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Math 112: Basic Mathematics Question Paper
Math 112: Basic Mathematics
Course:Bachelor Of Arts (Economics & Mathematics)
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
EXAMINATIONS FOR THE AWARD OF DEGREE BACHELOR OF ARTS (ECONOMICS & MATHEMATICS)
MATH 112: BASIC MATHEMATICS
STREAMS: BA (ECON&MATHS) Y2S1 TIME: 2 HOURS
DAY/DATE: WEDNESDAY24/4/2013 8.30 AM – 10.30 AM
INSTRUCTIONS:
Answer Question One (Compulsory) and any other Two Questions
Adhere to the Instructions on the Question Paper
Do Not write on the Question Paper.
QUESTION ONE (30 MARKS)
(a) In the Venn diagrams below shade the indicated region. [4 Marks]
(i)
B B
A’nB
a
(ii)
AnB’
(iii) (a) AnB’ and
(b) AnB
(iv)
(AnB)’
(b) Construct a truth table to verify if statement p?q and ~ q?~p are equivalent statements. [4 Marks]
(c) Find the smallest positive integer n for which ((1+i)/(1-i))^n=1 [3 Marks]
(d) Let f(x)=1/x,g(x)=x^2,h(x)=vx find (fogoh)(x)and (hogof)(x) [4 Marks]
(e) Determine the domain and range of the following:
(i) f(x)=v(1-x^2 ) [2 Marks]
(ii) y=1/x [2 Marks]
(f) Solve the equation
2?sin ?^2 x-sin??x=0 0=x=360? [4 Marks]
(g) in how many ways can a set of 4 different mathematics books and 3 different physics books be placed on a shelf if all the books be placed on a shelf if all the books in the same subject must be placed next to each other. [3 Marks]
(h) Mutiso’s salary is 12 000/= per annum. His salary increases by 10% annually. Find the total amount he will have earned in 6 years. [4 Marks]
QUESTION 2 (20 MARKS)
(a) Prove that Tan (A+B)=(Tan A+Tan B)/(1-Tan A Tan B) [3 Marks]
(b) If sin??A=4/5& sin??B=3/5? ? find the values of
(i) Sin??(A+B)?
(ii) Tan??(A+B)? [3 Marks]
(c) Express the following in factors
(i) 35!+2(36!)
(ii) v(n-1)!+(n+1)! [4 Marks]
(d) The 2nd 4th and 7th terms of an A.P are the 1st three consecutive terms of a geometric progression.
Find:
(i) The common ratio. [4 Marks]
(ii) The sum of the 1st 8 terms of the G.P. [3 Marks]
(e) Determine the constant term in the binomial expansion(2x-5/x)^20 [3 Marks]
QUESTION 3 (20 MARKS)
(a) Prove the following ?n-1?_(C_r )+?n-1?_(C_(r-1) )=n_(C_r ) [4 Marks]
(b) A president, vice president and treasurer are to be selected from a group of 10 individuals. How many different choices are possible. [2 Marks]
(c) In how many ways can 6 men and 2 women be seated in a row when the men and women can sit anywhere. [2 Marks]
(d) Find the first four terms of the expansion of (1+x)^(1/2). Using your expansion find an approximation of v105 to 4 d.p. [4 Marks]
(e) Given f(x)=sinx and g(x)=(-x)/2 evaluate the following
(i) g(f(x)) [1 Mark]
(ii) f(g(x)) [1 Mark]
(f) Given two functions
f:?(x-1)/(x+2) and
g:x?5/(x-2)
Find: (fg)^(-1) [5 Marks]
QUESTION FOUR (20 MARKS)
(a) Using the analytical approach, prove the following laws of set algebra
(i) AU(BnC)=(AUB)n (AUC) [4 Marks]
(ii) (AUB)^''=A^'' nB^'' [4 Marks]
(iii) (A-B)U(B-A)=(AUB)-(AnB) [4 Marks]
(b) Make a truth table to show that P?(qvr)and (p?q)v(p?v) are logically equivalent. [5 Marks]
(c) If Z=1/2+1/2 i, express the complex number Z^10 in the form r(cos?+i sin?)
[3 Marks]
QUESTION 5 (20 MARKS)
(a) If n is a positive integer, prove that (v3+i)^n+(v3-i)^n=2^(n+1) cos??np/6?
[7 Marks]
(b) State and prove De-moivres theorem if a=cos?+i sin?,b=cosß+i sinß and c=cos?+i sin?. [10 Marks]
(c) Find x if sin??x-cos??x=1,0=x=360? ? [3 Marks]
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