Questions — OCR C1 (324 questions)

Browse by module
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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2012 June Q5
5
  1. Sketch the curve \(y = \sqrt { x }\).
  2. Describe the transformation that transforms the curve \(y = \sqrt { x }\) to the curve \(y = \sqrt { x - 4 }\).
  3. The curve \(y = \sqrt { x }\) is stretched by a scale factor of 5 parallel to the \(x\)-axis. State the equation of the transformed curve.
OCR C1 2012 June Q6
6 Find the equation of the normal to the curve \(y = \frac { 6 } { x ^ { 2 } } - 5\) at the point on the curve where \(x = 2\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
OCR C1 2012 June Q7
7 Solve the equation \(x - 6 x ^ { \frac { 1 } { 2 } } + 2 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2012 June Q8
8
  1. Find the coordinates of the stationary point on the curve \(y = x ^ { 4 } + 32 x\).
  2. Determine whether this stationary point is a maximum or a minimum.
  3. For what values of \(x\) does \(x ^ { 4 } + 32 x\) increase as \(x\) increases?
OCR C1 2012 June Q9
9
  1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
  2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).
OCR C1 2012 June Q10
10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  1. Find the coordinates of the centre \(C\) and the length of the diameter.
  2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
  3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
  4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR C1 2013 June Q1
1 Express each of the following in the form \(a \sqrt { 5 }\), where \(a\) is an integer.
  1. \(4 \sqrt { 15 } \times \sqrt { 3 }\)
  2. \(\frac { 20 } { \sqrt { 5 } }\)
  3. \(5 ^ { \frac { 3 } { 2 } }\)
OCR C1 2013 June Q2
2 Solve the equation \(8 x ^ { 6 } + 7 x ^ { 3 } - 1 = 0\).
OCR C1 2013 June Q3
3 It is given that \(\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } } + 2 x\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
OCR C1 2013 June Q4
4
  1. Express \(3 x ^ { 2 } + 9 x + 10\) in the form \(3 ( x + p ) ^ { 2 } + q\).
  2. State the coordinates of the minimum point of the curve \(y = 3 x ^ { 2 } + 9 x + 10\).
  3. Calculate the discriminant of \(3 x ^ { 2 } + 9 x + 10\).
OCR C1 2013 June Q5
5
  1. Sketch the curve \(y = \frac { 2 } { x ^ { 2 } }\).
  2. The curve \(y = \frac { 2 } { x ^ { 2 } }\) is translated by 5 units in the negative \(x\)-direction. Find the equation of the curve after it has been translated.
  3. Describe a transformation that transforms the curve \(y = \frac { 2 } { x ^ { 2 } }\) to the curve \(y = \frac { 1 } { x ^ { 2 } }\).
OCR C1 2013 June Q6
6 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 y - 24 = 0\).
  1. Find the centre and radius of the circle.
  2. The point \(A ( 2,2 )\) lies on the circumference of \(C\). Given that \(A B\) is a diameter of the circle, find the coordinates of \(B\).
OCR C1 2013 June Q7
7 Solve the inequalities
  1. \(3 - 8 x > 4\),
  2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\).
    \(8 \quad A\) is the point \(( - 2,6 )\) and \(B\) is the point \(( 3 , - 8 )\). The line \(l\) is perpendicular to the line \(x - 3 y + 15 = 0\) and passes through the mid-point of \(A B\). Find the equation of \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2013 June Q9
9
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 6\), giving the coordinates of all points of intersection with the axes.
  2. Find the set of values of \(x\) for which \(2 x ^ { 2 } - x - 6\) is a decreasing function.
  3. The line \(y = 4\) meets the curve \(y = 2 x ^ { 2 } - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(P Q\).
OCR C1 2013 June Q10
10 The curve \(y = ( 1 - x ) \left( x ^ { 2 } + 4 x + k \right)\) has a stationary point when \(x = - 3\).
  1. Find the value of the constant \(k\).
  2. Determine whether the stationary point is a maximum or minimum point.
  3. Given that \(y = 9 x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\).
OCR C1 2014 June Q1
1 Express \(5 x ^ { 2 } + 10 x + 2\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
OCR C1 2014 June Q2
2 Express each of the following in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  1. \(\frac { 6 } { \sqrt { 3 } }\)
  2. \(10 \sqrt { 3 } - 6 \sqrt { 27 }\)
  3. \(3 ^ { \frac { 5 } { 2 } }\)
OCR C1 2014 June Q3
3 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).
OCR C1 2014 June Q4
4 The curve \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates \(( 2,5 )\).
  1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( x ) + 2\).
  2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( 2 x )\).
  3. Describe the transformation that transforms the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { f } ( x + 4 )\).
OCR C1 2014 June Q5
5 Solve the following inequalities.
  1. \(5 < 6 x + 3 < 14\)
  2. \(x ( 3 x - 13 ) \geqslant 10\)
OCR C1 2014 June Q6
6 Given that \(y = 6 x ^ { 3 } + \frac { 4 } { \sqrt { x } } + 5 x\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    \(7 \quad A\) is the point \(( 5,7 )\) and \(B\) is the point \(( - 1 , - 5 )\).
  3. Find the coordinates of the mid-point of the line segment \(A B\).
  4. Find an equation of the line through \(A\) that is perpendicular to the line segment \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 2014 June Q8
8 A curve has equation \(y = 3 x ^ { 3 } - 7 x + \frac { 2 } { x }\).
  1. Verify that the curve has a stationary point when \(x = 1\).
  2. Determine the nature of this stationary point.
  3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\).
OCR C1 2014 June Q9
9 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.
  2. The point \(P\) has coordinates \(( 6 , k )\) and lies inside the circle. Find the set of possible values of \(k\).
  3. Prove that the line \(2 y = x\) does not meet the circle.
OCR C1 2014 June Q10
10 A curve has equation \(y = ( x + 2 ) ^ { 2 } ( 2 x - 3 )\).
  1. Sketch the curve, giving the coordinates of all points of intersection with the axes.
  2. Find an equation of the tangent to the curve at the point where \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\). \section*{OCR}
OCR C1 2015 June Q1
1 Express \(\frac { 8 } { \sqrt { 3 } - 1 }\) in the form \(a \sqrt { 3 } + b\), where \(a\) and \(b\) are integers.