Questions — SPS (686 questions)

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SPS SPS FM Pure 2021 June Q10
8 marks Challenging +1.8
A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]
SPS SPS FM Pure 2021 June Q11
7 marks Standard +0.8
  1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
  2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x} dx\) Fully justify your answer. [6 marks]
SPS SPS FM Pure 2021 June Q13
8 marks Standard +0.8
$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,
  1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
  2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]
SPS SPS FM Pure 2021 June Q14
6 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.
  1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
  2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]
SPS SPS FM Pure 2021 June Q15
7 marks Standard +0.3
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = -\frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
SPS SPS FM Pure 2021 June Q16
7 marks Challenging +1.8
Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]
SPS SPS SM Pure 2021 June Q1
5 marks Moderate -0.8
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]
SPS SPS SM Pure 2021 June Q2
4 marks Easy -1.2
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 \(\overrightarrow{AB}\) [2]
  2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]
SPS SPS SM Pure 2021 June Q3
5 marks Moderate -0.8
\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]
SPS SPS SM Pure 2021 June Q4
5 marks Moderate -0.3
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]
SPS SPS SM Pure 2021 June Q5
6 marks Moderate -0.8
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]
SPS SPS SM Pure 2021 June Q6
6 marks Moderate -0.8
  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]
SPS SPS SM Pure 2021 June Q7
8 marks Standard +0.3
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]
SPS SPS SM Pure 2021 June Q8
5 marks Moderate -0.3
  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]
SPS SPS SM Pure 2021 June Q9
7 marks Standard +0.3
\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]
SPS SPS SM Pure 2021 June Q10
10 marks Moderate -0.3
$$g(x) = 2x^3 + x^2 - 41x - 70$$
  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]
SPS SPS SM Pure 2021 June Q11
9 marks Standard +0.3
  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]
SPS SPS SM Pure 2021 June Q12
5 marks Standard +0.3
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]
SPS SPS SM Pure 2021 June Q13
3 marks Moderate -0.8
  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]
SPS SPS SM Pure 2021 June Q14
13 marks Moderate -0.3
\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]
SPS SPS SM Pure 2021 June Q15
9 marks Standard +0.8
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]
SPS SPS FM 2020 September Q1
3 marks Moderate -0.3
Vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are given by $$\overrightarrow{AB} = \begin{pmatrix} 2p \\ q \\ 4 \end{pmatrix} \quad \overrightarrow{BC} = \begin{pmatrix} q \\ -3p \\ 2 \end{pmatrix},$$ where \(p\) and \(q\) are constants. Given that \(\overrightarrow{AC}\) is parallel to \(\begin{pmatrix} 3 \\ -4 \\ 3 \end{pmatrix}\), find the value of \(p\) and the value of \(q\). [3]
SPS SPS FM 2020 September Q2
4 marks Challenging +1.2
A sequence of numbers \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 3$$ $$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$
  1. Find \(\sum_{r=1}^{100} a_r\) [3]
  2. Hence find \(\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r\) [1]
SPS SPS FM 2020 September Q3
4 marks Moderate -0.3
Using algebraic integration and making your method clear, find the exact value of $$\int_1^5 \frac{4x + 9}{x + 3} \, dx = a + \ln b$$ where \(a\) and \(b\) are constants to be found [4]
SPS SPS FM 2020 September Q4
5 marks Moderate -0.8
  1. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]
Adele uses \(\theta = 5°\) to test the approximation in part (a). Adele's working is shown below.
Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\)
Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\)
    1. Identify the mistake made by Adele in her working.
    2. Show that \(8 - 5\theta^2\) can be used to give a good approximation to \(1 + 4\cos \theta + 3\cos^2 \theta\) for an angle of size \(5°\) [2]