Gauhati University B Sc mathematics first sem algebra question paper 2021 (MAT-HC-1026)

11 Pages·2021·0.466 MB·English

by  Gauhati University

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3 (Sem-1/CBCS) MAT HC2
2021
(Held in 2022)
MATHEMATICS
(Honours)
Paper: MAT-HC-1026
(Algebra )
Pull Marks: 80
Time:T hree hours
The figures in the margin indicate
full marks for the questions.
1. Answer the following as directed
1x10-10
(a) Find the polar representation of z= 2i.
(b) Ifx -0 and y> 0, then what is the value
of r?
Write the negation of the statement 'For
any integer n, n2> n' in plain English
then formulate the negation using set
of and quantifier.
context
Contd.
usingc ounter
(d) Disapprove the statement
examplc:
"For any x.yeE, x =y implies
from
function
(e) Suppose fis a constant
X to Y. The inverse image of a subset
of Y cannot be
an empty set
(i) the whole set X
subset of X
(tui) a non-empty proper
(Choose the cormect option)
n Let X =Y =R. Let Ac X, BY. Draw
the picture for A x B where A=-1,1]
and B-12, 31.
of linear equations
(g) Suppose a system
in echelon form has a 3 * 5 augmented
matrix whose fifth column is a pivot
column.
Is the system consistent ? Justify.
(h) Ifa set S - , D, i n
contains the o vector, is the set
linearly independent ? Justily.
3I Sen-1/CBCS) MAT HC2/ 2
HA compute
(AR
0 What is the determinant of an nn
elementary matris E that has been
scaled by 7.
Answer the following questions: 25-10
2.
al tf z=-2N3-2i, find the polar radius
and polar argument of z
b) 1s the function g :8-- given by
gx)-x-2 one-one and onto ?
Explain
e Let universal set be and index set be
.Fora natural number n..
identity with justification J
neN
3 Sem-1/CBCes MAT HC2/G 3 Contd
d) Show that Tis a linear transformation
by finding a matrix that impletnents
the mapping
TNg, 3. a)=(0, x +2, Xg +X3, X3+X4)
fe) A is a 2x4 matrix with two pivot
positions. Answer the folowing with
justification:
Does Ax -0 have a non-trivial
solution?
i) Does AR -b have at least one
solution for every b?
3. Answer any four questions from the
following
5x4-20
(a) Find the polar representation of the
complex number
zl-cosa+ isina a e| 0, 27).
(b) Let A and B be subsets of universal
an
set U. Prove
0 (AnB-A° UB
f) (AUB-AnB°
5
3 (Sem-1/CBCS MAT HC2/0
Detine bijection.
fc)
Let f-be fm)-m-1. ifm is
even fm)-m+1, if m is odd. Show f
is a bijection and f-f. 1+4-5
(d) For vectors , z, ,e2" define
,
, iz m pconstruct a 33
spun
matrix A with non-zero elements and a
vector b on such that b is not in
the set spanned by the columns of A.
2+35
fe) Alka-Seltzer contains sodium bi
carbonate (NaHCO) and citric acid
(H,CHO). When a tablet is dissolved
in water the following reaction produces
sodium citrate, water and carbon
dioxide:
NaHco, + H,C,H,O, Na,C,H,o,+H,0 C0,
Balance the chemical equation using
vector equation approach.
Prowe that matrix Ai s invertible
an nxn
if and only if A is row equivalent to
and in this case any scquence of
eiementary row operations that reduces
A to I also transforms , into A
3 Bem-1/CBCs MAT HC2/G 55 Contd.
Answer any four from the following
4
10x4-40
of the number
a) Find the cube roots
represent them in the
z=1+iand
5
complex plane.
(i) Firnd the number of ordered pairs
a, b) of real numbers such that
(a +ih=a-ib. 2
ii) 1f , y, z. be real numbers
such that sin x + stny + sin z =0
and cos y+ 0,
cosx+ cos z = prove
that sin 2x + sin 2y+ sin 2z - 0
and cos2x + cos 2y+ cos 2z - 0.
(b) Solve the cquation
2-2iz--2 -0.
5
(1) Find the inverse of the matrix if
it exists by performing
suitable
row operations on the augmented
matrix [A:I]
0-21
A33
5
3 (Scm-1/CBCS) MAT HC2/G
je f f:X->Y be a map and BY,
then prove B°)=U(B)
9A
where N. Find
ne
UA and 4
2
neN neN
i) Let f:>R be given by
f)-x.
Find f 0) Es(o, 1}).
4
(d) )State the induction principle and
use it to show that for positive
any
integer
1+2+3+... +n -"
2
fi) Write as an implication 'square of
an even number is divisible by 4'.
Then use direct proof to prove it.
3
3(Sem-1/CBCS) MAT HC2/G
Contd.
contrapositive
(ut) Give proof using
x-6x+5 is
if
For an integer x
3
even, then x is odd'.
the invertible matrix theorem
e)Use
to decide if A is invcrtible
021
A519
(ii) Compute det A where
2-86 8
3-95 10
A3o1-2
4
1 -4 0
(i) What do you mean by equivalence
class for an equivalence relation?
For the relation a = b mod(5) on
z, find all the distinct cquivalence
classes of z 1+3-4
3 (Sem-1/CBcS| MAT HC2/0
Solve the system of cquations
A-3x3 8
2x+2x2 +9x3 =7
2+5X=-2
Choose h and k such that the
4
system has
a) no solution
(b) unique solution
a
(c) many solutions
X +hx2 =2
4x +8x2 =k
(i) Write the general solution of
10x1-3x2-2x37 in parametric
vector form. 2
(g) Prove that the indexed set
S=,2 pof two
or more
vectors is lincarly dependent if and
only if at least one of the vectors
in Sis a linear combination of the
others. In fact, if S is linearly
dependent and , 0, then
some (vith j> 1) is a linear
combination of the preceding
vectors Dzy, Dy-1
5
3 (Sem-1/CBCs MAT HC2/0 9
Contd.
the
compute
rule to
(it) Use Cramer's 3
of the system
solutions
-5x+3x2 =9
3x-X -5
(1i) Suppose T:R-
matrix
and T()= AX for some
A and each in R5
does
and columns
How many rows
2
A have? Justify.
T:22- be the
(h) ) Let
each
that rotates
transformation
point in R2 about the origin
through an angle with the
counter-clockwise direction taken
positive. Find the standard
as
matrix for this transformation.
(i) Let T:E" -> 5m be a linear
transformation.
Prove that Tis one-to-one if and
only if the equation T(E) =õh as
only the trivial solution. 4
3( Sem-1/CBcS) MAT HC2/G 10