B.Sc. (Data Science) SEM III 2023 2024 2024 IIILINEAR ALGEBRA AND DISCRETE MATHEMATICS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks

1. Q1 Attempt the following (Any Three) (15M)
   • a) Define subspace of a vector space V Verify the set W= {( x,0)/x is subspace of
   • b) Solve the non-homogenous linear equations
   • c) Prove all five addition properties of vector space for the set S = {(3x,x):x,y R}
   • d) Define linearly dependent and independent vectors Verify S = is linearly dependent or independent Define linear of vectors in vector space V. Verify (1,2,3) as a linear combination of (1,0,0), (1, 1,0) and (1,1,3)
   • f) Let > R? defined by T(x, y) = (x — y, 1). Show that T is not a linear
2. Q2 Attempt the following (Any Three) (15M)
   • a) Find Area of tetrahedron formed by (1,1
   • b) Find distance and angle between two vectors x= (-2,1.3) and y=(4.5,1) in /R?
   • c) Use the Gram-Schmidt process to construct an basis tor the set
   • d) Check whether the set 0); (= = ,0), is orthonormal
   • e) Using Crammer’s rule, Solve the non-homogenous linear equations
   • f) Find i) volume of Parallelopiped formed by (3,2,1),(-1,3,0), li) area of a triangle formed by (-4,1) (3,-3)
3. Q3 Attempt the following (Any Three) (15M)
   • a) Show that Similar matrices have same eigen values
   • b) Solve the following differential equation using Eigen values and Eigen vectors Prove that eigen value of A is same eigen value of A‘
   • d) Check whether the matrix A= is diagonalizable. If yes, then non-singular matrix P such that is diagonal matrix
   • e) Check whether the matrix A orthogonal or
   • f) Find Eigen values and Eigen vectors for the matrix A=| 0 4
4. Q4 Attempt the following (Any Three) (15M)
   • a) Find singular value decomposition of A = 1
   • b) Define norm of a matrix. Find the condition number of the positive definite
   • c) Find minimum value of the given quadratic form:
   • d) Find local maxima, local minima and saddle point of the function Check whether the following matrices are positive definite or not
   • f) Explain the types of Quadratic forms
5. Q5 Attempt the following (Any Three) (15M)
   • a) Find Optimal Solution by simplex method Subject to constrains S20
   • b) Solve the given linear programming problem graphically Following payoff matrix refers to two player game player A and player B player has four strategic options. Find optimal strategies for player A and B in the following game. Also find the value of the game
   • d) Explain Duality in L.P.P. Also Write the dual of the given problem:
   • e) Solve the given linear programming problem graphically. é
   • f) Two types of food i.e. A and B are available. Each contains vitamin N1 and N2. A person needs 4 gram of vitamin N1 and 12 grams vitamin N2 per day. Food packet A contain 2 gram of vitamin N1 and 4 grams of vitamin N2. Food packet B contain | gram of vitamin N1 and 4 grams of vitamin N2 . If the food packet cost Rs.15 and Rs.10 for A and B respectively, Then Formulate the LPP to minimize

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