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 2015 June Q2
2
  1. Sketch the curve \(y = - \frac { 1 } { x }\).
  2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
  3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).
OCR C1 2015 June Q3
3 Express each of the following in the form \(5 ^ { k }\).
  1. \(25 ^ { 4 }\)
  2. \(\frac { 1 } { \sqrt [ 4 ] { 5 } }\)
  3. \(( 5 \sqrt { 5 } ) ^ { 3 }\)
OCR C1 2015 June Q4
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } - 6 = 0\).
OCR C1 2015 June Q5
5 The points \(A\) and \(B\) have coordinates \(( 2,1 )\) and \(( 5 , - 3 )\) respectively.
  1. Find the length of \(A B\).
  2. Find an equation of the line through the mid-point of \(A B\) which is perpendicular to \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 2015 June Q6
6 Solve the simultaneous equations $$2 x + y - 5 = 0 , \quad x ^ { 2 } - y ^ { 2 } = 3$$
OCR C1 2015 June Q7
7
  1. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )\), find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find the gradient of the curve \(y = x ^ { - \frac { 1 } { 3 } }\) at the point where \(x = - 8\).
OCR C1 2015 June Q8
8
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
  2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
  3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).
OCR C1 2015 June Q9
9 The curve \(y = 2 x ^ { 3 } - a x ^ { 2 } + 8 x + 2\) passes through the point \(B\) where \(x = 4\).
  1. Given that \(B\) is a stationary point of the curve, find the value of the constant \(a\).
  2. Determine whether the stationary point \(B\) is a maximum point or a minimum point.
  3. Find the \(x\)-coordinate of the other stationary point of the curve.
OCR C1 2015 June Q10
10 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 4 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Show that the tangent to the circle at the point \(P ( 8,2 )\) has equation \(3 x + 4 y = 32\).
  3. The circle meets the \(y\)-axis at \(Q\) and the tangent meets the \(y\)-axis at \(R\). Find the area of triangle \(P Q R\).
OCR C1 2016 June Q1
1
  1. Simplify \(( 2 x - 3 ) ^ { 2 } - 2 ( 3 - x ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 x ^ { 2 } - 3 x + 4 \right) \left( 5 - 2 x - x ^ { 3 } \right)\).
OCR C1 2016 June Q2
2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
OCR C1 2016 June Q3
3 Solve the simultaneous equations $$x ^ { 2 } + y ^ { 2 } = 34 , \quad 3 x - y + 4 = 0$$
OCR C1 2016 June Q4
4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).
OCR C1 2016 June Q5
5 Express the following in the form \(2 ^ { p }\).
  1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
  2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)
OCR C1 2016 June Q6
6
  1. Express \(4 + 12 x - 2 x ^ { 2 }\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the maximum point of the curve \(y = 4 + 12 x - 2 x ^ { 2 }\).
OCR C1 2016 June Q7
7
  1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
  2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
  3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).
OCR C1 2016 June Q8
8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).
  1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
  2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
  3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).
OCR C1 2016 June Q9
9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.
OCR C1 2016 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).
  1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
  2. Find the equation of the tangent to the circle at \(A\).
  3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
  4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).
OCR C1 2016 June Q11
11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .
OCR C1 2007 January Q9
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2009 June Q9
  1. Calculate the length of \(A B\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q9
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. \end{enumerate} The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
  2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
  3. Show that \(O P = O R\).
OCR C1 Q2
  1. \(y = x - 2 x ^ { 2 }\),
  2. \(y = \frac { 3 } { x ^ { 2 } }\). \item (a) Express \(x ^ { 2 } - 10 x + 27\) in the form \(( x + p ) ^ { 2 } + q\).
    (b) Sketch the curve with equation \(y = x ^ { 2 } - 10 x + 27\), showing on your sketch
  3. the coordinates of the vertex of the curve,
  4. the coordinates of any points where the curve meets the coordinate axes. \item The straight line \(l _ { 1 }\) has gradient 2 and passes through the point with coordinates \(( 4 , - 5 )\).
  5. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). \end{enumerate} The straight line \(l _ { 2 }\) is perpendicular to the line with equation \(3 x - y = 4\) and passes through the point with coordinates \(( 3,0 )\).
  6. Find an equation for \(l _ { 2 }\).
  7. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.