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 2009 June Q10
10
  1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
  2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
  3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
  4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?
OCR C1 2009 June Q11
11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).
  1. Find the value of \(k\).
  2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT
OCR C1 2010 June Q2
2
  1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
  2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
  3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.
OCR C1 2010 June Q3
3
  1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
  2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.
OCR C1 2010 June Q4
4
  1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.
OCR C1 2010 June Q5
5 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).
OCR C1 2010 June Q6
6 Find the gradient of the curve \(y = 2 x + \frac { 6 } { \sqrt { x } }\) at the point where \(x = 4\).
OCR C1 2010 June Q7
7 Solve the simultaneous equations $$x + 2 y - 6 = 0 , \quad 2 x ^ { 2 } + y ^ { 2 } = 57 .$$
OCR C1 2010 June Q8
8
  1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
  2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
  3. State the equation of the normal to the curve at its minimum point.
  4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).
OCR C1 2010 June Q9
9
  1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\).
    \(A B\) is the diameter of a circle.
  2. Find the radius of the circle.
  3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).
OCR C1 2010 June Q10
10
  1. Find the coordinates of the stationary points of the curve \(y = 2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\).
  2. State the set of values for \(x\) for which \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\) is a decreasing function.
  3. Show that the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 }\) is \(10 x - 4 y - 7 = 0\).
  4. Hence, with the aid of a sketch, show that the equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x = \frac { 5 } { 2 } x - \frac { 7 } { 4 }\) has two distinct real roots.
OCR C1 2011 June Q1
1 Express \(3 x ^ { 2 } - 18 x + 4\) in the form \(p ( x + q ) ^ { 2 } + r\).
OCR C1 2011 June Q2
2
  1. Sketch the curve \(y = \frac { 1 } { x }\).
  2. Describe fully the single transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x } + 4\).
OCR C1 2011 June Q3
3 Simplify
  1. \(\frac { ( 4 x ) ^ { 2 } \times 2 x ^ { 3 } } { x }\),
  2. \(\left( 36 x ^ { - 2 } \right) ^ { - \frac { 1 } { 2 } }\).
OCR C1 2011 June Q4
4 Solve the simultaneous equations $$y = 2 ( x - 2 ) ^ { 2 } , \quad 3 x + y = 26$$
OCR C1 2011 June Q5
5
  1. Express \(\sqrt { 300 } - \sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  2. Express \(\frac { 15 + \sqrt { 40 } } { \sqrt { 5 } }\) in the form \(a \sqrt { 5 } + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
OCR C1 2011 June Q6
6 Solve the equation \(3 x ^ { \frac { 1 } { 2 } } - 8 x ^ { \frac { 1 } { 4 } } + 4 = 0\).
OCR C1 2011 June Q7
7 Solve the inequalities
  1. \(- 9 \leqslant 6 x + 5 \leqslant 0\),
  2. \(6 x + 5 < x ^ { 2 } + 2 x - 7\).
OCR C1 2011 June Q8
8
  1. Find the coordinates of the stationary point on the curve \(y = 3 x ^ { 2 } - \frac { 6 } { x } - 2\).
  2. Determine whether the stationary point is a maximum point or a minimum point.
OCR C1 2011 June Q9
9 The points \(A ( 1,3 ) , B ( 7,1 )\) and \(C ( - 3 , - 9 )\) are joined to form a triangle.
  1. Show that this triangle is right-angled and state whether the right angle is at \(A , B\) or \(C\).
  2. The points \(A , B\) and \(C\) lie on the circumference of a circle. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
OCR C1 2011 June Q10
10 A curve has equation \(y = ( 2 x - 1 ) ( x + 3 ) ( x - 1 )\).
  1. Sketch the curve, indicating the coordinates of all points of intersection with the axes.
  2. Show that the gradient of the curve at the point \(P ( 1,0 )\) is 4 .
  3. The line \(l\) is parallel to the tangent to the curve at the point \(P\). The curve meets \(l\) at the point where \(x = - 2\). Find the equation of \(l\), giving your answer in the form \(y = m x + c\).
  4. Determine whether \(l\) is a tangent to the curve at the point where \(x = - 2\).
OCR C1 2012 June Q1
1 Simplify \(( x - 5 ) \left( x ^ { 2 } + 3 \right) - ( x + 4 ) ( x - 1 )\).
OCR C1 2012 June Q2
2 Express each of the following in the form \(7 ^ { k }\) :
  1. \(\sqrt [ 4 ] { 7 }\),
  2. \(\frac { 1 } { 7 \sqrt { 7 } }\),
  3. \(7 ^ { 4 } \times 49 ^ { 10 }\).
OCR C1 2012 June Q3
3
  1. Find the gradient of the line \(l\) which has equation \(3 x - 5 y - 20 = 0\).
  2. The line \(l\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
OCR C1 2012 June Q4
4
  1. Express \(2 x ^ { 2 } - 20 x + 49\) in the form \(p ( x - q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 20 x + 49\).