Autumn 2005 Math Exam for Software & Networking Students - Section A, Exams of Mathematics

Download Autumn 2005 Math Exam for Software & Networking Students - Section A and more Exams Mathematics in PDF only on Docsity! Cork Institute of Technology Bachelor of Science (Honours) in Software Development and Computer Networking – Stage 2 (Bachelor of Science in Software Development and Computer Networking – Stage 2) (NFQ – Level 8) Autumn 2005 Mathematics (Time: 3 Hours) Answer FIVE questions, selecting three questions from section A and two questions from section B. All questions carry equal marks. Examiners: Mr. P. Ahern Dr. D. Chambers Mr. P. O’Connor Mr E. A. Parslow Section A Q1. (a) The time taken to repair a certain fault is normally distributed with a mean of 12.4 minutes and a standard deviation of 2.8 minutes. Find the proportion of faults that can be repaired in less than 10 minutes. [5 marks] (b) The number of defects per metre of manufactured cable is Poisson distributed with a mean of 0.05 per m. Calculate the probability that there will be at most one defect in a 10 m run of this cable. [5 marks] (c) The time in days between breakdowns in a computer system is believed to be a random variable t with probability distribution tetf 1.01.0)( −= . Find the probability that the time between breakdowns will be less than 6 days. Find the mean time between breakdowns. [6 marks] (d) One per cent of the disks produced by a certain company are defective in some way. Calculate the probability that in a batch of ten such disks there will be more than one defective. [4 marks] Q2. (a) The diameters of 50 mass-produced components have been measured, correct to the nearest 0.1 mm, as follows: 33.0 33.9 33.9 34.6 35.7 36.0 36.2 36.5 36.7 36.8 37.1 37.3 37.3 37.4 37.6 37.6 37.7 38.2 39.0 39.0 39.1 39.1 39.2 39.3 39.3 39.4 39.4 39.6 39.6 39.6 39.8 39.8 39.9 39.9 40.0 40.4 40.5 40.5 40.6 40.8 40.9 41.2 41.5 41.8 42.0 42.1 42.3 42.7 43.5 44.4 Organise the data into 7 classes, beginning with 32.0-33.9 etc. Calculate the mean diameter and the standard deviation from this mean. Assuming a normal distribution, use these values of mean and standard deviation to estimate the proportion of components with diameter less than 38 mm. [14 marks] (b) A consignment of memory sticks consists of three batches. Batch A contains 200 cards of which 5 are defective; batch B contains 100 cards with 4 defective; batch C contains 150 cards with 8 defectives. A memory stick is chosen and found to be defective. Find the probability that it came from batch C. [6 marks] Q3. (a) Given that 4 2 4 5 1 2 6 1 1 1 − − = − , write down the values of 2 1 2 5 1 2 1 1 1 − − − and 4 2 4 5 2 1 1 1 x x x − − − [6 marks] (b) Describe the type of solution set of the equations 1 2 3 2 1 2 1 5 1 2 1 1 1 2 x x t x −         − = −               (i) when 1t = − (ii) when 2t = − . [6 marks] (c) Use Cramer’s Rule to solve the equations 1 2 3 2 1 2 2 5 1 2 0 1 1 1 4 x x x −         − =         −     Check your solutions. [8 marks] Q8. (a) A voltage )(tv in mV at time t s is described by the differential equation 12)()(01.0 =+′ tvtv Solve the equation subject to the initial condition 0)0( =v . What is the steady state voltage? State, approximately, the time taken to reach this steady state. [10 marks] (b) Solve the second order differential equation ( ) 8 ( ) 15 ( ) 60x t x t x t′′ ′+ + = subject to the initial conditions (0) (0) 0x x′= = . Draw a rough sketch of the solution. [10 marks] ∑ ∑ ∑ ∑ += = i ii i ii f uf cA f xf x ∑ ∑ ∑ ∑ += = i ii i ii f uf cA f xf µ ( ) ( )∑ ∑ ∑ ∑ ∑ ∑ ∑ − − − = − − = 11 1 )( 22 2 ii ii i ii i ii ff uf f uf c f xxf s 22 2)(         −= − = ∑ ∑ ∑ ∑ ∑ ∑ i ii i ii i ii f uf f uf c f xf µ σ ∑ = = n i ii ii i HEPHP HEPHPEHP 1 )|()( )|()( )|( ∑ = == n i ii XPXXE 1 )()(µ ∑ = −= n i ii XPX 1 22 )()( µσ P X n X p q X n X( ) =       − ! )( X eXP X λλ − = ktektf −=)( 2 2 2 1)( z exf − = π where σ µ− = xz Normal Distribution Tables Area under the Normal Curve P(z $ z,) z 0.0 ot 0.2 0.3 0.4 05 0.6 07 0.8 0.9 1.0 14 1.2 13 14 15 16 1 18 19 2.0 24 22 28 24 25 26 27 2.8 2.9 3.0 34 32 3.3 3.4 35 3.6 37 38 3.9 0.00 0.5000 0.5398 0.8793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 o.g918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.9990 0.9993 0.9995 0.9997 0.99977 o.g9984 0.99989 0.99993 0.99995 0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 og719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.9991 0.9993 0.9995 0.9997 0.99978 0.99985 0.99990 0.99993 0.99995, 0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 o.g212 0.8461 0.8686 0.8888 0.9068 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987 0.9991 0.9994 0.9995 0.9997 0.99978 0.99985 0.99990 0.99993 0.99996 0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.9991 0.9994 0.9996 0.9997 0.99979 0.99986 0.99990 0.99994 0.99998 0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7703 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 o.ge7t 0.9738 0.9793 0.9838 0.9875 0.9904 0.927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997 0.99980 o.g9986 0.99991 0.99994 0.99996 0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9878 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9980 0.9970 0.9978 0.9984 0.9989 o.gss2 0.9994 0.9998 0.9997 0.99981 0.99987 o.g9991 0.99994 0.99996 0.08 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9408 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9981 o.g971 0.9978 0.9985 0.9989 o.gsg2 0.9994 0.9996 0.9997 0.99981 0.99987 0.99992 0.99994 0.99996 0.07 0.5279 0.5675 0.6064 0.6443, 0.6808 07157 0.7488 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997 0.99982 0.99988 0.99992 0.99995 0.99996 1a 0.08 0.5319 os7i4 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8ss9 0.8810 0.8997 0.9162 0.9308 0.9429 0.9535 0.9625 o.g6s9 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9880 0.9993, 0.9995 0.9996 0.9997 0.99983 0.99988 0.9992 0.99995 0.99997 0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 o.g621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9708 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.99983 0.99989 o.ssss2 0.99995 0.99997