Mathematics-1 (Calculus & Linear Algebra)

CSET3102B.Tech CSE2023-27

Mahatma Gandhi Central University, Bihar

Faculty: Dr. Husna MoabMaximum Marks: 95

Mathematics-1 (Calculus & Linear Algebra) (CSET3102) - 2023

Multiple Choice Questions. Attempt all questions. Each question carries one mark.

1
If $1$ and $2$ are eigen values of a $2\times2$ matrix $A$ , what are the eigen values of $A^2$ and $A^{-1}$ ?
2
Find the Eigen values for the matrix $A = \begin{pmatrix} 1 & 2 \\ 8 & 1 \end{pmatrix}$
3
The inverse of a symmetric matrix (if it exists) is?
4
If every minor of order $'r'$ of a matrix is zero then $\rho(A) = ?$
5
Find the value of $p$ for which, the rank of the following given matrix is 1.

$A = \begin{pmatrix} 3 & p & p \\ p & 3 & p \\ p & p & 3 \end{pmatrix}$
6
Test for consistency and solve to find the value of x.

$5x+3y+7z=4$

$3x+26y+2z=9$

$7x+2y+10z=5$
7
Fit a straight line $y = a + bx$ into the given data. What is the value of $y$ when $x = 8$ .

x 1 2 3 4 5 6
y 20 21 22 23 24 25
8
What is the minimum angle by which the coordinate axes have to be rotated in anticlockwise sense (in Degrees), such that the function $f(x)=3x^3+5x+1016$ has at least one Rolle's point
9
What is meant by quadrature process in mathematics?
10
Which one of the following is an infinite curve?

x	1	2	3	4	5	6
y	20	21	22	23	24	25

Write briefly in 250 words. Attempt ANY FOUR questions out of six questions. Each question carries 5 marks.

1
Show that the equations $x+y+z=4$ , $2x+5y−2z=3$ , $x+7y−7z=5$ are not consistent.
2
Using Cayley-Hamilton theorem, find $A^8$ if $A = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}$
3
Show that vectors $(1,2,3)$ , $(3,−2,1)$ & $(1,−6,−5)$ are from a linearly dependent set.
4
If $x+y+z=u$ , $y+z=uv$ , $z=uvw$ then evaluate $\frac{\partial(x,y,z)}{\partial(u,v,w)}$
5
Verify if $u=2x−y+3z$ , $v=2x−y−z$ , $w=2x−y+z$ are functionally dependent and if so, find the relation between them.
6
Show that the rectangular solid of maximum volume that can be inscribed in a sphere is a cube.

Write in 500 words. Attempt ALL questions. Each carries 10 marks.

End of Question Paper