Questions — SPS SPS SM Pure (97 questions)

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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 SM Pure 2021 May Q1
7 marks Moderate -0.8
The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$
  1. Evaluate \(f(169)\). [2]
  2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
  3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]
SPS SPS SM Pure 2021 May Q2
4 marks Moderate -0.8
  1. Use the trapezium rule, with four strips each of width \(0.5\), to estimate the value of $$\int_0^2 e^{x^2} dx$$ giving your answer correct to 3 significant figures. [3]
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate. [1]
SPS SPS SM Pure 2021 May Q3
6 marks Moderate -0.8
Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
  2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
  3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]
SPS SPS SM Pure 2021 May Q4
3 marks Standard +0.3
Prove that \(\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta\), where \(\theta\) is measured in degrees. [3]
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
  1. Show that \(\sqrt{\frac{1-x}{1+x}} \approx 1 - x + \frac{1}{2}x^2\), for \(|x| < 1\). [5]
  2. By taking \(x = \frac{2}{7}\), show that \(\sqrt{5} \approx \frac{111}{49}\). [3]
SPS SPS SM Pure 2021 May Q6
4 marks Challenging +1.2
Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]
SPS SPS SM Pure 2021 May Q7
11 marks Standard +0.3
A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).
  1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
  2. Find the cartesian equation of the curve. [3]
  3. State the set of values that \(x\) can take and hence sketch the curve. [3]
SPS SPS SM Pure 2021 May Q8
12 marks Challenging +1.2
In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]
SPS SPS SM Pure 2021 May Q9
10 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]
SPS SPS SM Pure 2021 May Q1
5 marks Standard +0.3
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2\cos\theta + (1 - \tan\theta)^2 \approx 3 - 2\theta\). [3]
  2. Hence determine an approximate solution to \(2\cos\theta + (1 - \tan\theta)^2 = 28\sin\theta\). [2]