Beststudymaterials

In this question you must show detailed reasoning. Use Fermat’s Little Theorem to determine the least positive residue of 2180 (mod 23) 2 (4) ■■■■ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS ARE

4 The eleven numbers listed above are to be packed into bins of size n where n is a positive integer. When the first-fit bin packing algorithm is applied to the eleven numbers, the bins are packed as shown below. Bin 1: 17 8 12 Bin 2: 16 24 Bin 3: 19 11 4 Bin 4: 23 13 Bin 5: 20 (a) Explain why this packing means that the value of n must be 40 The original list of eleven numbers is to be sorted into descending order. (b) Use a quick sort to obtain the fully sorted list. You must make...

[In this question, i and j are horizontal perpendicular unit vectors.] A particle A has mass 3 kg and a particle B has mass 2 kg. The particles are moving on a smooth horizontal plane when they collide directly. Immediately before the collision, the velocity of A is (3i – j)ms–1 and the velocity of B is (–6i + 2j)ms–1 Immediately after the collision the velocity of A is  −2i + 2 j m s–1     (a) Find the total kinetic energy of the two particles before the co...

The discrete random variable X has the following probability distribution x – 1 0 1 3 5 P(X = x) 0.2 0.1 0.2 0.25 0.25 (a) Find Var(X ) (b) Find Var(X 2) 2 (3) (3) ■■■■ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQuestion 1 continued ■■■■ (Total for Question 1 is 6 marks) 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA

1. (a) Given that y = ln(3 + x2) complete the table with the value of y corresponding to x = 3, giving your answer to 4 significant figures. x 2 2.5 3 3.5 4 4.5 5 y 1.946 2.225 2.725 2.944 3.146 3.332 (1) In part (b) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (b) Use Simpson’s rule with all the values of y in the completed table to estimate, to 3 significant figures, the value of  ln(3 + x2)dx 2 (3) (c) ...

1. (a) Using the definition of sinhx in terms of exponentials, prove that 4sinh3 x + 3sinh x  sinh 3x (2) (b) Hence solve the equation sinh 3x = 19sinh x giving your answers as simplified natural logarithms where appropriate. 2 (5) ■■■■ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS ARE

Pearson Edexcel A Level GCE In Mathematics (9MA0) Paper 01,2,31 and 32 Pure Mathematics,Mathematics Pure,Statistics and Mechanics QP & MS JUNE 2024

1. f(z) = z4 − 6z3 + az2 + bz + 145 where a and b are real constants. Given that 2 + 5i is a root of the equation f(z) = 0 (a) determine the other roots of the equation f(z) = 0 (b) Show all the roots of f(z) = 0 on a single Argand diagram. 2 (7) (2) ■■■■ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS

P (0.5kg) Figure 1 Figure 1 shows a particle P of mass 0.5kg at rest on a rough horizontal plane. (a) Find the magnitude of the normal reaction of the plane on P. 2 (1) The coefficient of friction between P and the plane is 7 A horizontal force of magnitude X newtons is applied to P. Given that P is now in limiting equilibrium, (b) find the value of X. 2 (2) ■■■■ DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE I

Xian rolls a fair die 10 times. The random variable X represents the number of times the die lands on a six. (a) Using a suitable distribution for X, find (i) P(X = 3) (ii) P(X< 3) Xian repeats this experiment each day for 60 days and records the number of days when X = 3 (b) Find the probability that there were at least 12 days when X = 3 (c) Find an estimate for the total number of sixes that Xian will roll during these 60 days. (d) Use a normal approximation to estimate the probab...