VI Semester B.A/B.Sc.

Examination, June/July 2023

(CBSC) (2022-23 and Onwards) (Fresh)
Paper – VII : MATHEMATICS
Model Question paper-1

Time: 3 Hours Max. Marks: 70

Instructions: Answer all parts.
PART – A
I. Answer any Five questions : 5X2=10

1. Define vector space over a field.

2. Find a linear transformation T: R2 R2 such that T(1,0)=(1,2) and T(0,1)=(4,3)
3. Let T: R3 R3 be defined by T(x, y, z ) = (x+y-2z,x+2y+z,2x+2y-3z). Show that T is non
singular.
4. Write scale factors in spherical co-ordinate system
5. Define a coordinate surface.
6. Verify the condition of integrability for (y+z)dx+(x+y)dy+(x+y)dz=0
7. Form the partial differential equation by eliminating the arbitrary constants from

x2+y2=(z-c)2 tan2
8. Solve: p2+q2=1
PART-B
II. Answer any 3 of the following: 3X5=15

9. Let V(F) be a vector space over a field F. Prove that

i) �. −� =− � . � , Ɐ���, ���

ii) 0.� = 0, ∀ ��� , 0��

iii) �. 0 = 0, ∀ ���.

10. Show that the intersection of any two sub spaces of a vector space V(F) is also a subspace

of V(F). Is the union also a subspace. Justify your answer.

11 Find the dimension and basis of the subspace spanned by the vectors
(2, 4, 2), (1 ,-1, 0), (1 ,2, 1) and (0, 3, 1) in V3 ( R)
 12. Find a linear transformation of
T: R3 R3 such that T(1,0,0) = (4,5,8), T(1,-1,0) = (8,10,8) ,T(0,1,1) = (-3,-4,-7)
13. Find the range space, null space, rank, nullity and hence verify rank-nullity
theorem for T: V2(R) V2(R) defined by T(x ,y) = (x + y, x)

PART-C

III Answer any 3 of the following : 3X5=15

14. Express the following vectors in spherical words and find �� , �� , �∅

� = �� − 2�� + ��,
15. Express the following vectors in cylindrical co-ordinate and find �� , �∅ , ��

� = 2�� − �� + 3��,

16. Show that the cylindrical co-ordinate system is orthogonal co-ordinate system.
17. Show that the spherical coordinate system is orthogonal curvilinear coordinate
system and also prove that (fr , fθ , f∅ ) form a right handed basis.
18. Express the base vectors �1 , �2 , �3 �� ����� �� �, �, �

PART – D
IV Answer any 4 of the following 4X5=20
19. Verify the condition of integrebility and solve 3x2dx+3y2dy-(x3+y3+e2z)dz=0.
� �� �� ��
20. Solve �2 �
= ��
= �2
21. Solve 2p+2q=1.
22. Solve (D2-2DD’+D’2)z=ex+2y

23. Solve the equation subject to the condition

(i) u(0,t)=0 for all t≥ 0
(ii) u(x,0)=x2 –x ,0≤ � ≤ 1
24. Find the complete integral of p(1+q2)+(b-z)q=0 by Char pit’s method.
 PART-E
V. Answer any two of the following: 5X2=10

25. A scientist has found two solutions to a homogeneous system of 40 equations in

42 variables. The two solutions are not multiples, and all other solutions can be
constructed by adding together appropriate multiples of these two solutions. Can
the scientists be certain that an associated non homogeneous system (with the
same coefficients) has a solution?
26, A rectangular plate with an insulated surface is 8cm wide and so long compared to
its width that it may be considered as an infinite plate. If the temperature along short
edge y=0 is u(x, 0)=100 sin(px/8), 0≤ x ≤ 8, while two long edges x=0 and x=8 as well
as the other short edges are kept at 0° C. Find the steady state temperature at any point
of the plane.
27, A rod “l” cm with insulated lateral surface in initially at temperature f(x) at an inner
point of distance x cm from one end. If both the ends are kept at zero temperature at
any point of the rod at any subsequent time

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 VI Semester B.A/B.Sc. Examination, June/July 2023
(CBSC) (2022-23 and Onwards) (Fresh)
Paper – VII : MATHEMATICS
Model Question paper-2
Time: 3 Hours Max. Marks: 70
Instructions: Answer all parts.
PART – A
I. Answer any Five questions : 5X2=10
1. Prove that the set S={(1,0,0),(0,1,0),(0,0,1)} is linearly independent in V3®
2. Find a linear transformation T: R2→ R2 such that T(1,0)=(1,2) and T(0,1)=(4,3)
3. Define rank and nullity of linear transformation.
4. Write scale factors in cylindrical co-ordinate system
5. Define a coordinate surface.
6. Form the partial differential equation by eliminating the arbitrary constants from
x2+y2+(z-c)2=a2
7. � + � =x+y.
8. Solve (D2-2DD’+D’2)z=0

PART-B
II. Answer any 3 of the following: 3X5=15

9. Show that the intersection of any two sub spaces of a vector space V(F) is also
a subspace of V(F). Is the union also a subspace. Justify your answer.

10. Find a linear transformation T: R2→R2 such that T(1,2)=(3,0),T(2,1)=(1,2)

11. Verify whether the following are linear transformation or not

T: V1(R)→ V3(R) defined by T(x)=(x,2x,3x)
 −1 0
12. Find the linear transformation for the matrix A = 2 0 w.r.t the bases
1 3
B1={(1,2,0),(0,-1,0),(1,-1,1)} and B2={(1,0),(2,-1)}
13. State and prove rank-nullity theorem.
PART-C

III Answer any 3 of the following : 3X5=15

14. Express the following vectors in spherical words and find f r , f , f ,

� = 2�� − �� + 3��

15. Express the following vectors in spherical words and find f  , f , f z ,

� =2yi-zj+3xk

16. Show that the cylindrical co-ordinate system is orthogonal co-ordinate system.
17. Show that the spherical coordinate system is orthogonal curvilinear coordinate
system and also prove that (er , eθ , e∅ ) form a right handed basis.
18. Express the base vectors �1 , �2 , �3 �� ����� �� �, �, �

PART – D
IV Answer any 4 of the following 4X5=20
19. Verify the condition of integrebility and solve yz(1+x)dx+zx(1+y)dy+xy(1+z)dz=0.
�� �� ��
20. Solve �(�−�) = �(�−�) = �(�−�)
21. Solve p3+q3=27z
22. Solve (D2-2DD’+D’2)z= ex+2y +x3.
��
23. Form the partial differential equation by eliminating the arbitrary function from z=f( � )

24. Find the complete integral of p x+q y=p q by Charpit’s method.

 PART-E
V. Answer any two of the following 2X5=10

25. Company manufactures two products. For $1.00 worth of product B , the
company spends $0.45 on materials, $0.25 on labour, and $0.15 on overhead. For $1.00
worth of product C, the company spends $0.40 on materials, $0.30 on labour, and $0.15
on overhead. Let
0.45 0.40
b= 0.25 and c= 0.30
0.15 0.15
Then b and c represents the costs per dollar of income for the two products
a. What economic interpretation can be given to the vector 100b?
b.Suppose the company wishes to manufacture x1 dollars worth of product B and x2
dollars worth of product C . Give a vector that describe the various costs the company
will have (for materials, labour and overheads).
26. An infinitely long rectangular uniform plate with breadth π is bounded by two parallel
Edges maintained at 0° C. Base of the plate at a temperature �0 at all points. Determine the
Temperature at any point of the plate in the steady state.
27. A tightly stretched string with fixed end points x=0 and x=l is initially in a position given
By y(x,0)=y0 sin3(px/l). If it is released from this position, find the displacement y at any
Time and at any distance from the end x=0.

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 VI Semester B.A/B.Sc. Examination, June/July 2023
(CBSC) (2022-23 and Onwards) (Fresh)
Paper – VII : MATHEMATICS
Model Question paper-3

Time: 3 Hours Max. Marks: 70

Instructions: Answer all parts.
PART – A
I. Answer any Five questions : 5X2=10
1. Prove that in any vector space V over a field F ,C.O=O,∀ � ∈ �
2. Prove that T: V1(R)→ V3(R) defined by T(x)=(x,x2,x3) is not a linear transformation
3. T: R3→ R3 be defined by T(x,y,z)=(x+y-2z,x+2y+z,2x+2y-3z). Show that T is non singular.
4. Show that the spherical coordinate system is orthogonal curvilinear
coordinate system and also prove that (er , eθ , e∅ ) form a right handed basis.
5. In spherical system prove that eˆr  eˆ  eˆ

6. Form the partial differential equation by eliminating the arbitrary constants from
x2+y2=(z-c)2 tan2�.
7. Solve p+q= sinx+siny
8. Find the particular integral of the equation (D2-2DD’+D’2)z=ex+2y

PART-B
II. Answer any 3 of the following: 3X5=15

9. Prove that a non empty subset W of a vector space V(F) is a subspace of V(F) if and
only if ∀ �, � ∈ � ��� ��� ��� �, � ∈ �, �� + �� ∈ �
10. Find the dimension and basis of the subspace spanned by
(2,4,2),(1,-1,0),(1,2,1) and (0,3,1) in V3(R)
11. Find the matrix of linear transformation T: R2→R2 defined by T(x,y)=(x+4y,2x-3y)
relative to the basis B1={(1,0),(0,1)} and B2={(1,3),(2,5)}
 −1 0
12. Find the linear transformation for the matrix A = 2 0 w.r.t the bases
1 3
B1={(1,2,0),(0,-1,0),(1,-1,1)} and B2={(1,0),(2,-1)}
13. Find the range space,null space, rank,nullity and hence verify rank-nullity
theorem for T: V3(R)→ V2(R) defined by T(x,y,z)=(y-x,y-z)
14. Solve (y-z)p+(z-x)q = x-y.

PART-C

III Answer any 3 of the following : 3X5=15

15. Express the following vectors in spherical words and find f r , f , f ,

f  ziˆ  2 xjˆ  ykˆ
16. Express the following vectors in cylindrical co-ordinate and find f  , f , f z ,

f  2 y iˆ  zjˆ  3 xkˆ
17. Show that the cylindrical co-ordinate system is orthogonal co-ordinate system.
18. Show that the spherical coordinate system is orthogonal curvilinear coordinate
system and also prove that (er , eθ , e∅ ) form a right handed basis.

19. Express the base vectors �1 , �2 , �3 �� ����� �� �, �, �

PART – D
IV Answer any 4 of the following 4X5=20
20. Verify the condition of integrebility and solve (y2+z2-x2)dx-2xydy-2xzdz=0.

�� �� ��
21. Solve �2−�� = �2−�� = �2−��

22. Solve z2(p2+q2+1)=1.

23. Form the partial differential equation by eliminating the arbitrary functions
From z=f(x-at)+g(x+at).
24. Find the complete integral of p x+q y=p q by Char pit's method.
25. A tightly stretched string with fixed end points x=0 and x=l is initially in a position given by
�� .
y=yosin3 �
If it is released from the initial position, find the displacement y(x,t).
 PART-E
V. Answer any two of the following 2X5=10

26. A scientist has found two solutions to a homogeneous system of 40 equations in 42

variables. The two solutions are not multiples, and all other solutions can be constructed
by adding together appropriate multiples of these two solutions. Can the scientists be
certain that an associated non homogeneous system (with the same coefficients) has a
solution?
27. A rod of length l with insulated sides is initially at a uniform temperature �0 .
Its ends are suddenly cooled to 0° C and are kept at the temperature, find the
temperature formula u(x, t).
28. A string is stretched and fasted to two points x=0 and x=1 apart. Motion is stated by
Displacing the string into the form y=f(x )from which it is released at time t=0. Find the
displacement at any point on the string at a distance of x from one end at time t.

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