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UkaTarsadiaUniversity
IntegratedM.Sc.(Mathematics)(Semester1)
QuestionBank
SubjectCode:060090106
SubjectName:CC2ElementaryAlgebra
Unit:1Complexnumbers
1.
Define complex number.
2.
Define conjugate complex number.
3.
Express the following complex number in polar form:
(1) 2 + i (2) − 2 − i
4.
(4) 2 + 2i
(5) 3 + i (6) − 3 − i (7) 1 − i (8) 1 + 2i
Find the modulus of the following complex numbers:
(1)
(5)
5.
(3) 2 − i
!
!
+
!
!
!!! !
!!!
(2)
(6)
!!
!"
−i
!!!
!"
(7)
!!!
(3)
!"
!!!
!!!
!"
!"#
+
+i
!!
!"#
(4)
!!!"
!!!"
!!!
!!!
Find the polar coordinates of the points:
7.
M! 2, −2 , M! −2 3, −2 , M! 0,1 , M! −4 3, −4 , M! 0, −5 , M! (−2, −1)
Find the extended argument of following:
(1) z! = 2i 2 z! = −1 3 z! = 2 4 z! = −3i (5) z! = 1 − i√3
Compute 1 + i !""" .
8.
Find the following complex numbers where z! = 3 + 2i, z! = −2 + i, z! = − 2i
9.
Find (1) z! + z!
(2) z! − 2z!
(3) z! + z!
(4) z! + z! − z!
If z! = 1 + i and z! = 2 + i find z! z! and
10.
Prove that 1 + i
11.
Compute ; z =
12.
Find z for ; z =
6.
13.
14.
!
!
!
+ 1−i
(!!!)!" ( !!!)!
(!!!! !)!"
! !!!"
!!! !
!
!
= 2 !!! cos
.
!
+
!!! !
! !!!"
!
Find z , arg z, arg z for following:
(a) z = (1 − i)(6 + 6i)
b z = (7 − 7√3i)(−1 − i)
Compute the following products:
(1)
!
!
−i
√!
!
−3 + 3i 2 3 + 2i
(2) 1 + i −2 − 2i i
15.
(3) −2i(−4 + 4 3i)(3 + 3i)
(4) 3(1 − i)(−5 + 5i)
State and prove De Movire’s theorem
.
!!
!!
!!
!
.
.
2016
!"#$!!!"!#$! ! !"#$!!!"!#$! !
16.
Evaluate
17.
Prove that 3 + 4i
18.
Simplify: (1)
!"#$!!!"!#$! ! !"#!!!"!#! !
!
!
+ 3 − 4i
!
!
=2 5
!
!
cos
!
!
tan!!
!
!
.
!"#$!!!"!#$! ! !"#$!!!"!#$! !!
!"#$!!!"!#$! ! !"#$!!!"!#$! !!
!
! !
(2) 1 + cos + isin
!
!
!
! !
!
!
+ 1 + cos − isin
Unit:2FunctionsandIntegers
1.
Define divisibility.
2.
Define prime number.
3.
Define Greatest common divisor.
4.
Define Least common multiple.
5.
Define congruence.
6.
Define function.
7.
Define one to one function.
8.
Define onto function.
9.
Define injective.
10.
Define surjective.
11.
Define bijective.
12.
Define composition of function.
13.
If f, g, h: R → R be the functions defined by f x = x ! − 4x , g x =
!
!! !!
, h x = x ! then find
fog oh x and fo goh x and check they are equal.
14.
If f, g, h: R → R be the functions defined by f x = x + 2 , g x =
Find (1) gohof x
15.
2 hogof x
!
!! !!
, h x = 3 then
3 gof !! of(x) .
If f x = x + 2 , g x = x − 2 , h x = 3x for x ∈ R where R is set of real numbers.then prove that (1)
fog = gof (2) goh ≠ hog (3) fog oh = fo(goh).
16.
If f: A → B, g: B → C and h: C → D are composite functions then prove that
ho gof = (hog)of .
17.
Find gof(x) if f(x) = x + 4, g(x) = x ! .
18.
Let f(x) = x + 2, g(x) = x − 2, h(x) = 3x, x ∈ R then find foh, foh og, hog, fo hog .
19.
If A = a, b, c , B = x, y and C = {u, v, w} and if f: A → B, g: B → C are functions.
f = { a, x , b, y , (c, x)}, g = { x, u , y, w then find gof.
20.
If A = 1,2,3,4,5 , B = {1,2,3,8,9} and the functions f: A → B and g: A → A are defined by
f = { 1,8 , 3,9 , 4,3 , 2,1 , (5,2)} and g = { 1,2 , 3,1 , 2,2 , 4,3 , (5,2)}
find fog, gof, fof and gog if exists.
21.
If x = 1,2,3 , y = p, q , z = {a, b} and if f: x → y is defined by f =
and g: y → z is defined by g =
p, b , q, b
1, p , 2, p , 3, q
then find gof and fog .
22.
State and prove Fundamental theorem of arithmetic.
23.
When a and b are two integers, b > 0 there exist unique integer q and r such that a = bq + r
Where a ≤ r < b
24.
State and prove Euclid’s algorithm.
25.
Prove that gcd(a, b) can be expresses as an integral linear combination of a and b i.e. gcd(a. b) = ma +
nb,where m and n are integers.
26.
Use the Euclidian’s algorithm to find
(1) gcd(1819,3587)
(2) gcd(12345,54321)
If each case express the gcd as a linear combination of the given numbers.
27.
Find integers m and n such that (1) 512m + 198n = 9.
(2) 100996m + 20048n = 28.
(3) 154m + 260n = 3.
28.
Let s = {1,2,3,4,5} and let f, g, h: s → s be the functions defined by :
f = { 1,2 , 2,1 , 3,4 , 4,5 , (5,3)}
g=
1,3 , 2,5 , 3,1 , 4,2 , 5,4
h = { 1,2 , 2,2 , 3,4 , 4,3 , (5,1)}
(a) find fog and gof.Are these functions equal?
(b) Explain why f and g have inverse but h doesnot.find f !! and g !! .
(c) Show that fog
29.
!!
= g !! o f !!
Determine whether or not each of the following relation is a function with domain 1,2,3,4 .
If any relation in not a function explain why?
(1) R! = { 1,1 , 2,1 , 3,1 , 4,1 , (3,3)}
(2 )R ! = { 1,1 , 2,3 , 4,2 }
(3 )R ! = { 1,4 , 2,3 , 3,2 , 4,1 }
30.
Determine whether or not each of the following relation is a function. If a relation is a function then find
its range.
(1) R! =
2 R! =
x, y where x, y ∈ Z, y = x ! + 7 which is a relation from Z → Z.
x, y where x, y ∈ R, y ! = x which is a relation from R → R.
Unit:3
1
Define the following terms:
Square matrix
Hemitian matrix
Orthogonal matrix
Unitary matrix
Rank of matrix
1 0
0 −2
and 𝐵 =
4 −1
1 −1
2
Find AB where 𝐴 =
3
Write 𝐴! 𝑓𝑜𝑟 𝐴 =
4
Write any symmetric matrix of order 3.
5
Prove that for any matrix A, A+𝐴! is hermitian matrix.
6
Prove that for any square matrices A and B, 𝐴 + 𝐵
7
Prove that 𝑎𝑏!
𝑎
8
Prove that any square matrix can be expressed as a sum of symmetric and skew
symmetric matrices.
9
1 3 3
Find inverse of matrix 𝐴 = 1 4 3 using elementary row oprerations
1 3 4
10
1 −1 2 3
Find reduced row echelon form of A = 4 1 0 2
0 3 0 4
0 1 0 2
11
Solve the following systems of linear equations by Gauss elimination method :
1+𝑖
𝑖
2 − 3𝑖
2 − 4𝑖
4
0
[b] y - 4z = 8, 2x - 3y + 2z = 1, 5x - 8y + 7z = 1
Solve the following equations:
x +3 y + z = 0, -4x - 9y + 2z = 0 , -3y - 6z = 0
= 𝐴! + 𝐵! .
𝑏 ! is nilpotant matrix.
−𝑎𝑏
[a] x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6
12
!