Questions — OCR MEI (4333 questions)

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OCR MEI C1 2011 January Q8
3 marks Easy -1.8
8 Find the coordinates of the point of intersection of the lines \(x + 2 y = 5\) and \(y = 5 x - 1\).
OCR MEI C1 2011 January Q9
5 marks Moderate -0.8
9 Fig. 9 shows a trapezium ABCD , with the lengths in centimetres of three of its sides. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-3_464_878_347_632} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} This trapezium has area \(140 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(x ^ { 2 } + 2 x - 35 = 0\).
  2. Hence find the length of side AB of the trapezium.
OCR MEI C1 2011 January Q10
3 marks Easy -1.2
10 Select the best statement from $$\begin{aligned} & \mathbf { P } \Rightarrow \mathbf { Q } \\ & \mathbf { P } \Leftarrow \mathbf { Q } \\ & \mathbf { P } \Leftrightarrow \mathbf { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.
  1. P : WXYZ is a quadrilateral with 4 equal sides \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
  2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
  3. P: \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer
OCR MEI C1 2011 January Q11
13 marks Standard +0.3
11 The points \(A ( - 1,6 ) , B ( 1,0 )\) and \(C ( 13,4 )\) are joined by straight lines.
  1. Prove that the lines AB and BC are perpendicular.
  2. Find the area of triangle ABC .
  3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
  4. Find the coordinates of the point on this circle that is furthest from B .
OCR MEI C1 2011 January Q12
13 marks Moderate -0.8
12
  1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 2011 January Q13
10 marks Standard +0.8
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{754d34e4-2f47-48b7-9fbb-6caa7ac21eb7-4_686_878_936_632} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).
  1. Find the exact coordinates of the points of intersection of this curve with the axes.
  2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
  3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).
OCR MEI C1 2012 January Q1
3 marks Easy -1.2
1 Find the equation of the line which is perpendicular to the line \(y = 5 x + 2\) and which passes through the point \(( 1,6 )\). Give your answer in the form \(y = a x + b\).
OCR MEI C1 2012 January Q2
5 marks Easy -1.3
2
  1. Evaluate \(9 ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { \left( 4 x ^ { 4 } \right) ^ { 3 } y ^ { 2 } } { 2 x ^ { 2 } y ^ { 5 } }\).
OCR MEI C1 2012 January Q3
3 marks Easy -1.2
3 Expand and simplify \(( n + 2 ) ^ { 3 } - n ^ { 3 }\).
OCR MEI C1 2012 January Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 2012 January Q5
4 marks Easy -1.8
5 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$
OCR MEI C1 2012 January Q6
3 marks Moderate -0.5
6 Rearrange the following equation to make \(h\) the subject. $$4 h + 5 = 9 a - h a ^ { 2 }$$
OCR MEI C1 2012 January Q7
4 marks Moderate -0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f138ed97-09ca-488e-8651-1217ac2d7b21-2_684_734_1537_662} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the graph of \(y = \mathrm { g } ( x )\). Draw the graphs of the following.
  1. \(y = \mathrm { g } ( x ) + 3\)
  2. \(y = \mathrm { g } ( x + 2 )\)
OCR MEI C1 2012 January Q8
5 marks Moderate -0.8
8 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).
OCR MEI C1 2012 January Q9
4 marks Moderate -0.8
9 Complete each of the following by putting the best connecting symbol ( \(\Leftrightarrow , \Leftarrow\) or ⇒ ) in the box. Explain your choice, giving full reasons.
  1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
  2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\) Section B (36 marks)
OCR MEI C1 2012 January Q10
11 marks Moderate -0.3
10 Point A has coordinates (4, 7) and point B has coordinates (2, 1).
  1. Find the equation of the line through A and B .
  2. Point C has coordinates ( \(- 1,2\) ). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
  3. Find the coordinates of D , the midpoint of AC . Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.
OCR MEI C1 2012 January Q11
13 marks Moderate -0.3
11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).
  1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
  2. Sketch the curve \(y = \mathrm { f } ( x )\).
  3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).
OCR MEI C1 2012 January Q12
12 marks Moderate -0.3
12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).
  1. Write down the radius of the circle and the coordinates of its centre.
  2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
  3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
  4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C1 2013 January Q1
4 marks Easy -1.8
1 Find the value of each of the following.
  1. \(\left( \frac { 5 } { 3 } \right) ^ { - 2 }\)
  2. \(81 ^ { \frac { 3 } { 4 } }\)
OCR MEI C1 2013 January Q2
3 marks Easy -1.2
2 Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).
OCR MEI C1 2013 January Q3
3 marks Easy -1.8
3 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).
OCR MEI C1 2013 January Q4
4 marks Moderate -0.8
4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).
OCR MEI C1 2013 January Q5
4 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).
OCR MEI C1 2013 January Q6
4 marks Moderate -0.3
6 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.
OCR MEI C1 2013 January Q7
5 marks Moderate -0.8
7
  1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
  2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.