SPS SPS SM Mechanics (SPS SM Mechanics) 2022 February

1. Answer all the questions.
Find $$\int \left( x ^ { 4 } - 6 x ^ { 2 } + 7 \right) \mathrm { d } x$$ giving your answer in simplest form.
(3)
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2. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
   • meets the \(y\)-axis at the point \(P\)
   • has a minimum turning point at the point \(Q\)
   • Write down
     1. the coordinates of \(P\)
     2. the coordinates of \(Q\)
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3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)

1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
2. Find the value of \(k\), giving a reason for your answer.
3. Find the value of \(u _ { 3 }\)
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4. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
• the point \(B\) has position vector \(7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\)
• the point \(C\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }\)
  1. Find \(| \overrightarrow { A B } |\) giving your answer as a simplified surd.

Given that \(A B C D\) is a parallelogram,

find the position vector of the point \(D\). The point \(E\) is positioned such that

• \(A C E\) is a straight line
• \(A C : C E = 2 : 1\)
• Find the coordinates of the point \(E\).
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\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-12_855_1104_340_589} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\)
The line \(l\) is the tangent to \(C\) at \(P\)

Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.

Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).

Use algebraic integration to find the exact area of \(R\).
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6. A scientist is studying the growth of two different populations of bacteria. The number of bacteria, \(N\), in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study.
Given that

• there were 1000 bacteria in this population at the start of the study
• it took exactly 5 hours from the start of the study for this population to double
  1. find a complete equation for the model.
  2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study.
Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of \(T\).
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Show that $$\frac { 1 - \cos 2 \theta } { \sin ^ { 2 } 2 \theta } \equiv k \sec ^ { 2 } \theta \quad \theta \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$ where \(k\) is a constant to be found.

Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) $$\frac { 1 - \cos 2 x } { \sin ^ { 2 } 2 x } = ( 1 + 2 \tan x ) ^ { 2 }$$ Give your answers to 3 significant figures where appropriate.
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8. Show that $$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$ [BLANK PAGE]

9. The function f is defined by $$\mathrm { f } ( x ) = \frac { ( x + 5 ) ( x + 1 ) } { ( x + 4 ) } - \ln ( x + 4 ) \quad x \in \mathbb { R } \quad x > k$$

1. State the smallest possible value of \(k\).
2. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { ( x + 4 ) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers to be found.
3. Hence show that f is an increasing function.
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10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ stating
   • the coordinates of the maximum point
   • the coordinates of any points where the graph cuts the coordinate axes
   • Find, in terms of \(k\), the set of values of \(x\) for which
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ [BLANK PAGE]
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11. The curve \(C\) has parametric equations $$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\)
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12. Answer all the questions.
Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track.
The cyclists are modelled as particles and the motion of the cyclists is modelled as follows:

• At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\)
• Cyclist \(A\) is moving in a straight line with constant acceleration \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
• At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\)
• Cyclist \(B\) moves with constant acceleration \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) along the same straight line and in the same direction as cyclist \(A\)
• At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\)

Using the model,

1. sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds,
   • a velocity-time graph for the motion of \(A\)
   • a velocity-time graph for the motion of \(B\)
   • explain why the two graphs must cross before time \(t = T\) seconds,
   • find the time when \(A\) and \(B\) are moving at the same speed,
   • find the distance \(O X\)
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13. A golfer hits a ball from a point \(A\) with a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(15 ^ { \circ }\) above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball first lands at a point \(B\) which is 4 m below the level of \(A\) (see diagram).

1. Determine the time taken for the ball to travel from \(A\) to \(B\).
2. Determine the horizontal distance of \(B\) from \(A\).
3. Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball.
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14.
\includegraphics[max width=\textwidth, alt={}, center]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-36_527_1123_118_269} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg . The other end of the string is attached to a second particle \(B\) of mass 3 kg . Particle \(A\) is in contact with a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and particle \(B\) is in contact with a rough horizontal plane. A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass 4 kg . Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of \(60 ^ { \circ }\) to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a \mathrm {~ms} ^ { - 2 }\).

1. By considering an equation involving \(\mu , a\) and \(g\) show that \(a < \frac { 1 } { 9 } g ( 2 \sqrt { 3 } - 1 )\).
2. Given that \(a = \frac { 1 } { 9 } g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to \(\mathbf { 3 }\) significant figures.
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