 160\).