Questions — OCR (4619 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 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. \(y ^ { 2 } + 2 y \geqslant 0\).
OCR C1 2008 June Q8
8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
  3. Determine whether this stationary point is a minimum or maximum point.
  4. Find the \(x\)-coordinate of the other stationary point.
OCR C1 2008 June Q9
9
  1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
  3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
  4. Find an equation of the tangent to the circle at the point ( 8,9 ).
OCR C1 2008 June Q10
10
  1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
  3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
  4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
  5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).
OCR C1 Specimen Q1
1 Write down the exact values of
  1. \(4 ^ { - 2 }\),
  2. \(( 2 \sqrt { } 2 ) ^ { 2 }\),
  3. \(\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\).
OCR C1 Specimen Q2
2
  1. Express \(x ^ { 2 } - 8 x + 3\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Hence write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 8 x + 3\).
OCR C1 Specimen Q3
3 The quadratic equation \(x ^ { 2 } + k x + k = 0\) has no real roots for \(x\).
  1. Write down the discriminant of \(x ^ { 2 } + k x + k\) in terms of \(k\).
  2. Hence find the set of values that \(k\) can take.
OCR C1 Specimen Q4
4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = 4 x ^ { 3 } - 1\),
  2. \(y = x ^ { 2 } \left( x ^ { 2 } + 2 \right)\),
  3. \(y = \sqrt { } x\)
OCR C1 Specimen Q5
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 Specimen Q6
6
  1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
  2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
  3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
  4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.
OCR C1 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{5fa27228-37b2-45d9-a8dc-355b2f7f6fa4-3_757_810_1050_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.
  1. Calculate the radius of the circle and the coordinates of the centre.
  2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
  3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).
OCR C1 Specimen Q8
8
  1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
  4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.
OCR C1 Q1
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8$$
OCR C1 Q2
  1. Express
$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.
OCR C1 Q3
3. A circle has the equation $$x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0$$
  1. Find the coordinates of the centre of the circle.
  2. Find the radius of the circle.