Questions C1 (1442 questions)

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OCR MEI C1 2007 January Q12
12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks.
\(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.
  1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
  2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_451_483_776_790} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Not to scale
OCR MEI C1 2007 January Q13
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_456_387_1539_833} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).
  1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
  2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
  3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).
OCR MEI C1 2010 January Q1
1 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
OCR MEI C1 2010 January Q2
2 Solve the inequality \(\frac { 5 x - 3 } { 2 } < x + 5\).
OCR MEI C1 2010 January Q3
3
  1. Find the coordinates of the point where the line \(5 x + 2 y = 20\) intersects the \(x\)-axis.
  2. Find the coordinates of the point of intersection of the lines \(5 x + 2 y = 20\) and \(y = 5 - x\).
OCR MEI C1 2010 January Q4
4
  1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
  2. Sketch the graph of \(y = x ^ { 2 } - 4\).
OCR MEI C1 2010 January Q5
5
  1. Find the value of \(144 ^ { - \frac { 1 } { 2 } }\).
  2. Simplify \(\frac { 1 } { 5 + \sqrt { 7 } } + \frac { 4 } { 5 - \sqrt { 7 } }\). Give your answer in the form \(\frac { a + b \sqrt { 7 } } { c }\).
OCR MEI C1 2010 January Q6
6 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).
OCR MEI C1 2010 January Q7
7 When \(x ^ { 3 } + 2 x ^ { 2 } + 5 x + k\) is divided by ( \(x + 3\) ), the remainder is 6 . Find the value of \(k\).
OCR MEI C1 2010 January Q8
8 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 2010 January Q9
9 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).
OCR MEI C1 2010 January Q10
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-3_590_780_347_680} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).
  1. Verify that the lines AB and DC are parallel.
  2. Prove that the trapezium is not isosceles.
  3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
  4. Show that neither diagonal of the trapezium bisects the other.
OCR MEI C1 2010 January Q11
11 A circle has equation \(( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  1. State the coordinates of the centre of this circle and its radius.
  2. Verify that the point A with coordinates \(( 6 , - 6 )\) lies on this circle. Show also that the point B on the circle for which AB is a diameter has coordinates \(( 0,2 )\).
  3. Find the equation of the tangent to the circle at A .
  4. A second circle touches the original circle at A . Its radius is 10 and its centre is at C , where BAC is a straight line. Find the coordinates of C and hence write down the equation of this second circle.
OCR MEI C1 2010 January Q12
5 marks
12 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_524_885_406_628} \captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{figure}
  1. State the value of \(x\) at A , where the arch meets the road.
  2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
  3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_529_871_1489_678} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
    \end{figure}
  4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.
    [0pt] [5]
OCR MEI C1 2011 January Q1
1 Find the equation of the line which is parallel to \(y = 5 x - 4\) and which passes through the point (2, 13). Give your answer in the form \(y = a x + b\).
OCR MEI C1 2011 January Q2
2
  1. Write down the value of each of the following.
    (A) \(4 ^ { - 2 }\)
    (B) \(9 ^ { 0 }\)
  2. Find the value of \(\left( \frac { 64 } { 125 } \right) ^ { \frac { 4 } { 3 } }\).
OCR MEI C1 2011 January Q3
3 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).
OCR MEI C1 2011 January Q4
4 Solve the inequality \(5 - 2 x < 0\).
OCR MEI C1 2011 January Q5
5 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.
OCR MEI C1 2011 January Q6
6 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.
OCR MEI C1 2011 January Q7
7
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2011 January Q8
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
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
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
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 .