Questions C1 (1442 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 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
3 marks Easy -1.2
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
12 marks Moderate -0.8
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 marks Moderate -0.8
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
3 marks Easy -1.2
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
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. \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}
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 .
OCR MEI C1 Q1
12 marks Moderate -0.8
1 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q2
12 marks Moderate -0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a94593ca-d84d-4747-ac19-b05389e85b3c-1_781_1462_1118_342} \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 \(( 0 , - 30 )\).
  1. Use the intersections with both axes to express the equation of the curve in a factorised form.
  2. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
  3. 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.
  4. 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.
OCR MEI C1 Q3
12 marks Moderate -0.8
3 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
  3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
    (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
    (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q4
5 marks Moderate -0.3
4 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 Q5
12 marks Moderate -0.8
5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .
  1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
  4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.
OCR MEI C1 Q1
5 marks Moderate -0.8
1 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 Q2
13 marks Moderate -0.3
2 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 Q3
12 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6be6c0b0-76b7-49c0-bf1b-dc6f8f79981b-2_836_906_361_675} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
    [0pt] [3]
  2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
    [0pt] [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
  4. 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 \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  5. 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.
  6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 Q1
12 marks Moderate -0.3
1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
  1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$