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OCR C1 2006 January Q1
4 marks Easy -1.8
1 Solve the equations
  1. \(x ^ { \frac { 1 } { 3 } } = 2\),
  2. \(10 ^ { \prime } = 1\),
  3. \(\left( y ^ { - 2 } \right) ^ { 2 } = \frac { 1 } { 81 }\).
OCR C1 2006 January Q2
5 marks Easy -1.2
2
  1. Simplify \(( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$
OCR C1 2006 January Q3
5 marks Easy -1.2
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
7 marks Easy -1.2
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
7 marks Moderate -0.8
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
11 marks Moderate -0.8
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
11 marks Standard +0.3
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
11 marks Moderate -0.3
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
11 marks Moderate -0.3
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 MEI S1 Q1
18 marks Standard +0.3
1 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.
  1. Copy and complete the tree diagram to show this information. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-1_833_1156_851_573}
  2. Find the probability that a randomly selected person tests negative and is clear.
  3. Find the probability that a randomly selected person has the disease.
  4. Find the probability that a randomly selected person tests negative given that the person has the disease.
  5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
  6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .
  1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
  2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
  3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
    (A) wears either a jacket or a tie (or both),
    (B) wears no tie or no jacket (or wears neither).
OCR MEI S1 Q3
8 marks Standard +0.3
3 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.
  • \(A\) is the event that Isobel's parents watch a match.
  • \(\quad B\) is the event that Isobel scores in a match.
You are given that \(\frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).
  1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
  2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
  3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
  4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.
OCR MEI S1 Q4
18 marks Moderate -0.5
4 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .
  1. Copy and complete the probability tree diagram below, to illustrate the information above. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-3_466_667_834_737} Calculate the probabilities of the following events.
  2. The test gives a positive result.
  3. The test gives a correct result.
  4. The painting is genuine, given a positive result.
  5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
  6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
  7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.
OCR C1 2007 January Q1
3 marks Easy -1.2
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
4 marks Easy -1.8
2 Evaluate
  1. \(6 ^ { 0 }\),
  2. \(2 ^ { - 1 } \times 32 ^ { \frac { 4 } { 5 } }\).
OCR C1 2007 January Q3
5 marks Easy -1.2
3 Solve the inequalities
  1. \(3 ( x - 5 ) \leqslant 24\),
  2. \(5 x ^ { 2 } - 2 > 78\).
OCR C1 2007 January Q4
5 marks Moderate -0.5
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } + 3 x ^ { \frac { 1 } { 3 } } - 10 = 0\).
OCR C1 2007 January Q5
6 marks Easy -1.2
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 marks Moderate -0.8
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
8 marks Easy -1.3
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
11 marks Moderate -0.3
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
12 marks Moderate -0.3
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 MEI S1 Q1
5 marks Moderate -0.8
1 A school athletics team has 10 members. The table shows which competitions each of the members can take part in.
Competiton
100 m200 m110 m hurdles400 mLong jump
\multirow{10}{*}{Athlete}Abel✓✓✓
Bernoulli✓✓
Cauchy✓✓✓
Descartes✓✓
Einstein✓✓
Fermat✓✓
Galois✓✓
Hardy✓✓✓
Iwasawa✓✓
Jacobi✓
An athlete is selected at random. Events \(A , B , C , D\) are defined as follows. \(A\) : the athlete can take part in exactly 2 competitions. \(B\) : the athlete can take part in the 200 m . \(C\) : the athlete can take part in the 110 m hurdles. \(D\) : the athlete can take part in the long jump.
  1. Write down the value of \(\mathrm { P } ( A \cap B )\).
  2. Write down the value of \(\mathrm { P } ( C \cup D )\).
  3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
  4. Show that events \(B\) and \(D\) are not independent.
OCR MEI S1 Q2
18 marks Standard +0.3
2 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.
  1. All 3 doughnuts eaten contain jam.
  2. All 3 doughnuts are of the same kind.
  3. The 3 doughnuts are all of a different kind.
  4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
  5. exactly 2 Saturdays out of the 5 ,
  6. at least 1 Saturday out of the 5 .
OCR MEI S1 Q5
3 marks Standard +0.3
5 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-5_480_771_452_655} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).