Questions — SPS SPS SM (125 questions)

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SPS SPS SM 2021 February Q3
8 marks Standard +0.3
The Venn diagram shows the probabilities associated with four events, \(A\), \(B\), \(C\) and \(D\) \includegraphics{figure_3}
  1. Write down any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) [1]
  2. Given that \(P(B) = 0.4\) find the value of \(p\) [1]
  3. Given also that \(A\) and \(B\) are independent find the value of \(q\) [2]
  4. Given further that \(P(B'|C) = 0.64\) find
    1. the value of \(r\)
    2. the value of \(s\)
    [4]
SPS SPS SM 2021 February Q4
10 marks Easy -1.3
Each member of a group of 27 people was timed when completing a puzzle. The time taken, \(x\) minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot. \includegraphics{figure_4}
  1. Find the range of the times. [1]
  2. Find the interquartile range of the times. [1]
  3. For these 27 people \(\sum x = 607.5\) and \(\sum x^2 = 17623.25\) calculate the mean time taken to complete the puzzle. [1]
  4. calculate the standard deviation of the times taken to complete the puzzle. [2]
  5. Taruni defines an outlier as a value more than 3 standard deviations above the mean. State how many outliers Taruni would say there are in these data, giving a reason for your answer. [1]
  6. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\). When their times are included with the data of the other 27 people
    Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions. [3]
  7. Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d). [1]
SPS SPS SM 2021 February Q5
10 marks Easy -1.3
Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day. [1 mark]
    2. Find the variance of the number of times he falls off in a day. [1 mark]
    1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
    2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
  3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
    2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]
SPS SPS SM 2021 February Q6
10 marks Standard +0.3
The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(P(D = d)\)\(\frac{k}{10}\)\(\frac{k}{20}\)\(\frac{k}{30}\)\(\frac{k}{40}\)\(\frac{k}{50}\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
  2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
  3. A single observation of \(D\) is made. The value obtained, \(d\), is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\) Find the exact probability that the smallest angle of \(Q\) is more than \(50°\) [5]
SPS SPS SM 2021 February Q7
15 marks Standard +0.3
A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.
  1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. [1]
  2. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients' complaint. [4]
  3. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim N(5, 3.5^2)\) Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes [1]
    2. find \(P(T < 2 | T > 0)\) [3]
    3. hence explain why this normal distribution may not be a good model for \(T\). [1]
  4. The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of \(T > 2\) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. [5]
SPS SPS SM 2021 February Q8
7 marks Standard +0.3
Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.
Type of defectColourFabricSewingSizing
Probability0.250.300.400.05
Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
  1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
  2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]
SPS SPS SM 2022 October Q1
4 marks Easy -1.2
  1. Sketch the curve \(y = 3^{-x}\) [2]
  2. Solve the inequality \(3^{-x} < 27\) [2]
SPS SPS SM 2022 October Q2
6 marks Easy -1.2
  1. Complete the square for \(1 - 4x - x^2\) [3]
  2. Sketch the curve \(y = 1 - 4x - x^2\), including the coordinates of any maximum or minimum points and the y intercept only. [3]
SPS SPS SM 2022 October Q3
7 marks Moderate -0.8
A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
  1. the number of houses built in 1986, the first year of the building programme, [5]
  2. the total number of houses built in the 25 years of the programme. [2]
SPS SPS SM 2022 October Q4
8 marks Standard +0.3
  1. Find the positive value of \(x\) such that $$\log_x 64 = 2$$ [2]
  2. Solve for \(x\) $$\log_2(11 - 6x) = 2\log_2(x - 1) + 3$$ [6]
SPS SPS SM 2022 October Q5
11 marks Moderate -0.3
The first term of a geometric series is 120. The sum to infinity of the series is 480.
  1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
  2. Find, to 2 decimal places, the difference between the 5th and 6th term. [2]
  3. Calculate the sum of the first 7 terms. [2]
The sum of the first \(n\) terms of the series is greater than 300.
  1. Calculate the smallest possible value of \(n\). [4]
SPS SPS SM 2022 October Q6
6 marks Easy -1.2
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
  2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]
SPS SPS SM 2022 October Q7
7 marks Standard +0.8
A sequence is defined by $$u_1 = 3$$ $$u_{n+1} = 2 - \frac{4}{u_n}, \quad n \geq 1$$ Find the exact values of
  1. \(u_2\), \(u_3\) and \(u_4\) [3]
  2. \(u_{61}\) [1]
  3. \(\sum_{i=1}^{99} u_i\) [3]
SPS SPS SM 2022 October Q8
8 marks Standard +0.3
The equation \(k(3x^2 + 8x + 9) = 2 - 6x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11k^2 - 30k - 9 > 0$$ [4]
  2. Find the range of possible values for \(k\). [4]
SPS SPS SM 2022 October Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]
SPS SPS SM 2022 February Q1
6 marks Easy -1.3
  1. Evaluate \(27^{-\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1-\sqrt{5}}{3+\sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]
SPS SPS SM 2022 February Q2
8 marks Moderate -0.3
  1. Solve the equation \(x^4 - 10x^2 + 25 = 0\). [4]
  2. Given that \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\), find \(\frac{dy}{dx}\). [2]
  3. Hence find the number of stationary points on the curve \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\). [2]
SPS SPS SM 2022 February Q3
8 marks Moderate -0.3
Solve each of the following equations, for \(0° \leqslant x \leqslant 180°\).
  1. \(2\sin^2 x = 1 + \cos x\). [4]
  2. \(\sin 2x = -\cos 2x\). [4]
SPS SPS SM 2022 February Q4
8 marks Easy -1.3
  1. By expanding the brackets, show that \((x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12\). [3]
  2. Sketch the curve \(y = x^3 - 6x^2 + 5x + 12\), giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve \(y = -x^3 + 6x^2 - 5x - 12\). Label this curve \(C_2\). [2]
SPS SPS SM 2022 February Q5
6 marks Easy -1.2
The gradient of a curve is given by \(\frac{dy}{dx} = 2x^{-\frac{1}{2}}\), and the curve passes through the point \((4, 5)\). Find the equation of the curve. [6]
SPS SPS SM 2022 February Q6
8 marks Moderate -0.3
The diagram shows the curve \(y = 4 - x^2\) and the line \(y = x + 2\). \includegraphics{figure_6}
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line. [2]
  2. Use integration to find the area of the shaded region bounded by the line and the curve. [6]
SPS SPS SM 2022 February Q7
10 marks Moderate -0.3
The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}
  1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
  2. Find the area of the shaded segment. [3]
  3. Find the perimeter of the shaded segment. [4]
SPS SPS SM 2022 February Q8
9 marks Moderate -0.8
The diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}
  1. Evaluate \(f(-3)\). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
SPS SPS SM 2021 November Q1
8 marks Moderate -0.8
Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.
  1. \(y = \cos x - 2 \sin 2x\) [2]
  2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
  3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]