Questions — OCR MEI (4455 questions)

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OCR MEI C1 Q17
5 marks Easy -1.3
17
  1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { \sqrt { 3 } } { 6 \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.
OCR MEI C1 Q2
11 marks Easy -1.2
2 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]{55e2d4f5-c84d-4577-988e-96071a220d60-2_689_811_430_662} \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 \(y = a x + b\).
OCR MEI C1 Q3
12 marks Standard +0.3
3 The circle \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C .
  2. Find the coordinates of the intersections of the circle with the \(x\) - and \(y\)-axes.
  3. Show that the points \(\mathrm { A } ( 1,6 )\) and \(\mathrm { B } ( 7,4 )\) lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.
OCR MEI C1 Q4
3 marks Easy -1.8
4 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 Q5
12 marks Moderate -0.3
5 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.
OCR MEI C1 Q3
11 marks Moderate -0.3
3 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]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_520_873_478_675} \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]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717} \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.
OCR MEI C1 Q4
11 marks Moderate -0.3
4 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 Q2
13 marks Moderate -0.8
2 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).
  1. Show that the centre of this circle is \(C ( 4,2 )\) and find the radius of the circle.
  2. Show that the origin lies inside the circle.
  3. Show that AB is a diameter of the circle, where A has coordinates ( 2,7 ) and B has coordinates \(( 6 , - 3 )\).
  4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).
OCR MEI C1 Q3
12 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50cfc73d-850e-4a9b-b088-cc9741b66ffb-2_445_617_1008_741} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Not to scale A circle has centre \(\mathrm { C } ( 1,3 )\) and passes through the point \(\mathrm { A } ( 3,7 )\) as shown in Fig. 11.
  1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
  2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
  3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.
OCR MEI C1 Q1
11 marks Moderate -0.5
1 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 Q2
2 marks Easy -1.2
2 A line has gradient 3 and passes through the point \(( 1 , - 5 )\). The point \(( 5 , k )\) is on this line. Find the value of \(k\).
OCR MEI C1 Q3
3 marks Easy -1.2
3 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 Q4
3 marks Easy -1.8
4 Find the coordinates of the point of intersection of the lines \(x + 2 y = 5\) and \(y = 5 x - 1\).
OCR MEI C1 Q5
5 marks Standard +0.3
5 Fig. 9 shows a trapezium ABCD , with the lengths in centimetres of three of its sides. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d22f53f5-ba80-4065-a94b-2a9c92c20dfb-1_462_877_1796_684} \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 Q6
13 marks Standard +0.3
6 The points \(\mathrm { A } ( - 1,6 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 13,4 )\) 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 Q7
10 marks Standard +0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d22f53f5-ba80-4065-a94b-2a9c92c20dfb-2_696_879_960_673} \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 Q8
3 marks Easy -1.2
8 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 Q1
12 marks Moderate -0.5
1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-1_520_1122_357_551} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).
  1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.
OCR MEI C1 Q2
4 marks Easy -1.2
2
  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 Q3
5 marks Standard +0.3
3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).
OCR MEI C1 Q4
13 marks Standard +0.3
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-2_592_782_322_730} \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 Q5
4 marks Moderate -0.8
5 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-3_433_835_353_715} \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 Q1
12 marks Moderate -0.3
1 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.
    [diagram]
OCR MEI C1 Q2
3 marks Easy -1.2
2 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).
OCR MEI C1 Q3
3 marks Moderate -0.8
3 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{13979d37-ea09-4d51-aff8-81fa611cc080-2_579_1012_441_706} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C . \(5 \mathrm {~A} ( 9,8 ) , \mathrm { B } ( 5,0 )\) an \(\mathrm { C } ( 3,1 )\) are three points.
  1. Show that AB and BC are perpendicular.
  2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle.
  3. BD is a diameter of the circle. Find the coordinates of D .