Questions — OCR MEI C1 (472 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 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 Q9
9 Find the coordinates of the points where the curve \(y = x ^ { 2 } - 2 x - 8\) meets the line \(y = x + 2\).
OCR MEI C1 Q10
10 The diagram shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .
  1. \(\quad y = \mathrm { f } ( x ) + 2\),
  2. \(y = \mathrm { f } ( x + 2 )\).
OCR MEI C1 Q11
11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Explain why a must be negative.
  2. State two factors of \(y = a x ^ { 2 } + b x + c\).
  3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
  4. Write this in completed square form.
  5. Explain why the graphs of these two functions never meet.
OCR MEI C1 Q12
12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the other factors of \(\mathrm { f } ( x )\).
  3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
  4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
  5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
    Solve this equation to find the \(x\)-coordinates of A and B .
OCR MEI C1 Q13
13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-5_409_768_383_604} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure}
  1. Find the coordinates of X .
  2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
  3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
    Write down the equation of the circumcircle of the triangle ABC .
  4. Find the coordinates of the points where the circle cuts the \(x\) axis.
OCR MEI C1 Q2
2 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3414ad8-7959-49f8-b43d-3972b2c03642-1_316_590_631_582} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Not to scale
  1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
  2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.
OCR MEI C1 Q3
3
  1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
  2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
  3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).
OCR MEI C1 Q4
4 Make \(a\) the subject of \(3 ( a + 4 ) = a c + 5 f\).
OCR MEI C1 Q5
5 Find the coordinates of the point of intersection of the lines \(y = 3 x - 2\) and \(x + 3 y = 1\).
OCR MEI C1 Q6
6 Express \(3 x ^ { 2 } - 12 x + 5\) in the form \(a ( x - b ) ^ { 2 } - c\). Hence state the minimum value of \(y\) on the curve \(y = 3 x ^ { 2 } - 12 x + 5\).
\(7 \quad\) Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).
OCR MEI C1 Q8
8 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).
OCR MEI C1 Q9
9 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.
OCR MEI C1 Q10
10 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$
OCR MEI C1 Q11
11 Rearrange the following equation to make \(h\) the subject. $$4 h + 5 = 9 a - h a ^ { 2 }$$
OCR MEI C1 Q1
1
  1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.
OCR MEI C1 Q2
2 Make \(x\) the subject of the equation \(y = \frac { x + 3 } { x - 2 }\).
OCR MEI C1 Q3
3 Solve the equation \(y ^ { 2 } - 7 y + 12 = 0\).
Hence solve the equation \(x ^ { 4 } - 7 x ^ { 2 } + 12 = 0\).
OCR MEI C1 Q4
4
  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. Simplify \(\frac { 1 } { 5 + \sqrt { 2 } } + \frac { 1 } { 5 - \sqrt { 2 } }\).
OCR MEI C1 Q5
5 Solve the equation \(\frac { 4 x + 5 } { 2 x } = - 3\).
OCR MEI C1 Q6
6 Make \(a\) the subject of the equation $$2 a + 5 c = a f + 7 c$$
OCR MEI C1 Q7
7 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 2 = 0\) has no real roots.
OCR MEI C1 Q8
8 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).
\(9 n\) is a positive integer. Show that \(n ^ { 2 } + n\) is always even.
OCR MEI C1 Q10
10 Make \(C\) the subject of the formula \(P = \frac { C } { C + 4 }\).
OCR MEI C1 Q11
11
  1. Find the range of values of \(k\) for which the equation \(x ^ { 2 } + 5 x + k = 0\) has one or more real roots.
  2. Solve the equation \(4 x ^ { 2 } + 20 x + 25 = 0\).
OCR MEI C1 Q2
4 marks
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e867e1a-db4e-4fc1-ad93-b5c49a9506e6-2_1263_1219_322_463} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).
  1. Draw accurately the graph of \(y = 2 x + 3\) 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 \(y = 2 x + 3\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
  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] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e867e1a-db4e-4fc1-ad93-b5c49a9506e6-3_1292_1404_364_353} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
  4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
  5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
  6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation.
  7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.