Questions — OCR MEI C1 (499 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 Q13
4 marks Moderate -0.5
Rearrange \(y + 5 = x(y + 2)\) to make \(y\) the subject of the formula. [4]
OCR MEI C1 Q1
5 marks Moderate -0.8
  1. Solve the equation \(2x^2 + 3x = 0\). [2]
  2. Find the set of values of \(k\) for which the equation \(2x^2 + 3x - k = 0\) has no real roots. [3]
OCR MEI C1 Q2
4 marks Moderate -0.5
Make \(x\) the subject of the equation \(y = \frac{x + 3}{x - 2}\). [4]
OCR MEI C1 Q3
4 marks Moderate -0.8
Solve the equation \(y^2 - 7y + 12 = 0\). Hence solve the equation \(x^4 - 7x^2 + 12 = 0\). [4]
OCR MEI C1 Q4
5 marks Easy -1.2
  1. Write \(\sqrt{48} + \sqrt{3}\) in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
  2. Simplify \(\frac{1}{5 + \sqrt{2}} + \frac{1}{5 - \sqrt{2}}\). [3]
OCR MEI C1 Q5
3 marks Moderate -0.8
Solve the equation \(\frac{4x + 5}{2x} = -3\). [3]
OCR MEI C1 Q6
3 marks Moderate -0.8
Make \(a\) the subject of the equation $$2a + 5c = af + 7c.$$ [3]
OCR MEI C1 Q7
4 marks Standard +0.3
Find the set of values of \(k\) for which the equation \(2x^2 + kx + 2 = 0\) has no real roots. [4]
OCR MEI C1 Q8
2 marks Moderate -0.8
One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]
OCR MEI C1 Q9
2 marks Moderate -0.8
\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]
OCR MEI C1 Q10
4 marks Moderate -0.5
Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]
OCR MEI C1 Q11
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]
OCR MEI C1 Q1
12 marks Moderate -0.8
\includegraphics{figure_1} 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]
  2. Hence show that the equation of the curve may be written as \(y = 2x^3 + 11x^2 - x - 30\). [2]
  3. Draw the line \(y = 5x + 10\) accurately on the graph. The curve and this line intersect at \((-2, 0)\); find graphically the \(x\)-coordinates of the other points of intersection. [3]
  4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2x^2 + 7x - 20 = 0.$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. [5]
OCR MEI C1 Q2
12 marks Standard +0.3
\includegraphics{figure_2} Fig. 12 shows the graph of \(y = \frac{1}{x-2}\).
  1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\). [3]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x-2}\). [4]
OCR MEI C1 Q3
13 marks Moderate -0.3
\includegraphics{figure_3} 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 Q4
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 Q5
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 Q6
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 Q1
4 marks Easy -1.3
Evaluate the following.
  1. \(200^0\) [1]
  2. \(\left(\frac{25}{9}\right)^{-\frac{1}{2}}\) [3]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{1}{27}\right)^{\frac{2}{3}}\). [2]
  2. Simplify \(\frac{(4a^2c)^3}{32a^4c^7}\). [3]
OCR MEI C1 Q3
4 marks Easy -1.2
You are given that \(n\), \(n + 1\) and \(n + 2\) are three consecutive integers.
  1. Expand and simplify \(n^2 + (n + 1)^2 + (n + 2)^2\). [2]
  2. For what values of \(n\) will the sum of the squares of these three consecutive integers be an even number? Give a reason for your answer. [2]
OCR MEI C1 Q4
5 marks Easy -1.3
  1. Evaluate \((0.2)^{-2}\). [2]
  2. Simplify \((16a^{12})^{\frac{3}{4}}\). [3]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of each of the following.
  1. \(\left(\frac{5}{3}\right)^{-2}\) [2]
  2. \(81^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
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 Q7
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]