Questions SPS SM Pure (200 questions)

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SPS SPS SM Pure 2020 February Q1
4 marks
1
  1. Given that $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ show that $$x ^ { 2 } - 4 x - 12 = 0$$ [4]
    1. Write down the roots of the equation \(x ^ { 2 } - 4 x - 12 = 0\).
    2. State which of these two roots is not a solution of $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ giving a reason for your answer.
SPS SPS SM Pure 2020 February Q2
2
  1. Using integration by parts, find the indefinite integral, with respect to \(x\), of $$x \cos x$$
  2. Using the substitution \(u ^ { 2 } = 2 x + 1\), find the indefinite integral, with respect to $$x , \text { of } \frac { 6 x } { \sqrt { 2 x + 1 } } .$$
SPS SPS SM Pure 2020 February Q3
5 marks
3 In this question you must show detailed reasoning.
  1. Write down the first 5 terms of the geometric series $$\sum _ { r = 0 } ^ { n } 20 \times 0.5 ^ { r }$$
  2. Find the smallest value of \(n\) for which the series $$\sum _ { r = 0 } ^ { n } 20 \times 0.5 ^ { r }$$ is greater than \(99.8 \%\) of its sum to infinity.
    [0pt] [5]
SPS SPS SM Pure 2020 February Q4
5 marks
4
  1. Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\tan ^ { 2 } \theta + 1 \equiv \sec ^ { 2 } \theta\).
    A curve is given parametrically by $$x = a \sec \theta , \quad y = a \tan \theta$$ where \(a\) is a constant.
  2. Find a Cartesian equation of the curve.
  3. Determine an equation of the tangent to the curve at the point \(\theta = \frac { \pi } { 3 }\), giving your answer in exact form.
    [0pt] [5]
SPS SPS SM Pure 2020 February Q5
1 marks
5
  1. Find the first three non-zero terms of the expansion, in ascending powers of \(x\), of \(( 4 + x ) ^ { \frac { 1 } { 2 } }\).
  2. State the range of values of \(x\) for which your expansion in part (a) is valid. [1]
  3. Use your expansion to determine an approximation to \(\sqrt { } 36.9\), showing all the figures on your calculator.
SPS SPS SM Pure 2020 February Q6
6 In this question you must show detailed reasoning.
  1. Given that \(y = \mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } - 16 x ^ { 2 } + 7 x - 3\), determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } < 0$$
  2. State the geometrical interpretation of your answer to part (a), in terms of the shape of the graph of \(y = \mathrm { f } ( x )\) (not the gradient of the graph).
SPS SPS SM Pure 2020 February Q7
2 marks
7 A scientist is studying the flight of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 140 } { A + 45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } } , \quad 0 \leq t \leq T ,$$ where \(t\) seconds is the time after the bird leaves its nest, and \(A\) and \(T\) are constants. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{022274c9-7ed2-4436-ae97-d410d7d566fc-10_559_679_513_678} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 is a sketch showing the graph of \(H\) against \(t\).
It is given that this seabird's nest is \(\mathbf { 2 0 } \mathbf { ~ m }\) above sea level.
  1. Show that \(A = 35\).
    It is also given that $$45 \sin 2 t ^ { \circ } - 28 \cos 2 t ^ { \circ } \equiv 53 \sin ( 2 t - \alpha ) ^ { \circ }$$ where \(\alpha\) is a constant in the range \(0 < \alpha < 90\).
  2. Find the value of \(\alpha\) to one decimal place.
    Find, according to this model,
  3. the minimum height of the sea bird above sea level, giving your answer to the nearest cm,
    [0pt] [2]
  4. the limitation on the value of \(T\).
SPS SPS SM Pure 2020 February Q8
8 Differentiate from first principles $$y = \frac { 1 } { x }$$
SPS SPS SM Pure 2020 February Q9
9
  1. Describe fully a sequence of transformations that map the line \(y = x\) onto the line $$y = 10 - 2 x$$ The function f is defined as \(\mathrm { f } : x \rightarrow 10 - 2 x , x \in R , x \geq 0\).
    The function ff is denoted by g .
  2. Find \(\mathrm { g } ( x )\), giving your answer in a form without brackets.
  3. Determine the domain of g .
  4. Explain whether \(\mathrm { fg } = \mathrm { gf }\).
  5. Find \(\mathrm { g } ^ { - 1 }\) in the form \(\mathrm { g } ^ { - 1 } : x \rightarrow \ldots\)
SPS SPS SM Pure 2020 February Q10
10 The diagram shows part of the graphs of \(y = \cos ^ { 2 } x\) and \(y = 5 \sin 2 x\) for small positive values of \(x\). The graphs meet at the point \(A\) with \(x\)-coordinate \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{022274c9-7ed2-4436-ae97-d410d7d566fc-14_652_561_402_735}
  1. Find the exact area contained between the two graphs (between \(x = 0\) and \(x = \alpha\) ) and the \(y\)-axis. Give your answer in terms of \(\alpha , \cos 2 \alpha\) and/or \(\sin 2 \alpha\).
  2. Using the fact that \(\alpha\) is a small positive solution to the equation \(\cos ^ { 2 } x = 5 \sin 2 x\), show that \(\alpha\) satisfies approximately the equation \(\alpha ^ { 2 } + 10 \alpha - 1 = 0\), if terms in \(\alpha ^ { 3 }\) and higher are ignored.
  3. Use the equation in part (b) to find an approximate value of \(\alpha\), correct to 3 significant figures.
SPS SPS SM Pure 2020 February Q11
11 Two circles have equations
\(x ^ { 2 } + y ^ { 2 } - 4 x = 0 \quad\) and
\(x ^ { 2 } + y ^ { 2 } - 6 x - 12 y + 36 = 0\).
  1. Find the centre and radius of each circle and hence show that the \(y\)-axis is a tangent to both circles.
  2. Find the equation of the line through the centres of both circles.
  3. Determine the gradient of a line other than the \(y\)-axis which is a tangent to both circles.
    \section*{ADDITIONAL ANSWER SPACE} If additional space is required, you should use the following lined page(s). The question number(s) must be clearly shown. nunber(s) must be clearly shown.
SPS SPS SM Pure 2021 June Q1
1. A curve has equation $$y = 2 x ^ { 3 } - 4 x + 5$$ Find the equation of the tangent to the curve at the point \(P ( 2,13 )\).
Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
Solutions relying on calculator technology are not acceptable.
SPS SPS SM Pure 2021 June Q2
2. Given that the point \(A\) has position vector \(3 \mathbf { i } - 7 \mathbf { j }\) and the point \(B\) has position vector \(8 \mathbf { i } + 3 \mathbf { j }\),
  1. find the vector \(\overrightarrow { A B }\)
  2. Find \(| \overrightarrow { A B } |\). Give your answer as a simplified surd.
SPS SPS SM Pure 2021 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{106e8e6b-912d-45ef-89e9-12ffc04bfd25-06_369_661_155_760} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shape \(A B C D O A\), as shown in Figure 1, consists of a sector \(C O D\) of a circle centre \(O\) joined to a sector \(A O B\) of a different circle, also centre \(O\). Given that arc length \(C D = 3 \mathrm {~cm} , \angle C O D = 0.4\) radians and \(A O D\) is a straight line of length 12 cm ,
  1. find the length of \(O D\),
  2. find the area of the shaded sector \(A O B\).
SPS SPS SM Pure 2021 June Q4
4. The function f is defined by $$\mathrm { f } ( x ) = \frac { 3 x - 7 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$
  1. Find \(f ^ { - 1 } ( 7 )\)
  2. Show that \(\mathrm { ff } ( x ) = \frac { a x + b } { x - 3 }\) where \(a\) and \(b\) are integers to be found.
SPS SPS SM Pure 2021 June Q5
5. A car has six forward gears. The fastest speed of the car
  • in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
  • in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,
  1. find the fastest speed of the car in \(3 ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
  2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.
SPS SPS SM Pure 2021 June Q6
6. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
SPS SPS SM Pure 2021 June Q7
7. Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)
  1. show that \(3 k + 5 \sqrt { k } - 12 = 0\)
  2. Hence, using algebra, find any values of \(k\) such that $$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$ [BLANK PAGE]
  3. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 2\), find \(\mathrm { f } ( 3 + h )\) Express your answer in the form \(h ^ { 2 } + b h + c\), where \(b\) and \(c \in \mathbb { Z }\).
  4. The curve with equation \(y = x ^ { 2 } - 4 x + 2\) passes through the point \(P ( 3 , - 1 )\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{106e8e6b-912d-45ef-89e9-12ffc04bfd25-14_895_1337_207_468} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.
  1. State the value of \(c\) and the value of \(d\).
  2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
    1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
    2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
  3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 June Q10
10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$
  1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
  2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
  3. Find, using algebraic integration, the exact value of the area of \(R\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 June Q11
11.
  1. A circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$ The line \(l\) is the tangent to \(C _ { 1 }\) at the point \(P ( - 5,7 )\).
    Find an equation of \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
  2. A different circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$ where \(k\) is a constant.
    Given that \(C _ { 2 }\) lies entirely in the 4th quadrant, find the range of possible values for \(k\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 June Q12
12. Solve the equation $$\sin \theta \tan \theta + 2 \sin \theta = 3 \cos \theta \quad \text { where } \cos \theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
Fully justify your answer.
SPS SPS SM Pure 2021 June Q13
13.
  1. Prove that for all positive values of \(x\) and \(y\) $$\sqrt { x y } \leqslant \frac { x + y } { 2 }$$
  2. Prove by counter example that this is not true when \(x\) and \(y\) are both negative. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{106e8e6b-912d-45ef-89e9-12ffc04bfd25-22_844_1427_278_406} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A town's population, \(P\), is modelled by the equation \(P = a b ^ { t }\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log _ { 10 } P\) for the population over a period of 100 years.
    The line \(l\) meets the vertical axis at \(( 0,5 )\) as shown. The gradient of \(l\) is \(\frac { 1 } { 200 }\).
  3. Write down an equation for \(l\).
  4. Find the value of \(a\) and the value of \(b\).
  5. With reference to the model interpret
    1. the value of the constant \(a\),
    2. the value of the constant \(b\).
  6. Find
    1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
    2. the number of years it takes the population to reach 200000 , according to the model.
  7. State two reasons why this may not be a realistic population model.
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 June Q15
15. A curve has equation \(y = \mathrm { g } ( x )\). Given that
  • \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
  • the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
  • the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
    1. find \(g ( x )\),
    2. prove that the stationary point at \(( 2,9 )\) is a maximum.
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2021 May Q1
  1. The function f is defined for all non-negative values of \(x\) by
$$\mathrm { f } ( x ) = 3 + \sqrt { x }$$
  1. Evaluate \(\mathrm { ff } ( 169 )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\).
  3. On a single diagram sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), indicating how the two graphs are related.
    [0pt] [BLANK PAGE]