MATHEMATICS(HONOURS)PAPER-III-2005
SECTION-1
1. Choose the correct answers of the following:
(a) Let A and B be two sets each having n elements, then the number of bijections from A
to B is
(i) n! (ii) n (iii) n(n-1) (iv) one
(b)let A={ 1,2,3} and let R= {(1,2),(2,3). (1,3), (3,1)}.Then R is x
(i) a relation in A (ii) not a relation A (iii) reflexive (iv)transitive
(c) If A = [an] be a square matrix of order n, then
(i) A. Adj A = 𝐴−1 (ii) A.Adj A = |A|I (iv) A. Adj A = IA-1
(d) If
1
A= [0
0
0
3
1
0
0
4
2
1
0
5
3
2
0
6
4] then rank (A)= (i)1 (ii)2 (iii)3 (iv)4
3
0
(e) A system of m homogeneous equations in n variables
(i) is always inconsistent (ii) is always consistent
(iii) is sometimes consistent under certain conditions
(iv) has only trivial solution
(f) The function f(x) = 1/𝑥 is
(i) both continuous and uniformly continuous in (0,1)
(ii) neither continuous nor uniformly continuous in (0,1)
(iii) uniformly, continuous but not continuous in (0,1)
(iv) continuous but not uniformly continuous in (0,1)
(g) The function f(x) defined on R by
2
f(x) = 𝑒 −1/𝑥 , when x ≠ 0
= 0
, when x = 0 then
(i)
(ii)
(iii)
(iv)
f(x) does not possess derivatives of all orders
f(x) has derivatives of all orders
f(x) has Maclaurin's series expansion.
remainder Rnin the expansion converges to 0
(h) If f(x) is R-integrable in [a,b] and |f(x)|≤ M for some constant M then
𝑏
(i)| ∫𝑎 𝑓(𝑥)𝑑𝑥| ≤ 𝑀
𝑏
(ii)|∫𝑎 𝑓(𝑥)𝑑𝑥| ≤ 𝑀| 𝑏 − 𝑎|
𝑏
𝑏
(iii)|∫𝑎 𝑓(𝑥)𝑑𝑥 |≥ 𝑀|𝑏 − 𝑎|
(iv) |∫𝑎 𝑓(𝑥)𝑑𝑥| ≤ |𝑏 − 𝑎|
(i) The function z = e-v (cosu + isin u) ceases to be analytic when
(i)z=1 (ii)z = -1 (Hi)z =I (iV) z = 0
(j) The mapping of z - plane upon w - plane given by W= 1/z represents
(i) inversion in the real axis
(ii) inversion in the unit circle
(iii) inversion in a real axis and the unit circle
(v)
translation into imaginary axis
SECTION-II GROUP-A
2.(a) State De Morgan's laws in general forms:
(b) prove thaf mapping f, g of set X into a set Y are equal if and only if
F(x)=g(x)for all x∈ X.
3. (a)Define transcendental numbers.
(b) Show that the set of all irrational numbers is uncountable.
4. (a) Define a relation induced by a partition of a set with a suitable example.
(b) State and prove fundamental theorem on equivalence relation.
5. (a) Define a lattice and give an example of a partial ordered set which is not a lattice.
(b) Define a partially ordered set and prove that for partially ordered set (x,≤), the
following statements are equivalent:
(i) Every non-empty subset of X which has a lower bound has a gratest lower bound.
(ii) Every non-empty subset of X which has an upper bound has laest upper bound.
GROUP-B
6. (a) Prove that every orthogonal matrix is non-singular.
(b) Prove that every square matrix is uniquely expressed as Hermitian and skew-Hermitian
matrix.
7. (a) What are normal forms of a matrix ?
(b) Using Cayley-Hamilton theorem, find the inverse of
𝟏
[𝟏
𝟏
𝟏 𝟏
𝟐 𝟑]
𝟑 𝟔
8. (a) Prove that three points (𝑥1 𝑦1 )𝑎𝑛𝑑 (𝑥2 , 𝑦2 )𝑎𝑛𝑑(𝑥3 , 𝑦3 ) are colinear if rank of the
matrix.
𝑥1
[𝑥2
𝑥3
𝑦1
𝑦2
𝑦3
1
1]
1
is less then 3.
(b) Prove that the elementary transformations do not alter the rank of a matrix.
9.(a) Define basis and dimensions of a vector space V.
(b) Show that the vectors e1 , e2 , e3 … … en are linearly independent and they
form a basis of 𝑣𝑛 (R), where
𝒆𝟏 = (𝟏, 𝟎, 𝟎 … … , 𝟎), 𝒆𝟐 = (𝟎, 𝟏, 𝟎 … … . , 𝟎), … … , 𝒆𝒏 = (𝟎, 𝟎, 𝟎 … … 𝟏)
10, (a) Define consistency and inconsistency of a system of linear equations.
(b) If A be non-singular n rowed matrix, prove that AX = B possesses a
unique solution.
11. (a) Write different types of elementary matrices and their symbols.
(b) Show that the equations -4 x + jy + az = c
5x -4y + bz = d always have a solution for all values of a, b,c, d
GROUP-C
12. (a) Define removable discontinuity of a function at a point with a suitable example.
(b) Prove that if a function is continuous in a closed interval it also is uniformly
continuous in that interval.
1
13. (a) Show that lim sin 𝑥−𝑎 does not exist.
𝜋
(h) for f(x) = x cos 𝑥 , when x ≠ 0
= 0
, when x = 0
on [0,1] and corresponding to the partition
1 1
1 1
P={0, ,
, … . . , 3 , 2 , 1} show that f not of bounded variation on [0,1]
𝑛 𝑛−1
though it is continous in [0,1].
14.(a)Write Lagrange's form of remainder after n terms in the expansion of 𝑒 1
(b)State and prove Cauchy's mean value theorem.
15.(a) Prove that every constant function is R-integrable in any interval [a,b].
(b) Show that bounded function f is R-integrable in [a.b] if the set of its points of
discontinuity is finite.
16.(a) Define primitive of a function f.
(b) If f(x) is Riemann integrable in [atb], then show that |f(x)|is R-integrable in [a.b] and
𝑏
𝑏
| ∫ 𝑓(𝑥)𝑑𝑥| ≤ ∫ |𝑓(𝑥) |𝑑𝑥
𝑎
𝑎
GROUP-D
17.(a) If a function is regular, it is independent of 𝑧̅ and is function of z . Prove it.
(b) Derive necessary condition for a function to be analytic.
18. (a) Describe Milne-Thomson method of constructing analytic function
F(z) = u+iv when one conjugate function be given.
(b) Show that the function
f{z)=V(\xy\)
is not regular at origin, although the Cauchy-Riemann equations are satisfied at that
point.
19.(a) State the rule of preservance of cross-ratio under the bilinear transformation.
(b) Find the condition that the transformation w=az+b/az+c transforms the unit circle in
the W- plane into a straight line in the z-plane.
20.(a) What does the equation
𝑧−𝑧
arg (𝑧−𝑧1 ) = 𝜆 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡?
2
(b) state and prove necessary condition for conformal mapping.
MATHEMATICS(HONOURS)PAPER-III-2006
SECTION-1
1.Choose the correct answer of the following :
(a) Let A = {1,2,3} and let R= {(1,2). (2,3),(I,3), (3,1)}, then which is not true?
(i) R is not a relation (ii) R is not reflexive (iii) R is not symmetric (iv) R is not
transitive'
(b) If card. R = C and card. N = 𝜆 then:
(i)C + 𝜆 = 𝜆 (ii)C + 𝜆 =C (iii)C + C = 2C (iv) 𝜆. 𝜆 = 𝜆 2
(c) The sum of the characteristic roots of the
1 0
martix[0 2
0 0
0
0 ]is: (i)0 (ii)1 (iii)2 (iv)4
−1
cos 𝛼 −sin 𝛼
(d) [
] is
sin 𝛼 cos 𝛼
(i) Singular (ii)Orthogonal (iii) Symmetric (iv) Hermitian
(e) If A and B be two matrices conformal for the product AB; then:
(i) P(AB) = P(A).p(B) (ii)p(AB)≤P(A).P(B) (iii)P(AB)≤P(A) and
P( AB) ≤ P(B) (iv) P( AB) > P(A) and P(AB) > P(B)
(f) if f(x) is continuous in [ a, b], then:
(i) fix) is bounded in [a, b] (ii) f(x) is not bounded in [a, b] (iii) f(x) necessarily
differentiable in [a, b] (iv) None of these
(g) Let f be bounded function defined on [a, b] and P be a partition of [a,b], if p* is a
refinement of P, then:
(i)L(P) < L(P*) ≤ U(P*)≤ U{P) (ii)L(P*) ≤ L(P) ≤U(P) ≤ U(P*)
(iii) L(P*)≤ L(P)≤ U(P*) ≤ U(P) (iv) L(P) ≥ L(P*) > U(P) > U(P*)
(h) Let f be bounded function defined on [a. b], then to every 𝜀 > 0,there corresponds a 𝛿> 0
for all partition P of [a. b] with norm 𝜇(P) ≤ 𝛿:
𝑏
𝑏
𝑏
𝑏
(i) LP<∫𝑎 𝑓 − 𝜀 𝑎𝑛𝑑 𝑈(𝑃) > ∫𝑎 𝑓 + 𝜀 (ii) LP<∫𝑎 𝑓 − 𝜀 𝑎𝑛𝑑 𝑈(𝑃) < ∫𝑎 𝑓 + 𝜀
𝑏
𝑏
𝑏
𝑏
(iii) LP>∫𝑎 𝑓 + 𝜀 𝑎𝑛𝑑 𝑈(𝑃) < ∫𝑎 𝑓 − 𝜀 (iv) LP=∫𝑎 𝑓 + 𝜀 𝑎𝑛𝑑 𝑈(𝑃) = ∫𝑎 𝑓 − 𝜀
(i)
(𝑖)
The inverse of the point z with respect to the circle | z| = r is:
𝑟2
z
(𝑖)
𝑧
r2
(𝑖)
z
r2
(𝑖)
𝑟2
z
(j) The fixed points of the transformation W = Z-1/Z+1 are :
(i) ± 3 (ii) l±i (iii)-l±i (iv) ±i
Section - II Group – A
2- (a) pefine indexed family of sets.
(b) State and prove generalized De-Morgan's Laws.
3. (a) Defme partition of a set.
(b) Show that the relation "congruence modulo m" it an equivalence relation in z. Describe
the equivalence classes and show that the collection distinct equivalence classes with
respect to "congruence modulo m" is a partition of z.
4. (a) Show that cancellation laws of addition and multiplication do not hold for
cardinal numbers of infinite set.
■
(b) State and prove Schroeder-Bernstein theorem.
5.(a)State Zom's Lemma.
(b)Define partially and totally ordered sets. Distinguish them by constructing example of a
partially ordered set which is not totally ordered.
Group - B
6. (a) Show that every square matrix can be expressed as the sum of a symmetric
and a skew-symmetric matrices,
(b) State and prove Caley-Hamilton theorem.
7 (a)Define elementary matrices.Give an Example of an elementary matrix.
(b) Reduced the matrix:
0 1 −3 −1
1 1]
[ 1 0
3 1
0 2
1 1 −2 0
To normal form and hence find its rank
8. (a) Define row and column equivalence of matrices.
(b)Prove that the rank of a product of two matrices cannot execute the rank of either
matrix.
9. (a) Show that the vectors 𝑥1 = (1,2,4)𝑎𝑛𝑑 𝑥2 = (3,6,12) are linearly dependent.
(b) Under what conditions on the scalars, do vectors
(1,1,1), (1, 𝜉, , 𝜉 2 )𝑎𝑛𝑑 (1, −𝜉, 𝜉 2 )𝑓𝑜𝑟𝑚 𝑎 𝑏𝑎𝑠𝑖𝑐 𝑜𝑓 𝐶 3 ?
10. ( a) Dcfine row rank of a matrix.
(b) Prove that the row rank of a matrix is the same as its rank.
11. (a) If A be an n-round non-singular matrix, X and bare n× 1 matrices then prove that the
system of equation AX=B has a unique solution.
(b) Investigate for what values of a and b the equation
x+2y+3z=4
x+3y+4z=5
x+3y+az=b
have (i) no solution, (ii) a unique solution and(iii) an infinite number of solutions.
Group – C
Lt
12. (a) If the function f(x) is defined by f(x) =
𝑒 1/𝑥
1
𝑒𝑥+ 1
does x ->0 f(x) exists ?
(b) Prove that a function which is continuous in a closed interval [a, b] is uniformally
continuous in [a, b].
13. (a)Examine the continuity of the function f(x) = | x | ; x ≠ 0 and f(0) = 0.
(b)Prove that product of two functions of bounded variation on [a, b] is also of bouonded
variation on [a, b].
1
14. (a)If f(x) = x cos𝑥; 0 and f(0) = 0, find' (x) if exists.
(b) Obtain power series expansion of log(l + x) using suitable form of remainder.
15. (a) Prove that the lower R-integral cannot exceed upper R-integral.
(b) Prove if f is monotonic on [a, b], then it is R integrable on [a, b]
16. (a) define oscillatory sum.
(b) If f𝜖R [a, b] and g𝜖R [a, b], then prove that fg 𝜖 R [a,b].
Group – D
17. (a) If f(x, y) =𝑥 2 +𝑦 2, show that the repeated limits exits at (0,0) and are equal but double
𝑥𝑦
limits does not exits
(b) Examine continuity and differentiability of the function f(x, y) at (0.0). 1f
𝑥𝑦 2
f (x, y)= 𝑥 2 +𝑦 2 ; (𝑥, 𝑦) ≠(0,0) and f (0,0) = 0.
18. (a) Show that u - x3 -3xy2 is harmonic arid find its harmonic conjugate,
(b) State and prove sufficient condition for. f(z) to be analytic.
19. (a) show that an analytic function with constant modulus is constant,
(b) if W = f(z) represents a conformal transformation of a domain D in the z-plane into a
domain D' of the W-plane, then f(z) is an analytic function z in D.
.
20. (a) Define bilinear transformation.
(b) Show that the transformation w = 2x+3/z-4 maps the circle x2 + y2-4x =0
onto
the straight line 4u + 3 = 0 and explain, why the curve obtained is not a circle.
MATHEMATICS(HONOURS)PAPER-III-2007
SECT1ON-I
1. Choose the correct answer of the following :
(a) Let X = {1,2,3,4,5,6,7,8}, then which of the following familes of subsets
of X define a partition of X ?
(i){{l,2,3}, {2,4,8}, {5,6}, {6,7,8}}
{ {1},{2,3},{4,6,8},{5},{7} }
{ {1},{2},{3},{8},{2,7,8},{5,6} }
{ {1,2,3,4},{2,4,6,8},{5,7} }
(b) The set R of real numbers is
(i) enumerable (ii) countable (iii) uncountable (iv) denumerabie
(c)The matrix [
cos 𝜃 − sin 𝜃
]is
sin 𝜃 cos 𝜃
(i) orthogonal (ii) Hermitian (iii) unit (iv) symmetric
(d) If rank of A = 𝑟1 rank of B = 𝑟2 and rank of AB = 𝑟1 then
(i) r = 𝑟1 , 𝑟2 (ii) 𝑟1 ≤ r (iii) r ≤ 𝑟1 (iv) 𝑟2 ≤ r
(e) If adj A = B, then
(i) B.adj A = | A | (ii) AB = BA (iii) AB ≠BA(iv) B.adj A=A-1
(f) The function f(x) = |x| defined on[-l,l]is
(i) continuous but not differentiate at x = 0
(ii) continuous and differentiate at x = 0
(iii) differentiable but not continuous x = 0
(iv) neither continuous nor differentiable at x = 0
(g) The Lagrangian form of remainder in Taylor's theorem is
(i) ℎ𝑛 𝑓 𝑛 (𝑎 + 𝜃ℎ)
(ii)
ℎ 𝑛−1
(1 −
𝑛−1
𝜃)𝑛−1 𝑓 𝑛 (𝑎 + 𝜃ℎ)
n
(iii) ℎ𝑛 𝑓 𝑛−1 (𝑎 + 𝜃ℎ) (ii)
ℎ 𝑛 (1−𝜃)𝑛−𝑝 𝑛
𝑓
𝑛−1
(𝑎 + 𝜃ℎ); 0 < 𝜃 < 1; 𝑝 > 0
n
(h) If f is bounded and monotonic on [a.b] and g is R-integrable on [a.b] and a≤c≤b, then
𝑏
𝑏
𝑏
𝑏
𝑐
𝑏
(i) ∫𝑎 𝑓. 𝑔 = ∫𝑎 𝑓. ∫𝑎 𝑔
(ii)∫𝑎 𝑓. 𝑔 = 𝑓(𝑎) ∫𝑎 𝑔. ∫𝑎 𝑔
𝑏
𝑏
𝑏
𝑏
(iii) ∫𝑎 𝑓. 𝑔 = ∫𝑎 𝑓 + ∫𝑎 𝑔
(i)
𝑐
𝑏
(iv) ∫𝑎 𝑓. 𝑔 = 𝑔(𝑎) ∫𝑎 𝑓 + 𝑓(𝑏) ∫𝑐 𝑔
Polar forms of Cauchy-Riemann equation are
(𝑖)
𝜕𝑢
(𝑖𝑖𝑖)
𝜕𝑣 𝜕𝑢
𝜕𝑣
𝜕𝑢
1 𝜕𝑣 1 𝜕𝑢
𝜕𝑣
= 𝑟 𝜕𝜃 , 𝜕𝜃 = −𝑟 𝜕𝑟 (𝑖𝑖) 𝜕𝑟 = 𝑟 𝜕𝜃 , 𝑟 𝜕𝜃 = − 𝜕𝑟
𝜕𝑟
𝜕𝑢
𝜕𝑟
𝜕𝑣 𝜕𝑢
1 𝜕𝑣
1 𝜕𝑢
= 𝑟 𝜕𝜃 , 𝜕𝜃 = − 𝑟 𝜕𝑟 (𝑖) 𝑟
𝜕𝑟
𝜕𝑣
𝜕𝑢
𝜕𝑣
= − 𝜕𝜃 , 𝑟 𝜕𝜃 = 𝜕𝑟
(/) The fixed points of the transformation W =2/2-z are
(i)
0,1 (ii) 0,2 (iii) i,2 (iV) 2,3
SECTION-II GROUP-A
2.(a)If f: X -> 1Y be a mapping and A⊆ X, B ⊆ X, then prove that f(A ∪B)=f(a) ∪f(b).
(b)If {𝐴𝑖 : 𝑖 ∈ 𝑙} be an indexed family of subsets of the universal set Ω and B⊆ Ω, then
show that
(i) Bᴗ( ∩ 𝐴𝑖 ) = ∩(B ᴗA) (ii) B ∩( U𝐴𝑖 ) = U (B∩𝐴𝑖 )
3.(a) Define relation. Give an example of a relation which is symmetric and
transitive but not reflexive,
(b) If f: X -> Y and g: Y —> Z be two one-one onto mappings, then prove
That g f: X —> Z is also one-one onto and (g f)−1=f1.𝑔−1
4.(a) Define denumerable set.
(b) Prove that denumerable union of denumerable sets is denumerable
5.(a) Define lattice and give an example of a poset which is not a lattice
(b) If A be any set, then show that
card P(A) = 2 card A where P(A) denotes power set of A.
GROUP-B
6. (a) Show that every square matrix can be uniquely expressed as the sum of Hermitian
and a skew-Hermitian matrix.
(b) Show that the matrix .
2 2 1
A=[1 3 1]
1 2 2
Satisfies Cayley-Hamilton theorem. Hence or otherwise and A-1.
7.(a) Define Echelon form of a matrix with example.
(b) Prove that interchange of a pair of rows or columns does not cha the rank of a matrix.
I
8.(a) If A and B be two equivalent matrices, then show that rank A = rank
(b) Find two non-singular matrices P and Q such that PAQ is in non form where
1 1
2
A= [1 2
3]
0 −1 −1
9. (a) Define column rank of a matrix.
i
(b) If r be the row rank of an m x n matrix A, then prove that there exists
𝐾
non-singular matrix P such that PA = [ ] I where K is an r x n matrix
𝑂
consisting of a set of r linearly independent rows of A.
10. (a) Define basis of a vector space.
(b) A set S consisting n, vectors 𝑒1 = (1,0,0,...,0), e2=(0,1,0,...,0).... 𝑒𝑛 =( 0,0.....1). show
that S is a basis of 𝑉𝑛 (F).
I
11. (a) If X, and X2 be any two solutions of the system of homogeneos equations AX = 0,
then show that K1X1 + K2 X2 is also a solution of AX=0.
(b) Show that the equations
x + 2y-z = 3
3x - y + 2z = 1
2x-2y + 3z = 2
x - v + z = -1 are consistent and hence solve them.
GROUP-C
12. (a) Show that f (x) = 1/x is coritinous for all real values of x except x =0.
(b) If (x) is continuous in the closed interval [a.b], then show that f
(x) attains its bounds at least once in [a,b].
13. (a) If f(x) is a function of bounded variation on [a,b] and k > 0 such that
1
f(x)\ ≥ k for all x ∈[a,b] then prove that𝑓(𝑥)is alos a function of
bounded variation over [a,b],
(b)Examine the continuity and differentiability of the function
𝑓(𝑥) =
f(0)=0
1
1
1
1
−
𝑒 𝑥 −𝑒 𝑥
−
𝑒 𝑥 +𝑒 𝑥
, when x≠ 0
14. (a) State Taylor's mean value theorem with Lagrange's form of remainder.
(b) ln the mean value theorem, prove that.
1
𝐿𝑇 𝜃 = 2
h->0
provided 𝑓 𝑛 (x) exists in]a, b[ and is continuous in [a,b] and does not vanish anywhere in
[a,b].
15. (a) If 𝑃1 and 𝑃2 be any two partitions of [a,b], then prove that U(P1) ≥ U(P2).
(b) If f(x) is continuous on [a.b], then prove that it is R-integrable on [a,b].
16. (a) 1f f ∈ R [a,b], then prove that |f| ∈ R [a.b].
(b) fe R [a.b], g ∈ R [a,b] and g(x)≥ 0 or g(x) ≤ 0 for all x ∈ [a,b], then show that there
exists a number k lying between the bounds m and M of f such that
𝑏
𝑏
∫𝑎 𝑓. (𝑥). 𝑔(𝑥)𝑑𝑥 = 𝑘 ∫𝑎 𝑔(𝑥)𝑑𝑥
GROUP-D
17. (a) Examine the continuity of
𝑓(𝑥, 𝑦) = 𝑥𝑦/√𝑥 2 + 𝑦 2 , x≠ 0, 𝑦 ≠ 0 𝑎𝑛𝑑 𝑓(0,0) = 0 𝑎𝑡 𝑜𝑟𝑖𝑔𝑖𝑛 (0,0)
(b) Let (a,b) be a point of the domain of a function f such that (i)𝑓𝑥 exists in a certain
neighbourhood of (a,b) and (ii) fxy (a,b)continuous at (a,b) then show that 𝑓𝑥𝑦 (𝑎, 𝑏)
exists and is equal to fxy(a,b).
18. (a) Show that an analytic function with constant modulus is constant,
(b) If a function f(z) = u (x,y) + iV (x,y) is differentiate at any point z = x + iy, then
show that the four partial derivatives 𝑢𝑥 , 𝑢𝑦 , 𝑣𝑥 , 𝑣𝑦 exist and satisfy the equation.
𝑢𝑥 = 𝑣𝑦 : 𝑢𝑦 = −𝑣𝑥
19. (a) Discuss the tranformation W = z + 𝛼.
I
(b) Determine the region in W-plane into which the rectanngular region bounded by the
lines x = 0, y =0, x = 1, y = 2 in the z-plane is mapped under the transaction
W=(1+i)z+2-i
20. (a) Define bilinear transformation. Give the condition when it said to be
normalised.
(b) Find the bilinear transformation and inverse bilinear transformation which maps the
points z1= 2, z2=i, and = z3=-2, into the points 𝑤1 = 1, 𝑤2 = 𝑖, 𝑤3 = −1 .
MATHEMATICS(HONOURS)PAPER-III-2008
Section-I
1. Chose the correct answer of the following :
(a) Let X = {1,2}, define R = {(1,1), (2,2), (1.2)}, then :
(i) R is reflexive, symmetric, but not transitive
(ii) R is reflexive, transitive, but not symmetric
(iii) R is symmetric, transitive but not reflexive
(iv) R is reflexive, symmetric and transitive
(b) If card N = 𝜆0 , card R=C, then :
(i) 𝜆0 = c (ii) 𝜆0 > c (iii) 𝜆0 + c =c (iv) 𝜆0 +c >c
2
(c) The rank of the matrix[ 1
3
4
4 6 8
2 3 4 ]is:
6 9 12
8 12 6
(i)l (ii) 2 (iii) 3 (iv) 4
(d) Set of Vectors {(1,0,1), (1,1,0), (1,1,-1)} is :
(i) Linearly dependent
(ii) Linearly independent
(iii) Neither linearly dependent nor linearly independent
(iv) Linearly dependent and linearly independent both
(e) The sum of the characteristic roots of the Matrix
1
[0
8
0 0
4 0]is
2 1
(i)4 (ii) 2 (iii) 8 (iv) 1
(f)
The function f defined in R by f (x) = |x| + |x-l| is :
(i) Differentiable at x = 0
(ii) Diff. but not continuous at x = 0
(iii)Continous but not diff. at x = 0
(iv) Continuous and diff. both at x = 0
(g) if F (x) = 1 when x is rational = -1 when x is irrational, then
(i) f is R-integrable but |f| is not R-integrable
(ii)
|f| is R-int. but f is not R-int.
(iii)f and |f| both are R-int.
(iv) None of these hold
(h) If be bounded function on [a, b], P be a partion of [a, b] and P* be a refinement of P, then:
(i) U(P) ≤ U(p*)≤L(P*)≤L(P)
(ii)L(P) ≤L(P*) ≤U(P*)≤U(P)
(iii)L(P*) ≤ L(P) ≤ U (P*) ≤ U(P)
(iv)L(P) ≤ L(P*) ≤ U(P) ≤U (P*)
(i) The system PF curves u (x, y) = 𝑐1 and v(x,Y) = c2 are
orthogonal if:
(𝑖)
𝜕𝑢 𝜕𝑣
.
(𝑖𝑖𝑖)
(j)
+
𝜕𝑢 𝜕𝑣
.
.
𝜕𝑥 𝜕𝑥
𝜕𝑢 𝜕𝑣
= 0 (𝑖𝑖) 𝜕𝑥 . 𝜕𝑥 +
𝜕𝑥 𝜕𝑥
𝜕𝑦 𝜕𝑦
𝜕𝑢 𝜕𝑣
𝜕𝑢 𝜕𝑣
− 𝜕𝑦 . 𝜕𝑦 (𝑖𝑣)
𝜕𝑢 𝜕𝑣
.
𝜕𝑥 𝜕𝑦
𝜕𝑢 𝜕𝑣
.
𝜕𝑦 𝜕𝑥
𝜕𝑢 𝜕𝑣
=0
− 𝜕𝑦 . 𝜕𝑥 = 0
Inverse of the point Z w.r.t. the circle |z| =𝑟 2 𝑖𝑠:
𝑟2
𝑟2
𝑧
𝑧̅
(𝑖𝑖)
(𝑖𝑖𝑖) 2 (𝑖𝑣) 2
𝑧
𝑧
𝑟
𝑟
Section - II Group-A
2. (a) Let f: x -» y; A ⊆ y, B ⊆ y, then prove that f-1 (A∩B)= f-1(A) ∩ 𝑓 −1 (𝐵 ).
(b) If (Ai: i ∈ I} and {Bj: j ∈ J) are indexed families of subsets of the universal set 𝛺
then :
(i)
(ii)
Bᴗ( ∩ 𝐴𝑖 ) ᴗ( ∩ 𝐵𝑗−1 ) = (i,j)∩(𝐴𝑖 ᴗB)
( U𝐴𝑖 ) ∩(U𝐵𝑗 ) = U (𝐴𝑖 ∩𝐵𝑗 )
3. (a) What do you mean by partition of a set ? Explain with example.
(b) If R is an equivalence relation in a non-empty set X, the collection of all equivalence
classes is a partition of X that induces the relation R and conversely if D be a partition
of X then the induced relation is an equivalence relation whose collection of
equivalence classes is, exactly D.
4. (a) If E and F are any two denurnerable sets, then so is their Cartesian product E.x F.
(b) State and prove Schfoeder-Bemsteine Theorm.
5. (a) Define Maximal and minimal elements of a. set.
(b) Define partially,ordered set. Prove that if (X, ≤) is a!2 ] partially ordered set then the
following statements are equivalent :
(i)Every non-empty subset of X, which has an upper bound has a least upper bound.
(ii)Every non-empty subset of X, which has a lower bounds has a greatest lower bound.
Group-B
6. (a)If A be and n * n non-singular matrix then prove that
(𝐴1 )−1 = (𝐴1 )−1 .
(b)prove that every square matrix satisfies its characterstics equation.
7. (a) Define elementary with example.
(b) Reducing, into normal form, find the rank of the matrix
1 2−1
[4 1 2
3 −1 1
1
2 0
3
1]
2
1
8.(a) Prove that invertible matrix possesses a unique inverse.
(b) Prove that the rank of the -product of, two matrices cannot exceed the rank of the
either matrix.
:
9. (a) Define Linear span.
(b) Prove that there exists a basis for each finite dimensional vector space.
10. (a) Define Row rank of a matrix.
(b) Prove that the row rank of a matrix is the same as its rank,
11. (a) State the condition under which a, system of non-homogeneous equations is sold be
consistent.
(b) Investigate for what values of a and b the system of equation : X + 2y + 3z =4,
X + 3y+4z =5, X+3y+az = b have (i) no solution, (ii) unique solution and (iii)
aninfinite, number of solutions,
Group-C
12.(a) If f is continuous in a closed interval I, then so is | f |.
(b) Prove that a function which is continuous in a closed interval [a, b] is uniformally
continuous there.
13. (a) If f (x)=sin x/x ; x≠ 0
=1; X=0
Discuss the continuity at X = 0/
(b) What do you mean by total variation of a function ? Prove that the product of two
functions of bounded variations on f[a,b] is also of bounded variation on [a,b].
14. (a) Discuss the differentiability of f (x) = | x| at x: = 0.
(b) Obtain power series-expansion-of log (1+ X) using suitable from of remainder after n
them.
15.(a) If 𝑃1 and P2 be any two partitions of [a, b], then prove that U (𝑃1 ) ≥ L(P2).
(b) State and prove darboun theorem.
16. (a) Prove that if f ∈ R[a, b], then | f | ∈ R [a, b].
(b) Prove that every monotonic increasing function on [a,b] is R-integrable.
Group-D
𝑋𝑌
17. (a) If f (x,y)=𝑋 2 +𝑦 2 ; (x,y)≠(0,0)
= 0
;(x,y)=(0,0)
Does it (xy)Lt(o,o))f(x,y) exists?
(b) Examine the continuity and differentiability of
F(x,y)=
𝑋𝑦 2
𝑋 2 +𝑦 2
= 0
at point (0, 0)
; (x, y)≠ (0,0)
; (x,Y) = (0,0)
18. (a) Show that the function u =1/2log (x2 + y2) is harmonic and find its harmonic
conjugate.
(b) If f (z) is an analytic function of Z, then prove
𝜕2
𝜕2
That(𝜕𝑋 2 + 𝜕𝑦 2 )| Rf(z)|2 = 2|𝑓 1 (𝑧)|2 .
19. (a) Discuss inversion W =1/z
i
(b) If W = f (z) represents a conformal transformation of a domain D in the Z-plane into a
domain D1 of the W- Plane, then f (z) is analytic function of Z in D.
20. (a) Define belinear trasformation. State when it is said to
be normalised.
(b) Find the bilinear trasformation which maps the points 𝑍1 = 2, 𝑍2 = 𝑖 𝑎𝑛𝑑 Z3 = -2
intop the points 𝑊1 = 1, 𝑊2 = 𝑖, 𝑊3 = −1.