Msc maths model question papers of previous years

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The M.Sc Mathematics (IDE) Examination question paper is as follows:

I. A) a) Show that a finite dimensional subspace of a normed space X is closed in X.
b) Show by an example that an infinite dimensional subspace of a normed
space X may not be closed in X.
c) Show that the closed unit ball ) p ( p ∞≤≤1 l is convex, closed and bounded
but not compact. (6+5+6)

B) a) Let X, Y be normed spaces and Y X : F →a linear map. Prove that F is
continuous if and only if there exists 0 such that x ) x ( F ≤for all
X x∈.
b) Show that a linear functional f on a normed space X is continuous if and only
if z(f) is closed in X.
c) Give an example of a discontinuous linear map. (6+6+5)
II. A) a) Show that a normed space X is Banach if and only if every absolutely
summable series of elements in X is summable in X.
b) Let Y be a closed subspace of a normed space X. Show that X is Banach if
and only if Y and
Y
X
are Banach spaces in the induced norm and quotient
norm respectively. (8+9)

B) a) Show that a nonzero linear functional on a normed space is an open map.
b) State and prove Hahn-Banach extension theorem.
c) Let X = K2 with norm || ||∞and } ) ( x : K )) ( x ), ( x {( Y 0 2 2 1 2 ∈.

Define Y g ′∈by g (x(1), x(2)) = x(1). Show that the only Hahn Banach
extension of a g to X is given by f(x(1), (x(2)) = x(1). (5+7+5)
III. A) a) Let X be a normed space and E be a subset of X. Show that E is bounded in
X if and only if, f(E) is bounded in K for every X f ′∈
b) State and prove Closed Graph Theorem. (7+10)

B) a) Show that a linear functional f on a normed space is closed if and only if f is
continuous.
b) State and prove Open Mapping Theorem.
c) Let ] b , a [ C X ′with ∞∞x x x and ] b , a [ C Y ′with supreum
norm. Show that the map Y X : F →defined by F(x) = x is linear and
continuous but not open. (5+7+5)
IV. A) a) Let X be a normed space and ) X ( BL A∈. Show that A is invertible if and
only if A is bounded below and surjective.
b) If X is a normed space and ) X ( BL A∈define the spectrum ) A ( , eigen
spectrum ) A ( e and approximate eigen spectrum ) A ( a . Show also that
) A ( ) A ( ) A ( a e .
c) If X is a nonzero Banach space over C and ) X ( BL A∈prove that ) A ( is
nonempty. (5+7+5)

B) a) Let p X l with norm || ||p ∞p 1 .
For X = X .....) ) ( x ), ( x ( ∈2 1 let

Show that ) X ( BL A∈. Also find ) A ( e , ) A ( a and ) A ( .
______ -3- 4145
b) Define the transpose F′of a bounded linear map ) Y , X ( BL F∈show that
F F F ′.
c) If X is a Banach space and ) X ( BL A∈show that ) A ( ) A ( . (6+5+6)
V. A) a) Define reflexive normed space. Prove that a reflexive normed space is Banach.

Is the converse true ? Justify.
b) Define a compact linear map and give an example. Show that the set CL(X, Y)
of all compact linear maps from a normed space X to a Banach space Y is
closed in BL(X, Y).
c) Let X be a Banach space and ) X ( BL P∈be a projection. Show that
) X ( CL P∈if and only if P is of finite rank. (5+7+5)
B) Let X be a normed space and ) X ( CL A∈. Prove that
a) every nonzero spectral value of A is an eigen value of A.
b) the eigen spectrum of A is countable.
c) every eigen space of A corresponding to a nonzero eigen value of a A is finite
dimensional. (7+5+5)

I. A) a) Define roundoff error and truncation error.
b) Find a root of the equation x3 – x – 1 = 0 by bisection method.
c) Find a double root of the equation f(x)= x3 – x2 – x + 1 = 0. (3+10+4)
OR
B) a) Show that the order of Newton-Rapshon method is atleast two.
b) Find all roots of the equation 0 6 x 18 x 9 x 2 3 by Graeffe method
(root squaring method, 3 times).
c) Explain matrix Inversion method to solve a system of linear equation. (4+10+3)

II. A) a) Find the cubic polynomial which takes following values y(0) = 1, y(1) = 0,
y(2) = 1, y(3)=10. Also obtain y(4).
b) Apply Gauss central difference formula and estimate f(32) from following
table.
x 25 30 35 40
y=f(x) 0.2707 0.3027 0.3386 0.3794 (5+12)

OR

B) a) Find the polynomial of degree two which takes the values
x : 1 2 3 4 5 6 7
y : 1 2 4 7 11 16 22
b) Using Lagrange’s interpolation formula and R(x). Find the form of the function
y(x) from the following table.
x 0 1 3 4
y –12 0 12 24 (7+10)

III. A) a) Find
dx
dy
and 2
2
dx
y d
at x = 51, using Newton’s forward formula for derivatives
for the data.
x : 50 60 70 80 90
y : 19 – 96 36 – 65 58 – 81 77 – 21 94 – 61
b) Evaluate ∫
2
0
dx x sin , by Simpsons 3

1 rd rule dividing the range into six equal
parts. (8+9)

OR
B) a) Evaluate ∫
−
3
3
4dx x by using Trapezoidal rule take h =1.
b) Evaluate dy dx e
1
0
1
0
y x ∫∫using, a) Trapezoidal rule and Simpsons’ rule. (5+12)

IV. A) a) From Taylors’ series for y(x), find y(0.1) correct to four decimal places if
y(x) satisfies 2 y x y −and y(0) =1.
b) Given
dx
dy
= 1+y2 where y = 0 when x = 0. Find y(0.2), y(0.4), y(0.6) by
using Rungekutta method. (8+9)
OR
______ -3- 4154

B) a) Find the value of y(0.1) by Picards’ method given x y
x y
dx
dy

−, y(0) =1.
b) Solve the differential equation 2 y 1 y with y(0) = 0 by Milne’s-Thomson
method. Also find y(0.8) and y(1.0). (8+9)

V. A) a) Write a C/C++ program to find the positive root of 0 1 x x ) x ( f 3 by
bisection method.
b) Write a C/C++ program to find root of 0 5 x 2 x3 by Newtons – Raphson
method. (9+8)

OR
B) a) Write a program in C/C++ to compute the solution of y , x ( F
dx
dy ), y(x0) = y0
using Eulers’ method.
b) Write a program in C/C++ to solve a system of equations using
Gauss-Elimination method. (9+8)

__________________
Answered By StudyChaCha Member