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OCR MEI C1 Q12
12 marks Standard +0.3
12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the other factors of \(\mathrm { f } ( x )\).
  3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
  4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
  5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
    Solve this equation to find the \(x\)-coordinates of A and B .
OCR MEI C1 Q13
12 marks Standard +0.3
13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-5_409_768_383_604} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure}
  1. Find the coordinates of X .
  2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
  3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
    Write down the equation of the circumcircle of the triangle ABC .
  4. Find the coordinates of the points where the circle cuts the \(x\) axis.
OCR C1 Q1
3 marks Easy -1.2
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8$$
OCR C1 Q2
3 marks Easy -1.2
  1. Express
$$\frac { 2 } { 3 \sqrt { 5 } + 7 }$$ in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational.
OCR C1 Q3
4 marks Moderate -0.8
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.
OCR C1 Q4
5 marks Easy -1.2
4.
  1. Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
  2. State the coordinates of the vertex of the curve \(y = x ^ { 2 } + 6 x + 7\).
OCR C1 Q5
7 marks Standard +0.3
5. Solve the simultaneous equations $$\begin{aligned} & x + y = 2 \\ & 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$
OCR C1 Q6
8 marks Moderate -0.3
6.
\includegraphics[max width=\textwidth, alt={}]{e90356f2-7485-4a25-80c5-84e48ceddd62-2_472_753_248_456}
The diagram shows the curve with equation \(y = 3 x - x ^ { \frac { 3 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) and has a maximum at the point \(B\).
  1. Find the \(x\)-coordinate of \(A\).
  2. Find the coordinates of \(B\).
OCR C1 Q7
9 marks Moderate -0.8
7.
  1. Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
  2. State the number of real roots of the equation \(x ^ { 2 } - 6 x + 12 = 0\) and hence, explain why \(x ^ { 2 } - 6 x + 12\) is always positive.
  3. Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).
OCR C1 Q8
9 marks Moderate -0.3
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).
OCR C1 Q9
10 marks Moderate -0.3
9. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2 .
  1. Find the gradient of the curve at \(A\).
  2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
  3. Find the coordinates of \(B\).
OCR C1 Q10
14 marks Standard +0.3
10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).
  1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
    Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
  2. find the coordinates of \(B\) and \(C\),
  3. show that \(\angle B A C = 90 ^ { \circ }\),
  4. find the area of triangle \(A B C\).
OCR S2 2007 January Q1
4 marks Moderate -0.8
1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
OCR S2 2007 January Q2
5 marks Moderate -0.8
2 A school has 900 pupils. For a survey, Jan obtains a list of all the pupils, numbered 1 to 900 in alphabetical order. She then selects a sample by the following method. Two fair dice, one red and one green, are thrown, and the number in the list of the first pupil in the sample is determined by the following table.
\cline { 3 - 8 } \multicolumn{2}{c|}{}Score on green dice
\cline { 3 - 8 } \multicolumn{2}{c|}{}123456
Score on
red dice
1,2 or 3123456
For example, if the scores on the red and green dice are 5 and 2 respectively, then the first member of the sample is the pupil numbered 8 in the list. Starting with this first number, every 12th number on the list is then used, so that if the first pupil selected is numbered 8 , the others will be numbered \(20,32,44 , \ldots\).
  1. State the size of the sample.
  2. Explain briefly whether the following statements are true.
    1. Each pupil in the school has an equal probability of being in the sample.
    2. The pupils in the sample are selected independently of one another.
    3. Give a reason why the number of the first pupil in the sample should not be obtained simply by adding together the scores on the two dice. Justify your answer.
OCR S2 2007 January Q3
6 marks Moderate -0.5
3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.
OCR S2 2007 January Q4
7 marks Moderate -0.8
4 A set of observations of a random variable \(W\) can be summarised as follows: $$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$
  1. Calculate an unbiased estimate of the variance of \(W\).
  2. The mean of 70 observations of \(W\) is denoted by \(\bar { W }\). State the approximate distribution of \(\bar { W }\), including unbiased estimate(s) of any parameter(s).
OCR S2 2007 January Q5
12 marks Standard +0.3
5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)
  1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
  2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
  3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.
OCR S2 2007 January Q6
13 marks Standard +0.3
6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(2 a + 2 b = 1\).
  2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
  3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).
OCR S2 2007 January Q7
11 marks Standard +0.3
7 A television company believes that the proportion of households that can receive Channel C is 0.35 .
  1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
  2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
OCR S2 2007 January Q8
14 marks Challenging +1.8
8 The quantity, \(X\) milligrams per litre, of silicon dioxide in a certain brand of mineral water is a random variable with distribution \(\mathrm { N } \left( \mu , 5.6 ^ { 2 } \right)\).
  1. A random sample of 80 observations of \(X\) has sample mean 100.7. Test, at the \(1 \%\) significance level, the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 102\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 102\).
  2. The test is redesigned so as to meet the following conditions.
    • The hypotheses are \(\mathrm { H } _ { 0 } : \mu = 102\) and \(\mathrm { H } _ { 1 } : \mu < 102\).
    • The significance level is \(1 \%\).
    • The probability of making a Type II error when \(\mu = 100\) is to be (approximately) 0.05 .
    The sample size is \(n\), and the critical region is \(\bar { X } < c\), where \(\bar { X }\) denotes the sample mean.
    1. Show that \(n\) and \(c\) satisfy (approximately) the equation \(102 - c = \frac { 13.0256 } { \sqrt { n } }\).
    2. Find another equation satisfied by \(n\) and \(c\).
    3. Hence find the values of \(n\) and \(c\).
OCR C1 Q1
3 marks Moderate -0.8
  1. Find the set of values of the constant \(k\) such that the equation
$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.
OCR C1 Q2
4 marks Moderate -0.8
2. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively. Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 Q3
4 marks Easy -1.2
3.
  1. Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
  2. Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q4
4 marks Moderate -0.5
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
7 marks Moderate -0.5
  1. Given that
$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).