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 2006 January Q3
3 Given that \(y = 3 x ^ { 5 } - \sqrt { x } + 15\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2006 January Q4
4
  1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
  2. Hence sketch the curve \(y = \frac { 1 } { ( x - 3 ) ^ { 2 } }\).
  3. Describe fully a transformation that transforms the curve \(y = \frac { 1 } { x ^ { 2 } }\) to the curve \(y = \frac { 2 } { x ^ { 2 } }\).
OCR C1 2006 January Q5
5
  1. Express \(x ^ { 2 } + 3 x\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Express \(y ^ { 2 } - 4 y - \frac { 11 } { 4 }\) in the form \(( y + p ) ^ { 2 } + q\). A circle has equation \(x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0\).
  3. Write down the coordinates of the centre of the circle.
  4. Find the radius of the circle.
OCR C1 2006 January Q6
6
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } + 4\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. For what values of \(x\) does \(x ^ { 3 } - 3 x ^ { 2 } + 4\) increase as \(x\) increases?
OCR C1 2006 January Q7
7
  1. Solve the equation \(x ^ { 2 } - 8 x + 11 = 0\), giving your answers in simplified surd form.
  2. Hence sketch the curve \(y = x ^ { 2 } - 8 x + 11\), labelling the points where the curve crosses the axes.
  3. Solve the equation \(y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0\), giving your answers in the form \(p \pm q \sqrt { 5 }\).
OCR C1 2006 January Q8
8
  1. Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
  2. Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
  3. What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
  4. Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
  5. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 2006 January Q9
9 The points \(A , B\) and \(C\) have coordinates \(( 5,1 ) , ( p , 7 )\) and \(( 8,2 )\) respectively.
  1. Given that the distance between points \(A\) and \(B\) is twice the distance between points \(A\) and \(C\), calculate the possible values of \(p\).
  2. Given also that the line passing through \(A\) and \(B\) has equation \(y = 3 x - 14\), find the coordinates of the mid-point of \(A B\).
OCR C1 2007 January Q1
1 Express \(\frac { 5 } { 2 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
OCR C1 2007 January Q2
2 Evaluate
  1. \(6 ^ { 0 }\),
  2. \(2 ^ { - 1 } \times 32 ^ { \frac { 4 } { 5 } }\).
OCR C1 2007 January Q3
3 Solve the inequalities
  1. \(3 ( x - 5 ) \leqslant 24\),
  2. \(5 x ^ { 2 } - 2 > 78\).
OCR C1 2007 January Q4
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } + 3 x ^ { \frac { 1 } { 3 } } - 10 = 0\).
OCR C1 2007 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{82ae6eec-3007-467c-90df-18f2adb9ccb1-2_634_926_1242_612} The graph of \(y = \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\) is shown above.
  1. Sketch the graph of \(y = - \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\).
  2. The point \(P ( 1,1 )\) on \(y = \mathrm { f } ( x )\) is transformed to the point \(Q\) on \(y = 3 \mathrm { f } ( x )\). State the coordinates of \(Q\).
  3. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x + 2 )\).
OCR C1 2007 January Q6
6
  1. Express \(2 x ^ { 2 } - 24 x + 80\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } - 24 x + 80\).
  3. State the equation of the tangent to the curve \(y = 2 x ^ { 2 } - 24 x + 80\) at its minimum point.
OCR C1 2007 January Q7
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases.
  1. \(y = 5 x + 3\)
  2. \(y = \frac { 2 } { x ^ { 2 } }\)
  3. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)
OCR C1 2007 January Q8
8
  1. Find the coordinates of the stationary points of the curve \(y = 27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Determine, in each case, whether the stationary point is a maximum or minimum point.
  3. Hence state the set of values of \(x\) for which \(27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\) is an increasing function.
    \(9 \quad A\) is the point \(( 2,7 )\) and \(B\) is the point \(( - 1 , - 2 )\).
OCR C1 2007 January Q10
10 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 8 = 0\).
  1. Find the centre and radius of the circle.
  2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
  3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).
OCR C1 2008 January Q1
1 Express \(\frac { 4 } { 3 - \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2008 January Q2
2
  1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
  2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.
OCR C1 2008 January Q3
3 Given that \(3 x ^ { 2 } + b x + 10 = a ( x + 3 ) ^ { 2 } + c\) for all values of \(x\), find the values of the constants \(a , b\) and \(c\).
OCR C1 2008 January Q4
4 Solve the equations
  1. \(10 ^ { p } = 0.1\),
  2. \(\left( 25 k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } = 15\),
  3. \(t ^ { - \frac { 1 } { 3 } } = \frac { 1 } { 2 }\).
OCR C1 2008 January Q5
5
  1. Sketch the curve \(y = x ^ { 3 } + 2\).
  2. Sketch the curve \(y = 2 \sqrt { x }\).
  3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).
OCR C1 2008 January Q6
6
  1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
  2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
  3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).
OCR C1 2008 January Q7
7
  1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
  2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$
OCR C1 2008 January Q8
8
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. For what values of \(x\) does \(x ^ { 3 } + x ^ { 2 } - x + 3\) decrease as \(x\) increases?
OCR C1 2008 January Q9
9 The points \(A\) and \(B\) have coordinates \(( - 5 , - 2 )\) and \(( 3,1 )\) respectively.
  1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
  2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
  3. Calculate the length of \(A C\), giving your answer in simplified surd form.
  4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.