SPS SPS SM Pure (SPS SM Pure) 2021 June

A curve has equation $$y = 2x^3 - 4x + 5$$ Find the equation of the tangent to the curve at the point \(P(2, 13)\). Write your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers to be found. Solutions relying on calculator technology are not acceptable. [5]

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 \(\overrightarrow{AB}\) [2]
2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]

\includegraphics{figure_1} The shape \(ABCDOA\), as shown in Figure 1, consists of a sector \(COD\) of a circle centre \(O\) joined to a sector \(AOB\) of a different circle, also centre \(O\). Given that arc length \(CD = 3\) cm, \(\angle COD = 0.4\) radians and \(AOD\) is a straight line of length 12 cm,

1. find the length of \(OD\), [2]
2. find the area of the shaded sector \(AOB\). [3]

The function \(\mathbf{f}\) is defined by $$\mathbf{f}(x) = \frac{3x - 7}{x - 2} \quad x \in \mathbb{R}, x \neq 2$$

1. Find \(\mathbf{f}^{-1}(7)\) [2]
2. Show that \(\mathbf{f}(x) = \frac{ax + b}{x - 3}\) where \(a\) and \(b\) are integers to be found. [3]

A car has six forward gears. The fastest speed of the car • in 1st gear is 28 km h⁻¹ • in 6th gear is 115 km 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 3rd gear. [3]

Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,

1. find the fastest speed of the car in 5th gear. [3]

1. Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$(1 + kx)^{10}$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. [3]

Given that in the expansion of \((1 + kx)^{10}\) the coefficient \(x^3\) is 3 times the coefficient of \(x\),

1. find the possible values of \(k\). [3]

Given that \(k\) is a positive constant and \(\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4\)

1. show that \(3k + 5\sqrt{k} - 12 = 0\) [4]
2. Hence, using algebra, find any values of \(k\) such that $$\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$$ [4]

1. Given that \(\mathbf{f}(x) = x^2 - 4x + 2\), find \(\mathbf{f}(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
2. The curve with equation \(y = x^2 - 4x + 2\) 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 \(P\). [3 marks]

\includegraphics{figure_3} Figure 3 shows part of the curve with equation \(y = 3\cos x^2\). 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 \(d\). [1]
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3\cos x^2\) to the curve with equation
   1. \(y = 3\cos\left(\frac{x^2}{4}\right)\)
   2. \(y = 3\cos(x - 36)^2\)
   [2]
3. Solve, for \(450° \leq \theta < 720°\), $$3\cos\theta = 8\tan\theta$$ giving your solution to one decimal place. [4]

$$g(x) = 2x^3 + x^2 - 41x - 70$$

1. Use the factor theorem to show that \(g(x)\) is divisible by \((x - 5)\). [2]
2. Hence, showing all your working, write \(g(x)\) as a product of three linear factors. [4]

The finite region \(R\) is bounded by the curve with equation \(y = g(x)\) and the \(x\)-axis, and lies below the \(x\)-axis.

1. Find, using algebraic integration, the exact value of the area of \(R\). [4]

1. A circle \(C_1\) has equation $$x^2 + y^2 + 18x - 2y + 30 = 0$$ The line \(l\) is the tangent to \(C_1\) at the point \(P(-5, 7)\). Find an equation of \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found. [5]
2. A different circle \(C_2\) has equation $$x^2 + y^2 - 8x + 12y + k = 0$$ where \(k\) is a constant. Given that \(C_2\) lies entirely in the 4th quadrant, find the range of possible values for \(k\). [4]

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 \(0° < \theta < 180°\) Fully justify your answer. [5 marks]

1. Prove that for all positive values of \(x\) and \(y\) $$\sqrt{xy} \leq \frac{x + y}{2}$$ [2]
2. Prove by counter example that this is not true when \(x\) and \(y\) are both negative. [1]

\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), 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 \(\frac{1}{200}\).

1. Write down an equation for \(l\). [2]
2. Find the value of \(a\) and the value of \(b\). [4]
3. With reference to the model interpret
   1. the value of the constant \(a\),
   2. the value of the constant \(b\).
   [2]
4. 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.
   [3]
5. State two reasons why this may not be a realistic population model. [2]

A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)

1. find \(g(x)\), [7]
2. prove that the stationary point at \((2, 9)\) is a maximum. [2]