Questions — OCR (4619 questions)

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OCR C1 Q8
\begin{enumerate} \setcounter{enumi}{7} \item \(f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }\).
  1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Determine whether each stationary point is a maximum or minimum point.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions. \item The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.
OCR C1 Q10
10.
\includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).
  1. Find the gradient of 1 .
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the centre of \(C\).
  4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$
OCR C1 Q1
\begin{enumerate} \item (i) Express \(\frac { 21 } { \sqrt { 7 } }\) in the form \(k \sqrt { 7 }\).
(ii) Express \(8 ^ { - \frac { 1 } { 3 } }\) as an exact fraction in its simplest form. \item Find \(\frac { d y } { d x }\) when
OCR C1 Q5
5. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,
  1. show that $$k ^ { 2 } - 16 k + 48 \geq 0$$
  2. find the set of possible values of \(k\),
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.
OCR C1 Q6
6. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\),
  2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0$$ The point \(R\) has coordinates (2, 7).
  3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.
OCR C1 Q7
7. (i) Describe fully the single transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( x - 1 )\).
(ii) Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac { 1 } { x - 1 }\).
(iii) Find the \(x\)-coordinates of any points where the graph of \(y = \frac { 1 } { x - 1 }\) intersects the graph of \(y = 2 + \frac { 1 } { x }\). Give your answers in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational.
OCR C1 Q8
8.
\includegraphics[max width=\textwidth, alt={}, center]{98667bd4-a612-4b16-a75b-8d8637e5976d-3_611_828_251_392} The diagram shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
  2. Find an equation for \(l\).
  3. Find the coordinates of the point where \(l\) intersects \(C\) again.
OCR C1 Q9
9. The curve with equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 8 x ^ { \frac { 1 } { 2 } }\) has a minimum at the point \(A\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the \(x\)-coordinate of \(A\). The point \(B\) on the curve has \(x\)-coordinate 2 .
  3. Find an equation for the tangent to the curve at \(B\) in the form \(y = m x + c\).
OCR C1 Q1
  1. (i) Calculate the discriminant of \(2 x ^ { 2 } + 8 x + 8\).
    (ii) State the number of real roots of the equation \(2 x ^ { 2 } + 8 x + 8 = 0\).
  2. Find the set of values of \(x\) for which
$$( x - 1 ) ( x - 2 ) < 20 .$$
OCR C1 Q3
  1. (i) Solve the equation
$$x ^ { \frac { 3 } { 2 } } = 27 .$$ (ii) Express \(\left( 2 \frac { 1 } { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an exact fraction in its simplest form.
OCR C1 Q4
4. Differentiate with respect to \(x\) $$\frac { 6 x ^ { 2 } - 1 } { 2 \sqrt { x } } .$$
OCR C1 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{129a65ac-e77c-4274-a2b9-18825ea2302c-1_547_936_1407_351}
The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations
  1. \(\quad y = 2 \mathrm { f } ( x )\),
  2. \(y = - \mathrm { f } ( x )\).
OCR C1 Q6
6. \(f ( x ) = 2 x ^ { 2 } - 4 x + 1\).
  1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\mathrm { f } ( x ) = a ( x + b ) ^ { 2 } + c$$
  2. State the equation of the line of symmetry of the curve \(y = \mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 3\), giving your answers in exact form.
OCR C1 Q7
7. A curve has the equation $$y = x ^ { 3 } + a x ^ { 2 } - 15 x + b$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \(( - 1,12 )\),
  1. find the values of \(a\) and \(b\),
  2. find the coordinates of the other stationary point of the curve.
OCR C1 Q8
8. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),
  1. find the value of \(k\),
  2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
  3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).
OCR C1 Q9
9. A curve has the equation \(y = x + \frac { 3 } { x } , x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate 1 .
  1. Show that the gradient of the curve at \(P\) is - 2 .
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = m x + c\).
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again.
OCR C1 Q10
10. The straight line \(l _ { 1 }\) has equation \(2 x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). The straight line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(B ( - 6,6 )\).
  2. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
  3. Find the coordinates of \(C\). The point \(D\) lies on \(l _ { 1 }\) and is such that \(C D\) is perpendicular to \(l _ { 1 }\).
  4. Show that \(D\) has coordinates \(( 5,4 )\).
  5. Find the area of triangle \(A C D\).
OCR S2 2012 January Q1
1 A random sample of 50 observations of the random variable \(X\) is summarised by $$n = 50 , \Sigma x = 182.5 , \Sigma x ^ { 2 } = 739.625 .$$ Calculate unbiased estimates of the expectation and variance of \(X\).
OCR S2 2012 January Q2
2 The random variable \(Y\) has the distribution \(\mathrm { B } ( 140,0.03 )\). Use a suitable approximation to find \(\mathrm { P } ( Y = 5 )\). Justify your approximation.
OCR S2 2012 January Q3
3 The random variable \(G\) has a normal distribution. It is known that $$\mathrm { P } ( G < 56.2 ) = \mathrm { P } ( G > 63.8 ) = 0.1 \text {. }$$ Find \(\mathrm { P } ( G > 65 )\).
OCR S2 2012 January Q4
4 The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that \(\mathrm { E } ( H ) = 2.5\) and \(\operatorname { Var } ( H ) = 1.25\). The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar { H }\).
Use a suitable approximation to find \(\mathrm { P } ( \bar { H } < 2.6 )\).
OCR S2 2012 January Q5
5
  1. Six prizes are allocated, using random numbers, to a group of 12 girls and 8 boys. Calculate the probability that exactly 4 of the prizes are allocated to girls if
    (a) the same child may win more than one prize,
    (b) no child may win more than one prize.
  2. Sixty prizes are allocated, using random numbers, to a group of 1200 girls and 800 boys. Use a suitable approximation to calculate the probability that at least 30 of the prizes are allocated to girls. Does it affect your calculation whether or not the same child may win more than one prize? Justify your answer.
OCR S2 2012 January Q6
6 The number of fruit pips in 1 cubic centimetre of raspberry jam has the distribution \(\operatorname { Po } ( \lambda )\). Under a traditional jam-making process it is known that \(\lambda = 6.3\). A new process is introduced and a random sample of 1 cubic centimetre of jam produced by the new process is found to contain 2 pips. Test, at the \(5 \%\) significance level, whether this is evidence that under the new process the average number of pips has been reduced. Find (a) \(\mathrm { E } ( X )\),
(ii) The continuous random variable \(Y\) has the probability density function $$g ( y ) = \left\{ \begin{array} { l r } \frac { 1.5 } { y ^ { 2.5 } } & y \geqslant 1
0 & \text { otherwise. } \end{array} \right.$$ Given that \(\mathrm { E } ( Y ) = 3\), show that \(\operatorname { Var } ( Y )\) is not finite.
OCR S2 2012 January Q8
8 In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.
  1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. Assume now that a Poisson model is appropriate.
  2. Find the probability that in 10 ml there are at least 5 bacteria.
  3. Find the probability that in 3.7 ml there are exactly 2 bacteria.
  4. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation.
OCR S2 2012 January Q9
9 It is desired to test whether the average amount of sleep obtained by school pupils in Year 11 is 8 hours, based on a random sample of size 64. The population standard deviation is 0.87 hours and the sample mean is denoted by \(\bar { H }\). The critical values for the test are \(\bar { H } = 7.72\) and \(\bar { H } = 8.28\).
  1. State appropriate hypotheses for the test, explaining the meaning of any symbol you use.
  2. Calculate the significance level of the test.
  3. Explain what is meant by a Type I error in this context.
  4. Given that in fact the average amount of sleep obtained by all pupils in Year 11 is 7.9 hours, find the probability that the test results in a Type II error. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}