Questions — SPS SPS FM (161 questions)

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All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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SPS SPS FM 2019 Q1
3 marks Moderate -0.3
In the question you must show detailed reasoning Solve the equation below, giving your answer in the simplest form $$x\sqrt{32} - \sqrt{24} = (3\sqrt{3} - 5)(\sqrt{6} + x\sqrt{2})$$ [3]
SPS SPS FM 2019 Q2
3 marks Easy -1.2
Find the coefficient of the \(x^4\) term in \((2 - 3x)^6\). [3]
SPS SPS FM 2019 Q3
4 marks Easy -1.2
A sequence \(u_1, u_2, u_3, ...\) is defined by \(u_n = 3n - 1\), for \(n \geq 1\).
  1. Find the values of \(u_1, u_2, u_3\). [1]
  2. Find $$\sum_{n=1}^{40} u_n$$ [3]
SPS SPS FM 2019 Q4
3 marks Easy -1.8
Show that $$\log_a(x^{10}) - 2\log_a\left(\frac{x^3}{4}\right) = 4\log_a(2x)$$ [3]
SPS SPS FM 2019 Q5
6 marks Standard +0.3
Solve the following inequalities giving your answer in set notation:
  1. \(|4x + 3| < |x - 8|\) [3]
  2. \(\frac{x}{x^2+1} < \frac{1}{2}\) [3]
SPS SPS FM 2019 Q6
3 marks Standard +0.8
If \(a\) and \(b\) are odd integers such that 4 is a factor of \((a - b)\), prove by contradiction that 4 cannot be a factor of \((a + b)\). [3]
SPS SPS FM 2019 Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a circle which passes through the points \(A(2, 9)\) and \(B(10, 3)\). \(AB\) is a diameter of the circle.
  1. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the exact coordinates of \(C\). [4]
  2. Find the exact area of the triangle formed by \(B\), \(C\) and the centre of the circle [3]
SPS SPS FM 2019 Q8
7 marks Standard +0.3
Sketch the curve \(y = 2^{2x+3}\), stating the coordinates of any points of intersection with the axes. [2] The point \(P\) on the curve \(y = 3^{3x+2}\) has \(y\)-coordinate equal to 180. Use logarithms to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. [2] The curves \(y = 2^{2x+3}\) and \(y = 3^{3x+2}\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as $$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$ [3]
SPS SPS FM 2019 Q9
9 marks Standard +0.3
  1. Given that \(u_{n+1} = 5u_n + 4\), \(u_1 = 4\), prove by induction that \(u_n = 5^n - 1\). [4]
  2. For all positive integers, \(n \geq 2\), prove by induction that $$\sum_{r=2}^{n} r^2(r-1) = \frac{1}{12}n(n-1)(n+1)(3n+2)$$ [5]
SPS SPS FM 2019 Q10
5 marks Standard +0.3
Show that, for any value of the real constant \(b\), the equation \(x^3 - (b + 1)x + b = 0\) has \(x = 1\) as a solution. Find all values of \(b\) for which this equation has exactly two real solutions [5]
SPS SPS FM 2019 Q11
10 marks Challenging +1.2
In the question you must show detailed reasoning Given that the coefficients of \(x\), \(x^2\) and \(x^4\) in the expansion of \((1 + kx)^n\) are the consecutive terms of a geometric series, where \(n \geq 4\) and \(k\) is a positive constant
  1. Show that $$k = \frac{6(n-1)}{(n-2)(n-3)}$$ [4]
  2. For the case when \(k = \frac{7}{2}\), find the value of \(n\). [2]
  3. Given that \(k = \frac{7}{5}\), \(n\) is a positive integer, and that the first term of the geometric series is the coefficient of \(x\), find the number of terms required for the sum of the geometric series to exceed \(1.12 \times 10^{12}\). [4]
SPS SPS FM 2019 Q12
5 marks Challenging +1.8
In the question you must show detailed reasoning Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\), $$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$ is $$\frac{1}{\ln(2\sqrt{2})}$$ [5] [Total marks: 65]
SPS SPS FM 2020 December Q1
4 marks Moderate -0.3
Solve \(2 \sin x = \tan x\) exactly, where \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). [4]
SPS SPS FM 2020 December Q2
4 marks Moderate -0.3
Let \(a, b\) satisfy \(0 < a < b\).
  1. Find, in terms of \(a\) and \(b\), the value of $$\int_a^b \frac{81}{x^4} dx$$ [2]
  2. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]
SPS SPS FM 2020 December Q3
4 marks Moderate -0.3
  1. Sketch the graph of \(y = |3x - 1|\). [1]
  2. Hence, solve \(5x + 3 < |3x - 1|\). [3]
SPS SPS FM 2020 December Q4
6 marks Moderate -0.8
The following diagram shows the curve \(y = a \sin(b(x + c)) + d\), where \(a, b, c\) and \(d\) are all positive constants and \(x\) is measured in radians. The curve has a maximum point at \((1, 3.5)\) and a minimum point at \((2, 0.5)\). \includegraphics{figure_4}
  1. Write down the value of \(a\) and the value of \(d\). [2]
  2. Find the value of \(b\). [2]
  3. Find the smallest possible value of \(c\), given that \(c > 0\). [2]
SPS SPS FM 2020 December Q5
4 marks Moderate -0.8
The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$
  1. Write down the matrix representing A. [1]
  2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]
SPS SPS FM 2020 December Q6
4 marks Challenging +1.2
Given that \(z\) is the complex number \(x + iy\) and satisfies $$|z| + z = 6 - 2i$$ find the value of \(x\) and the value of \(y\). [4]
SPS SPS FM 2020 December Q7
7 marks Standard +0.3
The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}
  1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
  2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]
SPS SPS FM 2020 December Q8
5 marks Standard +0.3
  1. The \(2 \times 2\) matrix A is given by $$\mathbf{A} = \begin{pmatrix} 7 & 3 \\ 2 & 1 \end{pmatrix}.$$ The \(2 \times 2\) matrix B satisfies $$\mathbf{BA}^2 = \mathbf{A}.$$ Find the matrix B. [3]
  2. The \(2 \times 2\) matrix C is given by $$\mathbf{C} = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix}.$$ By considering \(\mathbf{C}^2\), show that the matrices \(\mathbf{I} - \mathbf{C}\) and \(\mathbf{I} + \mathbf{C}\) are inverse to each other. [2]
SPS SPS FM 2020 December Q9
5 marks Moderate -0.3
Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]
SPS SPS FM 2020 December Q10
4 marks Challenging +1.2
The \(2 \times 2\) matrix M is defined by $$\mathbf{M} = \begin{pmatrix} 0 & 0.25 \\ 0.36 & 0 \end{pmatrix}$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by M. [4]
SPS SPS FM 2020 December Q11
6 marks Standard +0.3
In the triangle \(PQR\), \(PQ = 6\), \(PR = k\), \(P\hat{Q}R = 30°\).
  1. For the case \(k = 4\), find the two possible values of \(QR\) exactly. [3]
  2. Determine the value(s) of \(k\) for which the conditions above define a unique triangle. [3]
SPS SPS FM 2020 December Q12
7 marks Standard +0.3
Consider the binomial expansion of \(\left(1 + \frac{x}{5}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer.
  1. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). [1]
The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
  1. Show that \(n^3 - 33n^2 + 182n = 0\). [3]
  2. Hence find the possible values of \(n\) and the corresponding values of the common difference. [3]
SPS SPS FM 2020 December Q13
5 marks Standard +0.3
A series is given by $$\sum_{r=1}^k 9^{r-1}$$
  1. Write down a formula for the sum of this series. [1]
  2. Prove by induction that \(P(n) = 9^n - 8n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1. [4]