Questions — OCR MEI (4333 questions)

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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 MEI C4 Q1
8 marks Standard +0.3
1
  1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
    the expansion is valid.
  2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).
OCR MEI C4 Q2
5 marks Moderate -0.3
2 Find the first three terms in the binomial expansion of \(( 4 + x ) ^ { \frac { 3 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q3
8 marks Standard +0.3
3
  1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
  2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
    constants to be determined. constants to be determined. State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q4
5 marks Moderate -0.5
4 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q5
5 marks Moderate -0.5
5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q6
8 marks Standard +0.3
6
  1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).
OCR MEI C4 Q7
6 marks Moderate -0.3
7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q2
6 marks Standard +0.3
2 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]
OCR MEI C4 Q3
7 marks Standard +0.3
3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]
OCR MEI C4 Q4
7 marks Standard +0.3
4 Find the first three terms in the binomial expansion of \(\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]
OCR MEI C4 Q5
6 marks Moderate -0.3
5 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q6
8 marks Standard +0.3
6
  1. Find the first three terms in the binomial expansion of \(\frac { 1 } { \sqrt { 1 - 2 x } }\). State the set of values of \(x\) for which the expansion is valid.
  2. Hence find the first three terms in the series expansion of \(\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }\).
OCR MEI C4 Q7
7 marks Standard +0.3
7
  1. Find the first three non-zero terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }\) for \(| x | < 2\). [4]
  2. Use this result to find an approximation for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
    4 significant figures.
  3. Given that \(\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.
OCR MEI C4 Q1
5 marks Moderate -0.3
1 Solve the equation \(\frac { 5 x } { 2 x + 1 } - \frac { 3 } { x + 1 } = 1\).
OCR MEI C4 Q2
5 marks Standard +0.3
2 Express \(\frac { 3 x } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) } \quad\) in partial fractions.