Questions — OCR C1 (324 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 C1 2006 June Q1
1 The points \(A ( 1,3 )\) and \(B ( 4,21 )\) lie on the curve \(y = x ^ { 2 } + x + 1\).
  1. Find the gradient of the line \(A B\).
  2. Find the gradient of the curve \(y = x ^ { 2 } + x + 1\) at the point where \(x = 3\).
OCR C1 2006 June Q2
2
  1. Evaluate \(27 ^ { - \frac { 2 } { 3 } }\).
  2. Express \(5 \sqrt { 5 }\) in the form \(5 ^ { n }\).
  3. Express \(\frac { 1 - \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
OCR C1 2006 June Q3
3
  1. Express \(2 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Solve \(2 x ^ { 2 } + 12 x + 13 = 0\), giving your answers in simplified surd form.
OCR C1 2006 June Q4
4
  1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 ,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
  3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).
OCR C1 2006 June Q5
5 Solve the inequalities
  1. \(1 < 4 x - 9 < 5\),
  2. \(y ^ { 2 } \geqslant 4 y + 5\).
OCR C1 2006 June Q6
6
  1. Solve the equation \(x ^ { 4 } - 10 x ^ { 2 } + 25 = 0\).
  2. Given that \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Hence find the number of stationary points on the curve \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\).
  4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 4 , \quad y = x - 1$$
  5. State the number of points of intersection of the curve \(y = x ^ { 2 } - 5 x + 4\) and the line \(y = x - 1\).
  6. Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 4\).
OCR C1 2006 June Q8
8 A cuboid has a volume of \(8 \mathrm {~m} ^ { 3 }\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \mathrm {~m} ^ { 2 }\).
  1. Show that \(A = 2 x ^ { 2 } + \frac { 32 } { x }\).
  2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\).
  3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer.
OCR C1 2006 June Q9
9 The points \(A\) and \(B\) have coordinates \(( 4 , - 2 )\) and \(( 10,6 )\) respectively. \(C\) is the mid-point of \(A B\). Find
  1. the coordinates of \(C\),
  2. the length of \(A C\),
  3. the equation of the circle that has \(A B\) as a diameter,
  4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(a x + b y = c\).
OCR C1 2007 June Q1
1 Simplify \(( 2 x + 5 ) ^ { 2 } - ( x - 3 ) ^ { 2 }\), giving your answer in the form \(a x ^ { 2 } + b x + c\).
OCR C1 2007 June Q2
2
  1. On separate diagrams, sketch the graphs of
    1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
    2. \(y = x ^ { 4 }\).
  2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).
OCR C1 2007 June Q3
3 Simplify the following, expressing each answer in the form \(a \sqrt { 5 }\).
  1. \(3 \sqrt { 10 } \times \sqrt { 2 }\)
  2. \(\sqrt { 500 } + \sqrt { 125 }\)
OCR C1 2007 June Q4
4
  1. Find the discriminant of \(k x ^ { 2 } - 4 x + k\) in terms of \(k\).
  2. The quadratic equation \(k x ^ { 2 } - 4 x + k = 0\) has equal roots. Find the possible values of \(k\)
OCR C1 2007 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516} The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.
  1. Show that the enclosed area, \(\mathrm { Am } ^ { 2 }\), is given by $$A = 20 x - 2 x ^ { 2 } .$$
  2. Use differentiation to find the maximum value of A .
OCR C1 2007 June Q6
6 By using the substitution \(y = ( x + 2 ) ^ { 2 }\), find the real roots of the equation $$( x + 2 ) ^ { 4 } + 5 ( x + 2 ) ^ { 2 } - 6 = 0$$
OCR C1 2007 June Q7
7
  1. Given that \(f ( x ) = x + \frac { 3 } { x }\), find \(f ^ { \prime } ( x )\).
  2. Find the gradient of the curve \(\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }\) at the point where \(\mathrm { x } = 4\).
OCR C1 2007 June Q8
8
  1. Express \(x ^ { 2 } + 8 x + 15\) in the form \(( x + a ) ^ { 2 } - b\).
  2. Hence state the coordinates of the vertex of the curve \(y = x ^ { 2 } + 8 x + 15\).
  3. Solve the inequality \(x ^ { 2 } + 8 x + 15 > 0\).
OCR C1 2007 June Q9
9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .
  1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find an equation for the line \(A B\).
OCR C1 2007 June Q10
10
  1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
  2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
  3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.
OCR C1 2008 June Q1
1 Express each of the following in the form \(4 ^ { n }\) :
  1. \(\frac { 1 } { 16 }\),
  2. 64 ,
  3. 8 .
OCR C1 2008 June Q2
2
  1. The curve \(y = x ^ { 2 }\) is translated 2 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
  2. The curve \(y = x ^ { 3 } - 4\) is reflected in the \(x\)-axis. Find the equation of the curve after it has been reflected.
OCR C1 2008 June Q3
3 Express each of the following in the form \(k \sqrt { 2 }\), where \(k\) is an integer:
  1. \(\sqrt { 200 }\),
  2. \(\frac { 12 } { \sqrt { 2 } }\),
  3. \(5 \sqrt { 8 } - 3 \sqrt { 2 }\).
OCR C1 2008 June Q4
4 Solve the equation \(2 x - 7 x ^ { \frac { 1 } { 2 } } + 3 = 0\).
OCR C1 2008 June Q5
5 Find the gradient of the curve \(y = 8 \sqrt { x } + x\) at the point whose \(x\)-coordinate is 9 .
OCR C1 2008 June Q6
6
  1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
  2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.
OCR C1 2008 June Q7
7 Solve the inequalities
  1. \(8 < 3 x - 2 < 11\),
  2. \(y ^ { 2 } + 2 y \geqslant 0\).