Questions — SPS (686 questions)

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SPS SPS FM 2020 October Q1
7 marks Moderate -0.8
  1. Find the binomial expansion of \((2 + x)^5\), simplifying the terms. [4]
  2. Hence find the coefficient of \(y^3\) in the expansion of \((2 + 3y + y^2)^5\). [3]
SPS SPS FM 2020 October Q2
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z}{y})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS FM 2020 October Q3
8 marks Standard +0.3
  1. Give full details of a sequence of two transformations needed to transform the graph \(y = |x|\) to the graph of \(y = |2(x + 3)|\). [3]
  2. Solve \(|x| > |2(x + 3)|\), giving your answer in set notation. [5]
SPS SPS FM 2020 October Q4
5 marks Standard +0.3
Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^n r(3r + 1) = n(n + 1)^2\). [5]
SPS SPS FM 2020 October Q5
6 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows triangle \(ABC\), with \(AB = x\) cm, \(AC = (x + 2)\) cm, \(BC = 2\sqrt{7}\) cm and angle \(CAB = 60°\).
  1. Find the value of \(x\). [4]
  2. Find the area of triangle \(ABC\), giving your answer in an exact form as simply as possible. [2]
SPS SPS FM 2020 October Q6
5 marks Moderate -0.3
Prove by contradiction that \(\sqrt{7}\) is irrational. [5]
SPS SPS FM 2020 October Q7
7 marks Moderate -0.3
A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).
  1. Find \(\frac{dy}{dx}\). [1]
  2. Hence sketch the gradient function for the curve. [4]
  3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]
SPS SPS FM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS FM 2020 October Q9
8 marks Challenging +1.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]
SPS SPS SM 2020 October Q1
2 marks Easy -1.8
Simplify fully the following expressions:
  1. \(\frac{7y^{13}}{35y^7}\) [1]
  2. \(6x^{-2} \div x^{-5}\) [1]
SPS SPS SM 2020 October Q2
3 marks Easy -1.8
A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).
  1. State what type of sequence this is. [1]
  2. Find \(u_{17}\). [2]
SPS SPS SM 2020 October Q3
6 marks Moderate -0.3
  1. Write \(3x^2 - 6x + 1\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [2]
  2. Solve \(3x^2 - 6x + 1 \leq 0\), giving your answer in set notation. [4]
SPS SPS SM 2020 October Q4
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
  2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]
SPS SPS SM 2020 October Q5
3 marks Easy -1.2
Let \(a = \log_2 x\), \(b = \log_2 y\) and \(c = \log_2 z\). Express \(\log_2(xy) - \log_2(\frac{z^2}{x})\) in terms of \(a\), \(b\) and \(c\). [3]
SPS SPS SM 2020 October Q6
5 marks Moderate -0.8
  1. A student was asked to solve the equation \(2^{2x+4} - 9(2^x) = 0\). The student's attempt is written out below. $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ $$\text{Let } y = 2^x$$ $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$ Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2^{2x+4} - 9(2^x) = 0\), giving your answer in exact form. [3]
SPS SPS SM 2020 October Q7
11 marks Moderate -0.3
  1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
  3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]
SPS SPS SM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q9
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]
SPS SPS SM 2020 October Q10
8 marks Standard +0.8
In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]
SPS SPS FM 2021 March Q1
10 marks Moderate -0.8
Differentiate the following with respect to \(x\), simplifying your answers fully
  1. \(y = e^{3x} + \ln 2x\) [1]
  2. \(y = (5 + x^2)^{\frac{3}{2}}\) [2]
  3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [4]
  4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [3]
SPS SPS FM 2021 March Q2
5 marks Standard +0.3
  1. Express \(2 \tan^2 \theta - \frac{1}{\cos \theta}\) in terms of \(\sec \theta\). [1]
  2. Hence solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]
SPS SPS FM 2021 March Q3
9 marks Moderate -0.3
$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$
  1. Write down the domain and range of \(\text{f}^{-1}\) [2]
  2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
  3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = \frac{5}{3}\) [3]
SPS SPS FM 2021 March Q4
7 marks Standard +0.8
\includegraphics{figure_4} The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]
SPS SPS FM 2021 March Q5
13 marks Standard +0.3
  1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
The temperature \(T °C\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5 \sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 °C\). [6]
SPS SPS FM 2021 March Q6
8 marks Standard +0.3
$$\text{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$$ [2] Prove by induction that \(\text{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]