BSc Mathematics SEM III 2019 2020 Oct 2020 MATHS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to right indicate full marks
3. Q1 Choose correct alternative in each of the following (2 marks each) matrix unit is matrix with dimension and b) a matrix with dimension but without actual elements c) a matrix without dimension but with elements d) none of these 2)The matrix | 0 1S
   • a) elementary matrix b) not elementary identity matrix d) none of these
4. Q3 Rank of every nonzero matrix is al d) none of these 4)Determinant of square matrix is
   • a) a function from R > b)a function from c)a function from R
   • d) none of these
5. Q5 Consider system AX = with det(A) = then
   • a) system is consistent with infinitely many solutions. b) system is inconsistent
   • c) system has unique solution of these
   • a) 20 b) -20 none of these
6. Q7 The vectors < >,< —2,2 > is
   • a) linearly independent b) linearly dependent c) both of these 8)Which of the following is not an inner product on R? xX = =
   • a) + + + none of
7. Q9 Consider f(x) = sinkx k > 0 Then norm of f with
   • a) 27 b) d) none of these in IR2 then the set of all vectors orthogonal to v in IR* represents
   • a) A straight line through origin and v b) A straight line through origin and perpendicular to \
   • c) A plane through origin with normal none of these
8. Q2 a) Attempt any ONE question from the following.(8marks each)
9. Q1 i)Prove that elementary matrix is invertible
   • ii)Prove that A square matrix Ais invertible iff it is product of an elementary matrices
10. Q2 Prove that the row and the column ranks of m X n matrix are equal
    • b) Attempt any TWO question from the following.(6marks each)
11. Q1 State and prove rank nullity theorem for a linear transformation T:V W
12. Q2 Define the matrix associated with linear transformation T:V > W and find the matrix associated with following linear transformation as T: R? > R? as T =
13. Q3 be one-one linear transformation then prove that U is
    • ii)Express following as a product of elementary matrices
14. Q4 Investigate the consistency of following equations and find the solution if possible
    • a) Attempt any ONE question from the following.(8marks each) ) Define the determinant of a matrix of order n X n and derive the formula to find the determinant of a matrix of order 3 x 3
15. Q2 For each nif 3 ann x function then prove that it is unique
    • b) Attempt any TWO from the following.(6marks each)
16. Q1 Prove that determinant of any triangular matrix is the product of main diagonal elements Stae the Cramer’s Rule and using it Solve the following
17. Q3 Define Vandermonde determinant of order n and b {3 ay are exne 3 2 3
18. Q4 Explain Laplace expansion and 12 row
19. Q4 a) Attempt any ONE question from the following.(8marks each)
20. Q1 If V is an inner product space then prove that | satisfies following properties iv)
21. Q2 ) Let V be a set of continuous real valued functions on 7]. Define + in V as Define (f,g) = Show that V forms an inner product space with respect to
    • b) Attempt any TWO question from the following.(6marks each)
22. Q1 State and prove Cauchy Schwarz inequality for u,v V
23. Q2 Using Gram Schmidt Process, construct an orthonormal basis of from Define orthpgonal complement ofa set and prove that if W isa subspace of V then is also a subspace of V Prove that Every orthogonal set of nonzero vectors in an inner product space is
24. Q5 Attempt any FOUR question from the following.(Smarks each) ) Prove that a linear transformation — W is an isomorphism iff
25. Q2 Let T: — be a linear transformation and A be the m X n matrix associated with natural basis of and then prove that rank A =dim(Img T)
26. Q3 If A, B then prove that = +)i) the area triangle whose vertices are
    • ii) the area of parallelogram formed by the edges V,,V2 where
    • iii) Find volume of a parallelopipedbounded by three vectors V3
27. Q5 Suppose = is basis for an inner space V then prove that zero vector is the only vector perpendicular to every basis vector
28. Q6 Suppose u = V = Find which of the following defines an inner prod uct on uw — li) > = +

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