Download PTU B.Tech 2020 March CSE-IT 1st Sem MA 1300 Linear Algebra For Engineers Question Paper

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Roll No. Total No. of Pages : 03
Total No. of Questions : 09

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B.Tech. (Software Engineering) (Sem.-1)

LINEAR ALGEBRA FOR ENGINEERS

Subject Code : MA-1300
M.Code : 77256

Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION -B & C have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

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SECTION-A

1. Solve the following :
   1. Find the general solution of the linear system whose augmented matrix is
      

      1	3	-5	0
      0	1	-1	-1

   2. Reduce the matrix

      1	3	5
      2	-1	4
      -2	8	2

      to row echelon form.
   3. Find the inverse of the matrix

      1	3
      2	4

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   4. Examine whether the transformation T : R² — R² defined as T(x):

      1	0
      y	e

      is linear or not?
   5. Let A =

      a	b
      c	d

      and let k be a scalar. Find a formula that relates det kA to k and det A.
   6. Let a =

      2

      -5

      -1

      and b =

      -7

      -4

      6

      . Compute ||a + b||²
   7. Show that similar matrices have same eigen values.
   8. If ? is an eigen value of A, show that ?ⁿ is an eigen value of Aⁿ.

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   9. Check whether the vectors u =

      1

      3

      -5

      and v =

      2

      -3

      3

      are orthogonal or not?
   10. The characteristic roots of A=

       8	-6	2
       -6	k	-4
       2	-4	3

       are 0, 3, 15. Find the value of k.

SECTION-B

1. a) Determine if the following system is consistent :
   

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   y—4z=38
   2x-3y+2z=1
   4x—-8y+12z=1
2. b) Let u=

   1

   4

   -2

   , v=

   -2

   -3

   7

   and w=

   4

   1

   h

   . For what value(s) of h is w in the plane spanned by u and v ?
3. a) Given A=

   1	2	4
   0	1	-5
   -2	4	3

   and b=

   -2

   2

   9

   , write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. Then solve the system and write the solution as a vector.

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4. b) Find the inverse of the matrix

   0	1	2
   1	0	3
   4	3	8

   using row transformations.
5. Let T : R³ — R³ be a linear transformation defined by T

   x

   y

   z

   =

   x+y

   x+y+z

   . Find the matrix representation of T w.r.t. the ordered basis B₁ = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and B₂={(1,0,1),(1,1,0), (0, 1, 1)}.

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SECTION-C

1. a) Let v₁=(1,-1,0), v₂=(0,1,-1) and v₃ = (0, 0, 1) be elements of R³. Show that the set of vectors {v₁, v₂, v₃} is linearly independent.

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2. b) Prove that

   1	w	w2
   w	w2	1
   w2	1	w

   =0, where w is a cube root of unity.
3. a) Let u₁=

   2

   3

   -5

   , u₂=

   -4

   -5

   8

   , and v=

   8

   2

   -9

   . Determine whether v is in the subspace of R³ generated by u₁ and u₂.
4. b) Solve the following system of linear equations by Cramer’s rule :
   x+y+z=6, x—y+2z=5, 3x+y+z=8
5. Determine the eigen values and corresponding eigen vectors of the matrix

   6	2	2
   -2	3	-1
   2	-1	3

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6. Diagonalize the matrix

   1	6	1
   1	2	0
   0	0	3

7. Find an orthogonal basis for the column space of the matrix

   3	5	1
   1	5	3
   3	-7	8

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.