Advanced level 2025 North West Regional Mock Further mathematics 3

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Problem 1:

Forces $F ” />_{1}$ and $F ” />_{2}$ act at points $P_{1}$ and $P_{2}$ , respectively, where:
- $F ” />_{1} = (- 8 i^+ 8 j^- 4 k^)$ N, $P_{1} = (5 i^- 6 j^- k^)$ m
- $F ” />_{2} = (i^- j^- 3 k^)$ N, $P_{2} = (7 i^- 4 j^+ 2 k^)$ N
When a third force $F ” />_{3}$ acting through $R$ , where $R = (- i^- 4 j^- 4 k^)$ m is added, the system of the three forces is equivalent to a single force $R ” /> = (6 i^- 2 j^+ 3 k^)$ N.
- a) Show that $F ” />_{1}$ and $F ” />_{2}$ are concurrent.
- b) Find $F ” />_{3}$ .
- c) By finding the sum of the vector moments of these forces about the origin, show that $R ” />$ acts along the line with Cartesian equation: $-2 x + 29 = 2 y + 18 = 3 z + 6$

Problem 2:

A metal ball of mass $m$ is dropped into a large container filled with diesel. The resistive force acting on the ball is of magnitude $m g c v$ , where $v$ is the speed and $c$ is a constant.
- a) Find the time taken for the ball to attain a speed of $2 c$ .
- b) Find also, the distance covered during this time.
- c) Assuming that the ball reaches a terminal velocity, find this terminal velocity.

Problem 3:

i. A particle moves on a straight line Ox with simple harmonic motion. The particle passes through the points A and B, where OA = 4 m and OB = 5 m with speeds $5$ ms$^{-1}$ and $13$ ms$^{-1}$ respectively. Show that the amplitude of motion is $237$ m and find the periodic time for the motion.
ii. Another particle P moves along a horizontal straight line Ox such that that at time $t$ seconds, its displacement $x$ metres from O satisfies the differential equation: $d t ^{2} d ^{2} x + 12 d t d x + 13 x = 0$
- a) Show that P is performing damped harmonic motion.
- b) Find $x$ in the form $x = e^{α t} (A cos βt + B sin βt)$ given that $x = 4$ and $d t d x = - 6$ when $t = 0$ .
- c) Find the time $t > 0$ when $x = 0$ .

Problem 4:

The variables $x$ and $y$ satisfy the differential equation $d x d y = x^{2} - 2 y^{2} + 1$ .
- a) Find the first four non-zero terms in the Taylor series solution of this differential equation.
- b) Using your series solution, find, correct to 4 decimal places, the value of $y$ when $x = 0.1$ given that $y = 0$ when $x = 0$ .
- c) Use the approximation $h^{2} d x ^{2} d ^{2} y = y_{n + 1} - 2 y_{n} + y_{n - 1}$ and a step length of $0.1$ to find, correct to 4 decimal places, the value of $y$ when $x = 0.2$ .
- d) Obtain, correct to 4 decimal places, an estimate for $\int_{0 0.2} y d x$ , using Simpson’s rule with 3 ordinates.

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