Questions C1 (1562 questions)

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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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
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OCR MEI C1 Q10
4 marks Easy -1.8
  1. Find \(a\), given that \(a^3 = 64x^{12}y^3\). [2]
  2. Find the value of \(\left(\frac{1}{2}\right)^{-5}\). [2]
OCR MEI C1 Q11
4 marks Easy -1.8
Find the value of each of the following, giving each answer as an integer or fraction as appropriate.
  1. \(8^{\frac{1}{3}}\) [2]
  2. \(\left(\frac{7}{3}\right)^{-2}\) [2]
OCR MEI C1 Q12
5 marks Moderate -0.8
  1. Simplify \(6\sqrt{2} \times 5\sqrt{3} \times \sqrt{24}\). [2]
  2. Express \((2 - 3\sqrt{5})^2\) in the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are integers. [3]
OCR MEI C1 Q13
5 marks Easy -1.3
Simplify the following.
  1. \(\frac{16^{\frac{1}{3}}}{81^{\frac{1}{4}}}\) [2]
  2. \(\frac{12(a^3b^2c)^4}{4a^2c^6}\) [3]
OCR MEI C1 Q1
5 marks Moderate -0.8
Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]
OCR MEI C1 Q2
12 marks Moderate -0.8
A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.
  1. Show that AB and BC are perpendicular. [3]
  2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
  3. BD is a diameter of the circle. Find the coordinates of D. [3]
OCR MEI C1 Q3
10 marks Moderate -0.3
A circle has equation \(x^2 + y^2 = 45\).
  1. State the centre and radius of this circle. [2]
  2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]
OCR MEI C1 Q4
12 marks Moderate -0.3
\includegraphics{figure_1} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).
  1. Find the equation of the line through A and B. [3]
  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. [2]
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
  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. [4]
OCR MEI C1 Q5
14 marks Standard +0.3
  1. Points A and B have coordinates \((-2, 1)\) and \((3, 4)\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5x + 3y = 10\). [6]
  2. Points C and D have coordinates \((-5, 4)\) and \((3, 6)\) respectively. The line through C and D has equation \(4y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB. Find the coordinates of point E. [3]
  3. Find the equation of the circle with centre E which passes through A and B. Show also that CD is a diameter of this circle. [5]
OCR MEI C1 Q6
13 marks Moderate -0.3
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. [3]
  2. Find the area of triangle ABC. [3]
  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. [6]
  4. Find the coordinates of the point on this circle that is furthest from B. [1]
OCR MEI C1 Q1
5 marks Easy -1.2
A line \(L\) is parallel to \(y = 4x + 5\) and passes through the point \((-1, 6)\). Find the equation of the line \(L\) in the form \(y = ax + b\). Find also the coordinates of its intersections with the axes. [5]
OCR MEI C1 Q2
4 marks Moderate -0.8
Find the coordinates of the point of intersection of the lines \(y = 5x - 2\) and \(x + 3y = 8\). [4]
OCR MEI C1 Q3
3 marks Moderate -0.8
A is the point \((1, 5)\) and B is the point \((6, -1)\). M is the midpoint of AB. Determine whether the line with equation \(y = 2x - 5\) passes through M. [3]
OCR MEI C1 Q4
3 marks Moderate -0.8
Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]
OCR MEI C1 Q6
3 marks Easy -1.2
Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]
OCR MEI C1 Q8
11 marks Moderate -0.8
\includegraphics{figure_8} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).
  1. Show that AB = BC. [3]
  2. Show that the diagonals AC and BD are perpendicular. [3]
  3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]
OCR MEI C1 Q9
3 marks Moderate -0.8
Find the equation of the line which is perpendicular to the line \(y = 5x + 2\) and which passes through the point \((1, 6)\). Give your answer in the form \(y = ax + b\). [3]
OCR MEI C1 Q1
3 marks Easy -1.2
Find the equation of the line passing through \((-1, -9)\) and \((3, 11)\). Give your answer in the form \(y = mx + c\). [3]
OCR MEI C1 Q2
4 marks Easy -1.2
  1. Find the points of intersection of the line \(2x + 3y = 12\) with the axes. [2]
  2. Find also the gradient of this line. [2]
OCR MEI C1 Q3
12 marks Moderate -0.8
  1. Express \(x^2 - 6x + 2\) in the form \((x - a)^2 - b\). [3]
  2. State the coordinates of the turning point on the graph of \(y = x^2 - 6x + 2\). [2]
  3. Sketch the graph of \(y = x^2 - 6x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis. [2]
  4. Solve the simultaneous equations \(y = x^2 - 6x + 2\) and \(y = 2x - 14\). Hence show that the line \(y = 2x - 14\) is a tangent to the curve \(y = x^2 - 6x + 2\). [5]
OCR MEI C1 Q4
4 marks Moderate -0.5
Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2x - 5\) and \(6x + 2y = 7\). [4]
OCR MEI C1 Q5
5 marks Moderate -0.8
  1. Find the gradient of the line \(4x + 5y = 24\). [2]
  2. A line parallel to \(4x + 5y = 24\) passes through the point \((0, 12)\). Find the coordinates of its point of intersection with the \(x\)-axis. [3]
OCR MEI C1 Q6
11 marks Moderate -0.3
  1. \includegraphics{figure_1} Fig. 10 shows a sketch of the graph of \(y = \frac{1}{x}\). Sketch the graph of \(y = \frac{1}{x-2}\), showing clearly the coordinates of any points where it crosses the axes. [3]
  2. Find the value of \(x\) for which \(\frac{1}{x-2} = 5\). [2]
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac{1}{x-2}\). Give your answers in the form \(a \pm \sqrt{b}\). Show the position of these points on your graph in part (i). [6]
OCR MEI C1 Q7
3 marks Easy -1.2
Find, in the form \(y = ax + b\), the equation of the line through \((3, 10)\) which is parallel to \(y = 2x + 7\). [3]
OCR MEI C1 Q1
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3x^2 + 6x + 10\) and the line \(y = 2 - 4x\). [5]
  2. Write \(3x^2 + 6x + 10\) in the form \(a(x + b)^2 + c\). [4]
  3. Hence or otherwise, show that the graph of \(y = 3x^2 + 6x + 10\) is always above the \(x\)-axis. [2]