Questions FP2 (1279 questions)

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CAIE FP2 2019 November Q4
9 marks Challenging +1.2
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{(2ag)}\) so that it begins to move along a circular path. The string becomes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac{2}{3}\). [5]
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. [4]
CAIE FP2 2019 November Q5
12 marks Challenging +1.8
\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]
The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).
  1. Find the value of \(\lambda\). [6]
CAIE FP2 2019 November Q6
7 marks Challenging +1.2
A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, \(\mu\) metres, is calculated from the data as \(1.65 \leqslant \mu \leqslant 1.85\).
  1. Find an unbiased estimate for the population variance. [3]
  2. Denoting the height of a member of the club by \(x\) metres, find \(\Sigma x^2\) for this sample of 9 members. [4]
CAIE FP2 2019 November Q7
7 marks Standard +0.3
The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm{F}(t) = \begin{cases} 1 - e^{-at} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200.
  1. Write down the value of \(a\). [1]
  2. Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]
A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99.
  1. Find the smallest possible value of \(n\). [4]
CAIE FP2 2019 November Q8
9 marks Challenging +1.2
A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. (1 tonne = 1000 kg.) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x^2 = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes\(^2\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the 10% significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\). [9]
CAIE FP2 2019 November Q9
10 marks Standard +0.8
A random sample of five pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The values are shown in the following table, where \(p\) and \(q\) are constants.
\(x\)12345
\(y\)4\(p\)\(q\)21
The equation of the regression line of \(y\) on \(x\) is \(y = -0.5x + 3.5\).
  1. Find the values of \(p\) and \(q\). [7]
  2. Find the value of the product moment correlation coefficient. [3]
CAIE FP2 2019 November Q10
10 marks Standard +0.8
The random variable \(X\) has probability density function f given by $$\mathrm{f}(x) = \begin{cases} \frac{1}{30}\left(\frac{8}{x^2} + 3x^2 - 14\right) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
The random variable \(Y\) is defined by \(Y = X^2\).
  1. Find the probability density function of \(Y\). [4]
  2. Find the value of \(y\) such that \(\mathrm{P}(Y < y) = 0.8\). [3]
CAIE FP2 2019 November Q11
28 marks Challenging +1.2
Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).
  1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]
The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
  1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
  2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
  3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]
OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
Number of repairs in a week012345\(\geqslant 6\)
Number of weeks61596310
  1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]
Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
Number of repairs in a week012345\(\geqslant 6\)
Expected frequency8.07612.92110.3375.5132.205\(a\)\(b\)
  1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
  2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]
Edexcel FP2 Q1
6 marks Standard +0.8
  1. Express \(\frac{1}{t(t+2)}\) in partial fractions. [1]
  2. Hence show that \(\sum_{n=1}^{\infty} \frac{4}{n(n+2)} = \frac{n(3n+5)}{(n+1)(n+2)}\) [5]
Edexcel FP2 Q2
6 marks Standard +0.3
Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]
Edexcel FP2 Q3
8 marks Standard +0.8
Find the general solution of the differential equation $$\sin x \frac{dy}{dx} - y \cos x = \sin 2x \sin x$$ giving your answer in the form \(y = f(x)\). [8]
Edexcel FP2 Q4
8 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3\cos \theta, \quad a > 0, \quad 0 \leq \theta < 2\pi.$$ The area enclosed by the curve is \(\frac{10\pi}{2}\). Find the value of \(a\). [8]
Edexcel FP2 Q5
10 marks Challenging +1.2
\(y = \sec^2 x\)
  1. Show that \(\frac{d^2 y}{dx^2} = 6 \sec^4 x - 4 \sec^2 x\). [4]
  2. Find a Taylor series expansion of \(\sec^2 x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\), up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [6]
Edexcel FP2 Q6
10 marks Challenging +1.2
A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).
  1. Show that \(C\) is a circle and find its centre and radius. [8]
The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  1. Shade the region \(R\) on an Argand diagram. [2]
Edexcel FP2 Q7
12 marks Standard +0.8
  1. Sketch the graph of \(y = |x^2 - a^2|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes. [2]
  2. Solve \(|x^2 - a^2| = a^2 - x\), \(a > 1\). [6]
  3. Find the set of values of \(x\) for which \(|x^2 - a^2| > a^2 - x\), \(a > 1\). [4]
Edexcel FP2 Q8
15 marks Standard +0.8
$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$ Given that \(x = 0\) and \(\frac{dx}{dt} = 2\) at \(t = 0\),
  1. find \(x\) in terms of \(t\). [8]
The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0\), \(P\) is \(x\) metres from the origin \(O\).
  1. Show that the maximum distance between \(O\) and \(P\) is \(\frac{2\sqrt{3}}{9}\) m and justify that this distance is a maximum. [7]
Edexcel FP2 Q1
7 marks Standard +0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using your answer to part (a) and the method of differences, show that $$\sum_{r=1}^n \frac{3}{(3r-1)(3r+2)} = \frac{3n}{2(3n+2)}$$ [3]
  3. Evaluate \(\sum_{r=1}^{30} \frac{3}{(3r-1)(3r+2)}\), giving your answer to 3 significant figures. [2]
Edexcel FP2 Q2
5 marks Challenging +1.2
The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac{d^2 x}{dt^2} + x + \cos x = 0.$$ When \(t = 0\), \(x = 0\) and \(\frac{dx}{dt} = \frac{1}{2}\). Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t^4\). [5]
Edexcel FP2 Q3
7 marks Standard +0.8
  1. Find the set of values of \(x\) for which $$x + 4 > \frac{2}{x+3}$$ [6]
  2. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \left|\frac{2}{x+3}\right|$$ [1]
Edexcel FP2 Q4
10 marks Standard +0.3
\(z = -8 + (8\sqrt{3})i\)
  1. Find the modulus of \(z\) and the argument of \(z\). [3]
Using de Moivre's theorem,
  1. find \(z^3\). [2]
  2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]
Edexcel FP2 Q5
10 marks Standard +0.8
\includegraphics{figure_1} Figure 1 Figure 1 shows the curves given by the polar equations $$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$ and $$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$
  1. Find the coordinates of the points where the curves intersect. [3]
The region \(S\), between the curves, for which \(r > 2\) and for which \(r < (1.5 + \sin 3\theta)\), is shown shaded in Figure 1.
  1. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are simplified fractions. [7]
Edexcel FP2 Q6
10 marks Standard +0.8
A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
  1. Given that \(|z - 6| = |z|\), sketch the locus of \(P\). [2]
  2. Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4i| = 5\). [3]
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac{30}{z}\).
  1. Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. [5]
Edexcel FP2 Q7
7 marks Challenging +1.2
  1. Show that the transformation \(z = y^{\frac{1}{2}}\) transforms the differential equation $$\frac{dy}{dx} - 4y \tan x = 2y^{\frac{1}{2}}$$ [I] into the differential equation $$\frac{dz}{dx} - 2z \tan x = 1$$ [II]
  2. Solve the differential equation (II) to find \(z\) as a function of \(x\). [6]
  3. Hence obtain the general solution of the differential equation (I). [1]
Edexcel FP2 Q8
14 marks Challenging +1.2
  1. Find the value of \(z\) for which \(y = zx \sin 5x\) is a particular integral of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [4]
  2. Using your answer to part (a), find the general solution of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [3]
Given that at \(x = 0\), \(y = 0\) and \(\frac{dy}{dx} = 5\),
  1. find the particular solution of this differential equation, giving your solution in the form \(y = f(x)\). [5]
  2. Sketch the curve with equation \(y = f(x)\) for \(0 \leq x \leq \pi\). [2]
Edexcel FP2 Q1
7 marks Standard +0.3
Find the set of values of \(x\) for which $$\frac{3}{x+3} > \frac{x-4}{x}.$$ [7]