Questions — OCR MEI C1 (472 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C1 Q2
2 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).
OCR MEI C1 Q3
3 A is the point \(( 1,5 )\) and \(B\) is the point \(( 6 , - 1 )\). \(M\) is the midpoint of \(A B\). Determine whether the line with equation \(y = 2 x - 5\) passes through M.
OCR MEI C1 Q4
4 Find the equation of the line which is perpendicular to the line \(y = 2 x - 5\) and which passes through the point \(( 4,1 )\). Give your answer in the form \(y = a x + b\).
OCR MEI C1 Q6
6 Find the equation of the line with gradient - 2 which passes through the point \(( 3,1 )\). Give your answer in the form \(y = a x + b\). Find also the points of intersection of this line with the axes.
OCR MEI C1 Q7
7 Find the set of values of \(k\) for which the graph of \(y = x ^ { 2 } + 2 k x + 5\) does not intersect the \(x\)-axis.
OCR MEI C1 Q8
8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d2142d2d-661b-4340-893f-f97f828c6855-2_447_763_602_690} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 is a sketch of quadrilateral ABCD with vertices \(\mathrm { A } ( 1,5 ) , \mathrm { B } ( - 1,1 ) , \mathrm { C } ( 3 , - 1 )\) and \(\mathrm { D } ( 11,5 )\).
  1. Show that \(\mathrm { AB } = \mathrm { BC }\).
  2. Show that the diagonals AC and BD are perpendicular.
  3. Find the midpoint of AC . Show that BD bisects AC but AC does not bisect BD .
OCR MEI C1 Q9
9 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 Q1
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 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 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
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 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
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
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
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
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
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
3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).
OCR MEI C1 Q4
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
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
1 Find the equation of the line passing through \(( - 1 , - 9 )\) and \(( 3,11 )\). Give your answer in the form \(y = m x + c\).
OCR MEI C1 Q3
3
  1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
  2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
  3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
  4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).
OCR MEI C1 Q5
5
  1. Find the gradient of the line \(4 x + 5 y = 24\).
  2. A line parallel to \(4 x + 5 y = 24\) passes through the point \(( 0,12 )\). Find the coordinates of its point of intersection with the \(x\)-axis.
OCR MEI C1 Q7
7 Find, in the form \(y = a x + b\), the equation of the line through \(( 3,10 )\) which is parallel to \(y = 2 x + 7\).
OCR MEI C1 Q1
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. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{13979d37-ea09-4d51-aff8-81fa611cc080-1_449_873_843_856} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}