Questions — SPS (1106 questions)

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SPS SPS SM Pure 2020 October Q9
9. Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$
  1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\).
  2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures.
SPS SPS SM Pure 2020 October Q10
10.
  1. Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
  2. Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
SPS SPS FM 2022 October Q1
1.
a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 5 }$$ giving each term in its simplest form.
b) Hence write down the first 3 terms, in ascending powers of \(y\), of the binomial expansion of $$\left( 2 + 3 y ^ { \frac { 3 } { 2 } } \right) ^ { 5 }$$
SPS SPS FM 2022 October Q2
  1. 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.
  2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$ [BLANK PAGE]
SPS SPS FM 2022 October Q3
3.
  1. Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
  2. Sketch the graph of \(y = \frac { 3 } { x }\) in the space provided and write down the equations of any asymptotes.
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q4
4. Prove, from first principles, that if \(f ( x ) = 2 x ^ { 2 } - 5 x + 2\) then \(f ^ { \prime } ( x ) = 4 x - 5\).
(3)
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SPS SPS FM 2022 October Q5
5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$ [BLANK PAGE]
SPS SPS FM 2022 October Q6
6. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm}\), \(C D = 8 \mathrm {~cm} , A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).
  1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
  2. Find the size of \(\angle B C D\) in degrees.
  3. Find the area of quadrilateral \(A B C D\).
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SPS SPS FM 2022 October Q7
2 marks
7. (i) Solve the inequality \(| 2 x + 1 | \leqslant | x - 3 |\).
(ii) Given that \(x\) satisfies the inequality \(| 2 x + 1 | \leqslant | x - 3 |\), find the greatest possible value of \(| x + 2 |\).
[0pt] [2]
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SPS SPS FM 2022 October Q8
8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),
  1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
  2. find the value of \(a\),
    [0pt] [BLANK PAGE]
SPS SPS FM 2022 October Q9
9. Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
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SPS SPS FM 2022 October Q10
10. 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 .
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SPS SPS FM 2022 October Q1
  1. a) Solve the inequality:
$$\frac { x - 9 } { 2012 } + \frac { x - 8 } { 2013 } + \frac { x - 7 } { 2014 } + \frac { x - 6 } { 2015 } + \frac { x - 5 } { 2016 } \leq \frac { x - 2012 } { 9 } + \frac { x - 2013 } { 8 } + \frac { x - 2014 } { 7 } + \frac { x - 2015 } { 6 } + \frac { x - 2016 } { 5 }$$ b) Find all ( \(x , y , z\) ) such that: $$\frac { 1 } { x } + \frac { 1 } { y + z } = \frac { 1 } { 3 } , \quad \frac { 1 } { y } + \frac { 1 } { z + x } = \frac { 1 } { 5 } , \quad \frac { 1 } { z } + \frac { 1 } { x + y } = \frac { 1 } { 7 }$$ [Question 1 - Continued]
[0pt] [Question 1 - Continued]
SPS SPS FM 2022 October Q2
2. A function is defined by: $$f ( x ) = \sqrt { \frac { 1 - x } { 1 + x } } , x \in \mathbb { R } , | x | < 1$$ a) P and Q are points on the curve with \(x\)-coordinates \(x\) and \(x + h\) respectively. Find the gradient of the line segment PQ . Simplify your answer to a single fraction.
b) Use differentiation from first principles to show that: $$f ^ { \prime } ( x ) = - \frac { 1 } { ( 1 + x ) \sqrt { 1 - x ^ { 2 } } }$$ c) Sketch the curve on the axes provided over the page, showing clearly the behaviour of the curve near \(x = 0\) and \(x = \pm 1\).
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
[0pt] [Question 2 - Continued]
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SPS SPS FM 2022 October Q3
3. If for some \(x , y \in \mathbb { R }\) we have \(| x + y | + | x - y | = 2\), find the maximal value of \(x ^ { 2 } - 6 x + y ^ { 2 }\).
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]
[0pt] [Question 3 - Continued]
SPS SPS FM 2022 October Q4
4. A sequence is defined by \(u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }\) for \(n \geq 1\).
a) In the case where \(r = \frac { 6 } { 5 }\) find the smallest value of \(n\) such that \(u _ { n } > 10 ^ { 50 }\). A convergent sequence is defined by \(v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }\) for \(n \geq 1\).
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of \(r\), giving your answer in exact form.
c) Evaluate the infinite product: $$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$ [Question 4 - Continued]
[0pt] [Question 4 - Continued]
[0pt] [Question 4 - Continued]
SPS SPS FM 2022 October Q5
5. A cotangent function \(\cot x\) is defined as \(\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }\).
a) If \(- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }\) and \(\cot \alpha = - \frac { 12 } { 5 }\), find the exact value of \(\sin \alpha\) and \(\cos \alpha\).
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle.
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
SPS SPS FM 2022 October Q6
6. A function is defined by: $$f ( x ) = \frac { a x + b } { c x + d } , x \in \mathbb { R } , x \neq - \frac { d } { c }$$ a) Find and simplify an expression for \(f ^ { - 1 } ( x )\), stating the domain. A function is defined by: $$g ( x ) = \frac { x - 6 } { x - 4 } , x \in \mathbb { R } , x \neq 4$$ b) Find \(g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\), stating an appropriate domain for each function.
c) Find \(g ^ { - 1 } ( x ) , g ^ { - 2 } ( x )\) and \(g ^ { - 3 } ( x )\), stating an appropriate domain for each function. NB: \(g ^ { - n } ( x ) = g ^ { - 1 } \left( g ^ { - 1 } \left( \cdots \left( g ^ { - 1 } ( x ) \right) \cdots \right) \right)\) with \(n\) copies of \(g ^ { - 1 }\).
d) State the range of \(g ( x ) , g ^ { 2 } ( x )\) and \(g ^ { 3 } ( x )\). A function is defined (over an appropriate domain) by \(h ( x ) = g ( x ) + g ^ { - 1 } ( x )\).
e) Solve the inequality \(h ( x ) \geq 4\).
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SPS SPS SM 2022 October Q1
  1. (a) Sketch the curve \(y = 3 ^ { - x }\)
    (b) Solve the inequality \(3 ^ { - x } < 27\)
  2. (a) Complete the square for \(1 - 4 x - x ^ { 2 }\)
    (b) Sketch the curve \(y = 1 - 4 x - x ^ { 2 }\), including the coordinates of any maximum or minimum points and the y intercept only.
    [0pt] [BLANK PAGE]
  3. 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
(a) the number of houses built in 1986, the first year of the building programme,
(b) the total number of houses built in the 25 years of the programme.
[0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q4
4. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$ [BLANK PAGE]
SPS SPS SM 2022 October Q5
5. 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 \(\frac { 3 } { 4 }\).
  2. Find, to 2 decimal places, the difference between the 5th and 6th term.
  3. Calculate the sum of the first 7 terms. The sum of the first \(n\) terms of the series is greater than 300 .
  4. Calculate the smallest possible value of \(n\).
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q6
6. \begin{displayquote} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Solve the equation \end{displayquote} $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
  2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$ [BLANK PAGE]
SPS SPS SM 2022 October Q7
7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 61 }\)
  3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)
    [0pt] [BLANK PAGE]
SPS SPS SM 2022 October Q8
8. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
  2. Find the range of possible values for \(k\).
    [0pt] [BLANK PAGE] \section*{9. 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 .
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SPS SPS SM 2021 November Q1
2 marks
  1. Do not use a calculator for this question
Find the value of \(x\) for which \(\sqrt { 3 } \times 3 ^ { x } = \frac { 1 } { 9 }\)
[0pt] [2 marks]