Questions — OCR MEI (4333 questions)

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OCR MEI C1 2013 January Q8
4 marks Easy -1.8
8 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.
OCR MEI C1 2013 January Q9
5 marks Moderate -0.8
9 You are given that \(\mathrm { f } ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).
OCR MEI C1 2013 January Q10
14 marks Standard +0.3
10
  1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
  2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
  3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.
OCR MEI C1 2013 January Q11
12 marks Moderate -0.3
11
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.
OCR MEI C1 2013 January Q12
10 marks Moderate -0.3
12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).
  1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
  2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
  3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C1 2009 June Q1
4 marks Easy -1.2
1 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes.
OCR MEI C1 2009 June Q2
3 marks Easy -1.8
2 Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 2009 June Q3
3 marks Moderate -0.8
3 When \(x ^ { 3 } - k x + 4\) is divided by \(x - 3\), the remainder is 1 . Use the remainder theorem to find the value of \(k\).
OCR MEI C1 2009 June Q4
2 marks Easy -1.2
4 Solve the inequality \(x ( x - 6 ) > 0\).
OCR MEI C1 2009 June Q5
4 marks Easy -1.3
5
  1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).
OCR MEI C1 2009 June Q6
3 marks Moderate -0.8
6 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.
OCR MEI C1 2009 June Q7
3 marks Easy -1.8
7 Find the value of each of the following.
  1. \(5 ^ { 2 } \times 5 ^ { - 2 }\)
  2. \(100 ^ { \frac { 3 } { 2 } }\)
OCR MEI C1 2009 June Q8
5 marks Easy -1.3
8
  1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
  2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).
OCR MEI C1 2009 June Q11
12 marks Moderate -0.3
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1209192c-655e-439d-be50-8747dbbb7672-3_444_846_351_648} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O .
  2. Find the coordinates of the point where this perpendicular meets AB .
  3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
  4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
  5. Find the area of triangle OAB .
OCR MEI C1 2009 June Q12
13 marks Moderate -0.3
12
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
    Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 2009 June Q13
11 marks Moderate -0.3
13 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).
  1. State the coordinates of the centre and the radius of this circle.
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
  3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
  4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.
OCR MEI C1 2010 June Q1
3 marks Easy -1.2
1 Find the equation of the line which is parallel to \(y = 3 x + 1\) and which passes through the point with coordinates \(( 4,5 )\).
OCR MEI C1 2010 June Q2
5 marks Easy -1.8
2
  1. Simplify \(\left( 5 a ^ { 2 } b \right) ^ { 3 } \times 2 b ^ { 4 }\).
  2. Evaluate \(\left( \frac { 1 } { 16 } \right) ^ { - 1 }\).
  3. Evaluate \(( 16 ) ^ { \frac { 3 } { 2 } }\).
OCR MEI C1 2010 June Q4
5 marks Easy -1.2
4 Solve the following inequalities.
  1. \(2 ( 1 - x ) > 6 x + 5\)
  2. \(( 2 x - 1 ) ( x + 4 ) < 0\)
OCR MEI C1 2010 June Q5
5 marks Easy -1.2
5
  1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
  2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).
OCR MEI C1 2010 June Q6
5 marks Moderate -0.3
6 You are given that
  • the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
  • when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .
Find the values of \(k\) and \(m\).
OCR MEI C1 2010 June Q7
4 marks Easy -1.8
7 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.
OCR MEI C1 2010 June Q8
4 marks Easy -1.2
8 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).
OCR MEI C1 2010 June Q9
2 marks Easy -1.8
9 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
10
  1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
  2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
  3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).