STUPIDSID
Choose the appropriate answer for the following MCQs.
1 (a) 1
If \(|\vec{a} \times \vec{b}| \ = \ \vec{a} \cdot \vec{b}\) then the angle between two vectors \(\vec{a}\) and \(\vec{b}\) is
(a) 30° (b) 45° (c) 60° (d) 90°
(a) 30° (b) 45° (c) 60° (d) 90°
1 M
1 (a) 2
If \( \bar{a}=2\bar{i}-3\bar{j}+\bar{k} \) then \(|\bar{a}|= \)
\( (a) \ \sqrt{-4} \ (b) \ \sqrt{4} \ (c) \ \sqrt{13}\ (d) \ \sqrt{14} \)
\( (a) \ \sqrt{-4} \ (b) \ \sqrt{4} \ (c) \ \sqrt{13}\ (d) \ \sqrt{14} \)
1 M
1 (a) 3
If \(\bar{F}\) is conservative then
\( (a) \ \nabla \times \bar{F}=0 \ \ (b) \ \nabla \times \bar{F} \ne 0 \ \ (c) \ \nabla \bar{F} = 0 \ \ (d) \ \nabla \cdot \bar{F}=0 \)
\( (a) \ \nabla \times \bar{F}=0 \ \ (b) \ \nabla \times \bar{F} \ne 0 \ \ (c) \ \nabla \bar{F} = 0 \ \ (d) \ \nabla \cdot \bar{F}=0 \)
1 M
1 (a) 4
\[ If \ A=\begin{bmatrix} 1 &-2 \\0 &1 \end{bmatrix} \] then the determine of A is
(a) -2 (b) 1 (c) -1 (d) 0
(a) -2 (b) 1 (c) -1 (d) 0
1 M
1 (a) 5
\[ If \ A=\begin{bmatrix} -5 &-3 \\2 &1 \end{bmatrix} \] then the determine of A is
(a) 0 (b) -1 (c) 1 (d) 2
(a) 0 (b) -1 (c) 1 (d) 2
1 M
1 (a) 6
The characteristic equation for the matrix \[ A= \begin{bmatrix} 2 &0 \\4 &1 \end{bmatrix} \] \[ (a) \ (\lambda -2)^2=0 \ \ (b) \ \lambda +2 =0 \ \ (c) \ (\lambda - 2 )(\lambda +2) =0 \ \ (d) \ \lambda -2=0 \]
1 M
1 (a) 7
If A is a matrix with 5 columns and mulity of A=2 then rank (A) is
(a) 5 (b) 2 (c) 3 (d) 4
(a) 5 (b) 2 (c) 3 (d) 4
1 M
1 (b) 1
\[ If \ |\bar{a}+ \bar{b}| = |\bar{a}- \bar{b}| \] then the angle between two vectors a and b
(a) 30° (b) 45° (c) 60° (d) 90°
(a) 30° (b) 45° (c) 60° (d) 90°
1 M
1 (b) 2
If \(\vec{r}=x\vec{i}+y\vec{j}+ z \vec{k}\), then the divergence of \(\vec{r}\) is
(a) 2 (b) -2 (c) 3 (d) -3
(a) 2 (b) -2 (c) 3 (d) -3
1 M
1 (b) 3
If A and kA have same rank then what can be said about k? (a) zero (b) non-zero (c) positive (d) negative
1 M
1 (b) 4
If V is a vector space having a basis B with n elements then dim(V) =
(a) < n (b) > n (c) n (d) none of these
(a) < n (b) > n (c) n (d) none of these
1 M
1 (b) 5
For a n×n matrix A, Which one of the following statements does not imply the other?
(a) A is not invertible (b) det ( A ) ≠ 0 (c) rank ( A ) =n (d) λ=0 is not an eigen-value of A
(a) A is not invertible (b) det ( A ) ≠ 0 (c) rank ( A ) =n (d) λ=0 is not an eigen-value of A
1 M
1 (b) 6
If a complex number λ0 is an eigen value of 2×2 real matrix A, then which one of the following is not true?
(a) λ is also an eigen-value of A (b) det (A) ≠ 0 (c) rank (A)=2 (d) A is not invertible
(a) λ is also an eigen-value of A (b) det (A) ≠ 0 (c) rank (A)=2 (d) A is not invertible
1 M
1 (b) 7
If a 3×3 matrix A is diagonalizable then which one of the following is true?
(a) A has 2 distinct eigen-values.
(b) A has 2 linearly independent eigen-vectors.
(c) A has 3 linearly independent eigen-vectors.
(d) none of these
(a) A has 2 distinct eigen-values.
(b) A has 2 linearly independent eigen-vectors.
(c) A has 3 linearly independent eigen-vectors.
(d) none of these
1 M
2 (a)
\[ Show \ that \ A= \begin{bmatrix} \cos \theta & - \sin \theta &0 \\ \sin \theta &\cos \theta &0 \\0 &0 &1
\end{bmatrix} \ is \ orthogonal \]
3 M
2 (b)
Is T:R3→R3 defined by T(x,y,z)=(x+3y,y,z+2x) linear? Is it one-to-one, onto or both? Justify.
4 M
2 (c)
Define rank of a matrix. Determine the rank of the matrix \[ A= \begin{bmatrix}3 &4 &5 &6 &7 \\4 &5 &6 &7 &8 \\5
&6 &7 &8 &9 \\10 &1 1&12 &13 &14 \\15 &16 &17 &18 &19 \end{bmatrix} \]
7 M
3 (a)
\[ Find \ A^{-1} \ for \ A=\begin{bmatrix} 1 &0 &1 \\-1 &1 &1 \\0 &1 &0 \end{bmatrix} , \ if \ exits \]
3 M
3 (b)
Obtain the reduced row echelon from of the matrix \[ A=\begin{bmatrix} 1 &3 &2 &2 \\1 &2 &1 &3 \\2 &4 &3 &4 \\3 &7 &4 & 8 \end{bmatrix} \] and hence find the rank of the matrix A.
4 M
3 (c)
State rank-nullity theorem. Also verify it for the linear transformation T: R3 → R2 defined by T ( x , y , z ) = ( x + y + z , x + y ).
7 M
4 (a)
\[ If \ A= \begin{bmatrix} 1 &2 &-3 \\0 &2 &3 \\0 &0 &2 \end{bmatrix} \] then find the eigen values of AT and 5A
3 M
4 (b)
Solve the system of linear equations by Cramer's Rule:
x+2y+z=5
3x-y+z=6
x+y+4z=7
x+2y+z=5
3x-y+z=6
x+y+4z=7
4 M
4 (c)
Verify Green's Theorem in the plane for \[ \oint_C(3x^2-8y^2)dx+(4x-6xy)dy \] where C is the boundary of the region defined by y2=x and x2=y
7 M
5 (a)
Find grad (φ), if φ =log (x2+y2+z2) at the point (1, 0, -2).
3 M
5 (b)
Find the angle between the surfaces x2+y2+z2 and x2+y2-z=3 at the point (2, -1, 2).
4 M
5 (c) 1
Let T: R2 → R3 be the linear transformation defined by T(x,y)=(y, -5x+13y, -7x+16y). Find the matrix for the transformation T with respect to the basic \[ B= \left \{ (3,1)^T, \ (5,5)^T \right \} \] for R2 and \[ B= \left \{(1, 0, -1)^T, (-1, 2, 2)^T, (0, 1, 2)^T \right \} \] for R3
5 M
5 (c) 2
Find a basis for the orthogonal complement of the subspace W of R3 defined as W = {(x,y,z) in R3 | -2x + 5y - z = 0}
2 M
6 (a)
Show that \[ \bar{F} = (y^2 -z^2 +3yz - 2x) \bar{i} \\+ (3xz + 2xy) \bar{j} + (3xy - 2xz+2z)\bar{k} \] is both solenoidal and irrotational.
3 M
6 (b)
A vector field is given by \[ \bar{F} = (x^2 +xy^2)\bar{i} + (y^2 + x^2 y) \bar{j} \] Find the scalar potential.
4 M
6 (c) 1
Show that the set of all pairs of real number of the form (1, x) with the operations defined as \[ (1, x_1)+ (1, x_2)= (1,x_1+x_2) \ and k(1, x)= (1, kx) \]
5 M
6 (c) 2
Verify Caylay-Hamilton Theorem for the matrix \[ A = \begin{bmatrix} 1 &-1 &2 \\0 &2 &1 \\0 &1 &-1 \end{bmatrix} \]
2 M
7 (a)
Find a basis for the subspace of P2 spanned by the vectors 1+x, x2, -2+2x2, -3x
3 M
7 (b)
Let R3 have the Euclidean inner product. Transform the basis = S { ( 1, 0, 0 ) , ( 3, 7, -2 ) , ( 0, 4,1 ) } into an orthonormal basis using the Gram-Schmidt ortho-normalization process
4 M
7 (c)
Evaluate \[ \iint_s \bar{F}\cdot \bar{n}dS \] where \[ \bar{F} =yz\bar{i}+ xz\bar{j}+ xy \bar{k} \] and S is the surface of the sphere \[x^2 +y^2 + z^2=1\] in the first octant.
7 M
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