Guides d'étude, Notes de cours & Résumés

ig. 1.1 I A B K a The diagram in Fig. 1.1 represents a system of pipes through which a fluid can flow from two sources to three sinks. It also shows a cut a. The weights on the arcs show the capacities of the pipes in gallons per minute. (a) Add a supersource S and a supersink T to the network in the Printed Answer Booklet, giving appropriate weightings and directions to the connecting arcs. [2] (b) Calculate the capacity of the cut a. [1] 3 © OCR 2023 Y413/01 Jun23 Turn over 10 ...

Throughout all parts of this question, the resistance to the motion of a car has magnitude kv2 N, where vm s -1 isthe speed of the car and k is a constant. At first, the car travels along a straight horizontal road with constant speed 20m s -1 . The power developed by the car at this speed is 5000W. (a) Show that k = 5 . [3] (b) Find the power the car must develop in order to maintain a constant speed of 28m s -1 when travelling along the same horizontal road. [1] The car climbs ...

The transformation R of the plane is reflection in the line x = 0. (a) Write down the matrix M associated with R. [1] (b) Find M2 . [1] (c) Interpret the result of part (b) in terms of the transformation R. [1] 2 In this question you must show detailed reasoning. The equation x 2 - kx + 2k = 0, where k is a non-zero constant, has roots a and b. Find a + b in terms of k, simplifying your answer. [4] b a 3 In this question you must show detailed reasoning. The function f(z) is given...

(a) The matrix P is given by P = K L 0 - 2 2 N 2 - 2 O. (i) Write down the dimensions of P. [1] (ii) Write down the transpose of P. [1] J 3 (b) The matrices Q, R and S are given by Q = ^1 2h, R = K L -4 N O and S = ^3 P -2h. Write down the sum of the two of these matrices which are conformable for addition. [1] (c) The dimensions of matrix A are 4 by 5. The matrices A and B are conformable for multiplication so that the matrix C = BA can be formed. The matrix C has 6 rows. (...

In this question you must show detailed reasoning. Determine the value of / 50 r 2 (16 -r). [3] r=1 2 In this question you must show detailed reasoning. The equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0 has roots a, b, c and d. (a) Show that a quartic equation whose roots are a + 1, b + 1, c + 1 and d + 1 is w 4 +3w 2 +2 = 0. [3] (b) Hence determine the exact roots of the equation z 4 + 4z 3 + 9z 2 + 10z + 6 = 0. [3] 3 (a) Show that -3+ 3 i = 3 e 5 ri . [2] (b) Hence determin...

(a) Express 205 in the form 7q+r for positive integers q and r, with 0 G r 1 7. [1] (b) Given that 7 | 205 # 8666 , use the result of part (a) to justify that 7 |8666. [2] 2 For all positive integers n, the terms of the sequence un are given by the formula un = 3n 2 + 3n + 7 (mod 10). (a) Show that un+5 = un for all positive integers n. [2] (b) Hence describe the behaviour of the sequence, justifying your answer. [2] 3 A surface has equation z = x 2 y 2 - 3xy + 2x + y for all real v...

Jane wants to travel from home to the local town. Jane can do this by train, by bus or by both train and bus. (a) Give an example of a problem that Jane could be answering that would give a construction problem. [1] A website gives Jane all the possible buses and trains that she could use. Jane finds 7 possible ways to make the journey. • 2 of the 7 journeys involve travelling by train for at least part of the journey • 6 of the 7 journeys involve travelling by bus for at least part o...

Two particles A, of mass mkg, and B, of mass 3mkg, are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. A is projected along the surface, directly away from B, with a speed of 2.4ms -1 . (a) Find the speed of B immediately after the string becomes taut. [2] (b) Find, in terms of m, the magnitude of the impulse exerted on B as a result of the string becoming taut. [2] 2 O A small body P of mass 3kg is at rest at t...

A radar device is used to detect flaws in motorway roads before they become dangerous. The number of flaws in a 1 km stretch of motorway is denoted by X. It may be assumed that flaws occur randomly. (a) State two further assumptions that are necessary for X to be well modelled by a Poisson distribution. [2] Assume now that X can be modelled by the distribution Po(5.7). (b) Determine the probability that in a randomly chosen stretch of motorway, of length 1 km, there are between 8 and 11 f...