Questions — OCR MEI C1 (499 questions)

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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 2014 June Q1
5 marks Easy -1.3
1
  1. Evaluate \(\left( \frac { 1 } { 27 } \right) ^ { \frac { 2 } { 3 } }\).
  2. Simplify \(\frac { \left( 4 a ^ { 2 } c \right) ^ { 3 } } { 32 a ^ { 4 } c ^ { 7 } }\).
OCR MEI C1 2014 June Q2
3 marks Easy -1.2
2 A is the point \(( 1,5 )\) and B is the point \(( 6 , - 1 )\). M is the midpoint of AB . Determine whether the line with equation \(y = 2 x - 5\) passes through M.
OCR MEI C1 2014 June Q3
4 marks Moderate -0.8
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-2_798_819_836_623} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows the graph of \(y = \mathrm { f } ( x )\). Draw the graphs of the following.
  1. \(y = \mathrm { f } ( x ) - 2\)
  2. \(y = \mathrm { f } ( x - 3 )\)
OCR MEI C1 2014 June Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 2014 June Q5
4 marks Easy -1.8
5 Make \(a\) the subject of \(3 ( a + 4 ) = a c + 5 f\).
OCR MEI C1 2014 June Q6
3 marks Moderate -0.8
6 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).
OCR MEI C1 2014 June Q7
4 marks Moderate -0.8
7 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 7 }\).
OCR MEI C1 2014 June Q8
4 marks Moderate -0.8
8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).
OCR MEI C1 2014 June Q9
4 marks Moderate -0.8
9 You are given that \(n , n + 1\) and \(n + 2\) are three consecutive integers.
  1. Expand and simplify \(n ^ { 2 } + ( n + 1 ) ^ { 2 } + ( n + 2 ) ^ { 2 }\).
  2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. Section B (36 marks)
OCR MEI C1 2014 June Q10
11 marks Moderate -0.8
10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-3_680_800_1146_628} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Write down the coordinates of B .
  2. Find the radius of the circle and hence write down the equation of the circle.
  3. AD is a diameter of the circle. Find the coordinates of D .
  4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
  5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
  6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
  7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
  8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
  9. Use the intersections with both axes to express the equation of the curve in a factorised form.
  10. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
  11. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
  12. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}
OCR MEI C1 2016 June Q1
5 marks Easy -1.8
1 Find the value of each of the following.
  1. \(3 ^ { 0 }\)
  2. \(9 ^ { \frac { 3 } { 2 } }\)
  3. \(\left( \frac { 4 } { 5 } \right) ^ { - 2 }\)
OCR MEI C1 2016 June Q2
4 marks Easy -1.8
2 Find the coordinates of the point of intersection of the lines \(2 x + 3 y = 12\) and \(y = 7 - 3 x\).
OCR MEI C1 2016 June Q3
4 marks Easy -1.8
3
  1. Solve the inequality \(\frac { 1 - 2 x } { 4 } > 3\).
  2. Simplify \(\left( 5 c ^ { 2 } d \right) ^ { 3 } \times \frac { 2 c ^ { 4 } } { d ^ { 5 } }\).
OCR MEI C1 2016 June Q4
4 marks Moderate -0.5
4 You are given that \(a = \frac { 3 c + 2 a } { 2 c - 5 }\). Express \(a\) in terms of \(c\).
OCR MEI C1 2016 June Q5
5 marks Easy -1.2
5
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { 5 + 2 \sqrt { 3 } } { 4 - \sqrt { 3 } }\) in the form \(c + d \sqrt { 3 }\), where \(c\) and \(d\) are integers.
OCR MEI C1 2016 June Q6
4 marks Easy -1.2
6 Find the binomial expansion of \(( 1 - 5 x ) ^ { 4 }\), expressing the terms as simply as possible.
OCR MEI C1 2016 June Q7
5 marks Easy -1.3
7
  1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.
OCR MEI C1 2016 June Q8
5 marks Moderate -0.8
8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).
OCR MEI C1 2016 June Q9
13 marks Moderate -0.3
9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-3_1255_1470_434_299} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
  2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
  3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
    (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
    (B) Find the equation of the translated curve, simplifying your answer.
OCR MEI C1 2016 June Q10
11 marks Moderate -0.3
10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-4_579_748_301_657} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Prove that angle ABC is \(90 ^ { \circ }\).
  2. Find the equation of the circle which has AC as a diameter.
  3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2016 June Q11
12 marks Standard +0.3
11
  1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
  2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
  3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}