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University Exam Question Paper
University Exam
Course:Bachelor Of Science
Institution: Laikipia University question papers
Exam Year:2012
Basic mathematics
Answer question one and any other two
Question one - 30 marks (compulsory)
(a).Given the sets :AUB={1,2,3,4,5,6,7,9} and AnB''={2,3,5}
Find
(i)The elements of set B (3 marks)
(ii)the number of subsets of AUB with five elements (3 marks)
(b)The functions f and g are defined in the set R such that f(x) =mx+c, m and c are constants, and g(x) =x2-2
(i)Evaluate g-1(5) (3 marks)
(ii)If f (1) =3 and f(x) =g(x) when x=3 find the values of m and c (3 marks)
(c) (i)Given that sin (x+k)°=3 cos (x+k)° and tan x= 2 tan k. Obtain the value of tan k ° (4 marks)
(ii)Calculate the conjugate of the complex number Z=2i/1-I (2 marks)
(d)The sum of the first n terms of a series is given by Sn=3n + 2
(i)Calculate the sum of the third and four terms. (3 marks)
(ii)Determine the least number of terms that can give a sum of at least 731(3 marks)
(e)(i)Find the value of n if (n-2)C3=2(n-4) (3 marks)
(ii)Calculate the coefficient of x2 in the expansion of (x2+1/x)13 (3 marks)
Question two- 20 marks
(a)Prove that v3 is irrational number. (5 marks)
(b) Solve for x in the range 0°=x =180° (5 marks)
24sin2x-7cos2x= (25v2)/2
(c) (i)Simplify the set operation.(3 marks)
(AnB’) U (AnB)
(ii) Prove that (3 marks)
(A''nB)UB=B
(iii)The set A has a total of 30 proper subsets. Calculate the number of singleton subsets of A (3 marks)
Question three- 20 marks
(a)(i)Determine the number of all the rearrangements in the letters of PROPOSITION. (2 marks)
(ii)Evaluate n in the equation. (3 marks)
(n-3)!-2(n-4)! =6! (n-5)
(b)A committee of 5 members is to be selected from 7 women and 6 men. Find the number of committees that would have 60% of either ender. (4 marks)
(c)Find out whether the given proposition is a tautology or not (4 marks)
(i)(~p->q)<-> (p v q)
(ii)(p^~q)->~q
(d)Solve for z member of c in the equation.z3+z2+z+1=0 (3 marks)
Question four-20 marks
(a)Solve for the equation (4 marks)
Sin2x = cos 4x for 0°=x= 180°
(b)The terms of a sequence are given by Tn= 2(6n)+3
(i)Calculate the sum of the first three terms of sequence (2 marks)
(ii)Prove by induction that all the terms of the sequence are divisible by 3 (4 marks)
(c)Given the function f(x)=2x ,x member of R
(i)Obtain the value of x which f-1(x)=0 (3 marks)
(ii)State the smallest number system in which f(-1) is found. (2 marks)
(d)Expand the binominal (x-1/x2)11 in ascending powers of x up to the fourth term (5 marks)
Question five- 20 marks
(a)Given that 1-2sin2x =k ,find in terms of k
(i) tan2 2x (3 marks)
(ii)cos 4x (4 marks)
(b)Solve for z member of c in the equation z2+iz-1=0 (4 marks)
(c)An operation * is defined on the set Z such that x*y=x+y+1
Determine, giving reasons whether
(i)Z is closed under *or not (3 marks)
(ii)* is commutative or not (3 marks)
(d)Show that the relation R:xRy ?x ?y is transitive on the set of real numbers (3 marks).
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