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 C1 2014 June Q7
7 marks Moderate -0.8
\(A\) is the point \((5, 7)\) and \(B\) is the point \((-1, -5)\).
  1. Find the coordinates of the mid-point of the line segment \(AB\). [2]
  2. Find an equation of the line through \(A\) that is perpendicular to the line segment \(AB\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 2014 June Q8
9 marks Moderate -0.8
A curve has equation \(y = 3x^3 - 7x + \frac{2}{x}\).
  1. Verify that the curve has a stationary point when \(x = 1\). [5]
  2. Determine the nature of this stationary point. [2]
  3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]
OCR C1 2014 June Q9
12 marks Moderate -0.8
A circle with centre \(C\) has equation \((x - 2)^2 + (y + 5)^2 = 25\).
  1. Show that no part of the circle lies above the \(x\)-axis. [3]
  2. The point \(P\) has coordinates \((6, k)\) and lies inside the circle. Find the set of possible values of \(k\). [5]
  3. Prove that the line \(2y = x\) does not meet the circle. [4]
OCR C1 2014 June Q10
12 marks Moderate -0.3
A curve has equation \(y = (x + 2)^2(2x - 3)\).
  1. Sketch the curve, giving the coordinates of all points of intersection with the axes. [3]
  2. Find an equation of the tangent to the curve at the point where \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [9]
OCR MEI C1 Q1
3 marks Easy -1.2
Solve the inequality \(2(x - 3) < 6x + 15\). [3]
OCR MEI C1 Q2
3 marks Easy -1.2
Make \(r\) the subject of \(V = \frac{4}{3}\pi r^3\). [3]
OCR MEI C1 Q3
2 marks Easy -1.8
In each case, choose one of the statements $$P \Rightarrow Q \quad\quad P \Leftarrow Q \quad\quad P \Leftrightarrow Q$$ to describe the complete relationship between P and Q.
  1. For \(n\) an integer: P: \(n\) is an even number Q: \(n\) is a multiple of 4 [1]
  2. For triangle ABC: P: B is a right-angle Q: \(AB^2 + BC^2 = AC^2\) [1]
OCR MEI C1 Q4
4 marks Moderate -0.8
Find the coefficient of \(x^3\) in the expansion of \((2 + 3x)^5\). [4]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of the following.
  1. \(\left(\frac{1}{3}\right)^{-2}\) [2]
  2. \(16^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
5 marks Moderate -0.8
The line \(L\) is parallel to \(y = -2x + 1\) and passes through the point \((5, 2)\). Find the coordinates of the points of intersection of \(L\) with the axes. [5]
OCR MEI C1 Q7
5 marks Easy -1.2
Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]
OCR MEI C1 Q8
5 marks Moderate -0.8
Find, in the form \(y = mx + c\), the equation of the line passing through A\((3, 7)\) and B\((5, -1)\). Show that the midpoint of AB lies on the line \(x + 2y = 10\). [5]
OCR MEI C1 Q9
5 marks Moderate -0.8
Simplify \((3 + \sqrt{2})(3 - \sqrt{2})\). Express \(\frac{1 + \sqrt{2}}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational. [5]
OCR MEI C1 Q10
12 marks Moderate -0.8
\includegraphics{figure_10} Fig. 10 shows a circle with centre C\((2, 1)\) and radius 5.
  1. Show that the equation of the circle may be written as $$x^2 + y^2 - 4x - 2y - 20 = 0.$$ [3]
  2. Find the coordinates of the points P and Q where the circle cuts the \(y\)-axis. Leave your answers in the form \(a \pm \sqrt{b}\). [3]
  3. Verify that the point A\((5, -3)\) lies on the circle. Show that the tangent to the circle at A has equation \(4y = 3x - 27\). [6]
OCR MEI C1 Q11
12 marks Moderate -0.3
A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).
  1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
  2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]
OCR MEI C1 Q12
12 marks Moderate -0.3
  1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
  2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
  3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]
OCR MEI C1 2006 January Q1
2 marks Easy -1.2
\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]
OCR MEI C1 2006 January Q2
4 marks Moderate -0.8
\includegraphics{figure_2} Fig. 2 shows graphs \(A\) and \(B\).
  1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
  2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]
OCR MEI C1 2006 January Q3
4 marks Easy -1.8
Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]
OCR MEI C1 2006 January Q4
4 marks Easy -1.8
Solve the inequality \(\frac{3(2x + 1)}{4} > -6\). [4]
OCR MEI C1 2006 January Q5
4 marks Moderate -0.8
Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]
OCR MEI C1 2006 January Q6
3 marks Easy -1.2
When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]
OCR MEI C1 2006 January Q7
5 marks Moderate -0.8
\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point 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]
OCR MEI C1 2006 January Q8
5 marks Easy -1.3
  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]
  2. Express \(\frac{\sqrt{3}}{6 - \sqrt{3}}\) in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are rational. [3]
OCR MEI C1 2006 January Q9
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]