SPS SPS SM Pure (SPS SM Pure) 2021 June

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.

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.

3. \begin{figure}[h] \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 \(A O B\).

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.

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.

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 \(k\).

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]

9. \begin{figure}[h] \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]

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]

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]

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.

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]

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]