Questions — SPS (686 questions)

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SPS SPS FM Pure 2024 February Q6
7 marks Standard +0.3
  1. Explain why \(\int_1^\infty \frac{1}{x(2x + 5)} dx\) is an improper integral. [1]
  2. Prove that $$\int_1^\infty \frac{1}{x(2x + 5)} dx = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. [6]
SPS SPS FM Pure 2024 February Q7
7 marks Standard +0.8
In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square, \(S\).
  1. Find the area of \(S\). [3]
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
SPS SPS FM Pure 2024 February Q8
8 marks Challenging +1.2
A linear transformation of the plane is represented by the matrix \(\mathbf{M} = \begin{pmatrix} 1 & -2 \\ \lambda & 3 \end{pmatrix}\), where \(\lambda\) is a constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin. [5]
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]
SPS SPS FM Pure 2024 February Q9
9 marks Standard +0.8
In this question you must show detailed reasoning. The complex number \(-4 + i\sqrt{48}\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. [6]
The points which represent the cube roots of \(z\) are denoted by \(A\), \(B\) and \(C\) and these form a triangle in an Argand diagram.
  1. Write down the angles that any lines of symmetry of triangle \(ABC\) make with the positive real axis, justifying your answer. [3]
SPS SPS FM Pure 2024 February Q10
11 marks Challenging +1.8
The diagram shows the polar curve \(C_1\) with equation \(r = 2\sin\theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_10}
  1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$ where \(\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]
SPS SPS FM Pure 2024 February Q11
7 marks Challenging +1.8
Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}
  1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]
SPS SPS FM Pure 2024 February Q12
7 marks Challenging +1.2
Find the general solution of the differential equation $$x\frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]
SPS SPS FM Pure 2024 February Q13
7 marks Challenging +1.2
In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh\frac{1}{2}x^2\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360°\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm. [7]
SPS SPS FM Pure 2024 February Q14
6 marks Challenging +1.8
Show that \(\int_0^{\frac{1}{\sqrt{3}}} \frac{4}{1-x^4} dx = \ln(a + \sqrt{b}) + \frac{\pi}{c}\) where \(a\), \(b\) and \(c\) are integers to be determined. [6]
SPS SPS FM Pure 2024 February Q15
8 marks Challenging +1.2
\(y = \cosh^n x\) \quad \(n \geq 5\)
    1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
    2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]
SPS SPS FM 2024 October Q1
6 marks Moderate -0.8
Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),
  1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
  2. Find the maximum value of \(f(x)\). [1]
  3. Explain why \(f^{-1}(x)\) does not exist. [1]
SPS SPS FM 2024 October Q2
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.3
  1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left(2 + \frac{1}{3}kx\right)^6\), where \(k\) is a constant. [3]
  2. In the expansion of \((3 - 4x)\left(2 + \frac{1}{3}kx\right)^6\), the constant term is equal to the coefficient of \(x^2\). Determine the exact value of \(k\), given that \(k\) is positive. [3]
SPS SPS FM 2024 October Q4
3 marks Moderate -0.8
The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]
SPS SPS FM 2024 October Q5
9 marks Standard +0.3
In this question you must show detailed reasoning. The polynomial \(f(x)\) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of \(f(x)\). [1]
    2. Hence find the exact roots of the equation \(f(x) = 0\). [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form \(f(x) = 0\). [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS FM 2024 October Q6
7 marks Standard +0.8
The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of \(k\), giving a reason for your answer. [4]
  2. Find the value of \(\sum_{r=2}^{\infty} u_r\). [3]
SPS SPS FM 2024 October Q7
9 marks Standard +0.8
The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}
  1. Determine the coordinates of \(B\). [6]
  2. Hence determine the exact value of \(\tan\alpha\). [3]
SPS SPS FM 2024 October Q8
5 marks Standard +0.3
Prove by induction that \(11 \times 7^n - 13^n - 1\) is divisible by \(3\), for all integers \(n > 0\). [5]
SPS SPS FM 2024 October Q9
9 marks Standard +0.8
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]
SPS SPS SM 2024 October Q1
3 marks Easy -1.2
A is inversely proportional to B. B is inversely proportional to the square of C. When A is 2, C is 8. Find C when A is 12. [3]
SPS SPS SM 2024 October Q2
5 marks Moderate -0.8
  1. Write \(3x^2 + 24x + 5\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
The finite region R is enclosed by the curve \(y = 3x^2 + 24x + 5\) and the \(x\)-axis.
  1. State the inequalities that define R, including its boundaries. [2]
SPS SPS SM 2024 October Q3
5 marks Moderate -0.8
The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]
SPS SPS SM 2024 October Q4
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q5
8 marks Moderate -0.8
\includegraphics{figure_5} Figure 4 The line \(l_1\) has equation \(y = \frac{3}{5}x + 6\) The line \(l_2\) is perpendicular to \(l_1\) and passes through the point \(B(8, 0)\), as shown in the sketch in Figure 4.
  1. Show that an equation for line \(l_2\) is $$5x + 3y = 40$$ [3]
Given that
  • lines \(l_1\) and \(l_2\) intersect at the point C
  • line \(l_1\) crosses the \(x\)-axis at the point A
  1. find the exact area of triangle \(ABC\), giving your answer as a fully simplified fraction in the form \(\frac{p}{q}\) [5]
SPS SPS SM 2024 October Q6
8 marks Moderate -0.8
In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$
  1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
  2. Find the time when the mass is 30 grams. [3]
  3. Sketch the graph of \(m\) against \(t\). [2]