Acmp 203: Mathematical Methods Of Computer Scientists Question Paper
Acmp 203: Mathematical Methods Of Computer Scientists
Course:Bachelor Of Science In Computer Science
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATION FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE (COMPUTER SCIENCE)
ACMP 203: MATHEMATICAL METHODS OF COMPUTER SCIENTISTS
STREAMS: B.SC (COMP. SCI) Y2S2 TIME: 2 HOURS
DAY/DATE: WEDNESDAY 7/8/2013 2.30 P.M. – 4.30 P.M.
INSTRUCTIONS:
Answer Question ONE (Compulsory) and any other two Questions.
Adhere to the instructions on the answer booklet.
Do not write on the question paper.
QUESTION ONE – COMPULSORY
Given that and g(x)=1/(x+1), evaluate
(i)
(ii) f^(-1) (x) [4 marks]
Find of the function [3 marks]
Evaluate
[2 marks]
Evaluate the gradient (dy/dx) of the function y=?tan?^(-1) (v(x-1)) [4 marks]
Using an integrating factor, solve the differential equation
( dy)/dx-2xy=x [3 marks]
Find the angle between the vectors a=i-2j+4k and b=-4i+j-2k. [3 marks]
Solve the system of linear equations using row reduction.
?-x?_1+x_2+?2x?_3=2
?3x?_1-x_2+x_3=6
?-x?_1+?3x?_2+?4x?_3=4
[3 marks]
Evaluate the determinant of the matrix
(¦(1&3&-2@4&-5&6@0&0&2)) [2 marks]
Evaluate the characteristic polynomial of the matrix
(¦(2&2&1@1&3&1@1&2&2)) [3 marks]
Using the ratio test, show that the series
is convergent. [2 marks]
QUESTION 2
Find the cross product a × b given that
a=i+3j-Rand
b=2i-j+R [4 marks]
Test the consistency of the system below. If found consistent solve it.
?-x?_1+?2x?_2-?3x?_3=4
?2x?_1-?4x?_2+?6x?_3=-8
? x?_1-?2x?_2+?3x?_3=-4 [5 marks]
Find the eigen values and eigen vectors of the matrix A below.
A=(¦(1&0&-1@1&2&1@2&2&3)) [11 marks]
QUESTION 3
(i) Use the trapezoidal rule with n = 5 to approximate to 4dp.
?_1^2¦dx/x [6 marks]
(ii) Evaluate the exact integral hence obtain the error of approximation. [4 marks]
(i) Define an exact differential equation. [2 marks]
(ii) Prove that the differential equation
(?5x?^4+?3x?^2 y^2-?2xy?^3 )dx+(?2x?^3 y-?3x?^2 y^2-?5y?^4 )dy=0
is exact hence solve it. [4 marks]
Find the particular solution of the differential equation
dy/dx=xy,y(0)=1 [4 marks]
QUESTION 4
(i) State four properties/characteristics of affine transformation. [4 marks]
(ii) Give examples of Affine transformations. [1 mark]
Evaluate the image co-ordinates of a 2 x 2 square centred at the origin after undergoing a rotation of 45o about its centre then a translation which moves the centre of the square to the point (3, 2) in 3D. [5 marks]
Prove that the series
is divergent
using the root test. [3 marks]
Using the ratio test, prove that the series
is divergent. [7 marks]
QUESTION 5
Evaluate the following limits
[3 marks]
[3 marks]
[3 marks]
(i) Prove that dy/dx of y=?Sin?^(-1) x=1/v(?1-x?^2 ) [4 marks]
(ii) Given that? y=2?^3x, evaluatedy/dx. [4 marks]
Apply the ratio test to the power series
to find the radius of convergence and the interval of convergence. [3 marks]
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