Questions C1 (1562 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 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 2011 June Q10
3 marks Moderate -0.8
Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]
OCR MEI C1 2011 June Q11
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
  2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
  3. For the curve \(y = 4x^2 + 24x + 31\),
    1. write down the equation of the line of symmetry, [1]
    2. write down the minimum \(y\)-value on the curve. [1]
OCR MEI C1 2011 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
  2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]
OCR MEI C1 2011 June Q13
13 marks Moderate -0.3
\includegraphics{figure_13} Fig. 13 shows the circle with equation \((x - 4)^2 + (y - 2)^2 = 16\).
  1. Write down the radius of the circle and the coordinates of its centre. [2]
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
  3. Show that the point A \((4 + 2\sqrt{2}, 2 + 2\sqrt{2})\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]
OCR MEI C1 2012 June Q1
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 2012 June Q2
3 marks Easy -1.8
Make \(b\) the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]
OCR MEI C1 2012 June Q3
4 marks Easy -1.8
  1. Evaluate \(\left(\frac{1}{5}\right)^{-2}\). [2]
  2. Evaluate \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\). [2]
OCR MEI C1 2012 June Q4
3 marks Easy -1.2
Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]
OCR MEI C1 2012 June Q5
5 marks Moderate -0.8
  1. Simplify \(\frac{10\sqrt{6}}{3}{\sqrt{24}}\). [3]
  2. Simplify \(\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}\). [2]
OCR MEI C1 2012 June Q6
5 marks Moderate -0.8
  1. Evaluate \(^5C_3\). [1]
  2. Find the coefficient of \(x^3\) in the expansion of \((3 - 2x)^5\). [4]
OCR MEI C1 2012 June Q7
4 marks Moderate -0.5
Find the set of values of \(k\) for which the graph of \(y = x^2 + 2kx + 5\) does not intersect the \(x\)-axis. [4]
OCR MEI C1 2012 June Q8
5 marks Standard +0.3
The function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\). Find the values of \(b\) and \(c\). [5]
OCR MEI C1 2012 June Q9
4 marks Moderate -0.8
Simplify \((n + 3)^2 - n^2\). Hence explain why, when \(n\) is an integer, \((n + 3)^2 - n^2\) is never an even number. Given also that \((n + 3)^2 - n^2\) is divisible by \(9\), what can you say about \(n\)? [4]
OCR MEI C1 2012 June Q10
11 marks Moderate -0.3
\includegraphics{figure_10} 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 2012 June Q11
12 marks Moderate -0.3
A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).
  1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
  4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]
OCR MEI C1 2012 June Q12
13 marks Moderate -0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).
  1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
  2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
  3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]
OCR MEI C1 2013 June Q1
3 marks Easy -1.2
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 2013 June Q2
4 marks Easy -1.2
Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]
OCR MEI C1 2013 June Q3
5 marks Easy -1.8
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{1}{4}}\). [3]
OCR MEI C1 2013 June Q4
3 marks Easy -1.2
Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac{1}{3}\pi r^2(a + b)$$ [3]
OCR MEI C1 2013 June Q5
3 marks Moderate -0.8
You are given that \(\text{f}(x) = x^5 + kx - 20\). When \(\text{f}(x)\) is divided by \((x - 2)\), the remainder is 18. Find the value of \(k\). [3]
OCR MEI C1 2013 June Q6
4 marks Moderate -0.8
Find the coefficient of \(x^3\) in the binomial expansion of \((2 - 4x)^5\). [4]
OCR MEI C1 2013 June Q7
5 marks Moderate -0.8
  1. Express \(125\sqrt{5}\) in the form \(5^t\). [2]
  2. Simplify \(10 + 7\sqrt{5} + \frac{38}{1 - 2\sqrt{5}}\), giving your answer in the form \(a + b\sqrt{5}\). [3]
OCR MEI C1 2013 June Q8
5 marks Moderate -0.8
Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]
OCR MEI C1 2013 June Q9
4 marks Moderate -0.8
\(n - 1\), \(n\) and \(n + 1\) are any three consecutive integers.
  1. Show that the sum of these integers is always divisible by 3. [1]
  2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3. [3]