STUPIDSID
1(a)(i)
Which of the following is orthogonal to (1, 2, -3)?
(a) (3, 6, 3) (b) (-3, 6, 3) (c) (-3, 6, -3) (d) (-3, -3, 6)
(a) (3, 6, 3) (b) (-3, 6, 3) (c) (-3, 6, -3) (d) (-3, -3, 6)
1 M
1(a)(ii)
If λ = 3 ,2 are eigen values of 2 × 2 matrix A , then one of the eigen value
of A4 is
(a) 0 (b) 3 (c) 9 (d) 81
(a) 0 (b) 3 (c) 9 (d) 81
1 M
1(a)(iii)
Which of the following is not a subspace of R2?
(a) {0} (b) line y = 5x (c) line y = 3x+2 (d) R2
(a) {0} (b) line y = 5x (c) line y = 3x+2 (d) R2
1 M
1(a)(iv)
Rank of the matrix \(\begin{bmatrix}
5 & -3 & 4\\
0 & 2 & 9\\
0 & 0 & -6
\end{bmatrix}\) is
(a) 0 (b) 1 (c) 2 (d) 3
(a) 0 (b) 1 (c) 2 (d) 3
1 M
1(a)(v)
If A is an 5 × 6 matrix and rank of A is 4 then nullity of A is
(a) 0 (b) 1 (c) 2 (d) 3
(a) 0 (b) 1 (c) 2 (d) 3
1 M
1(a)(vi)
If A is any square matrix then, A + AT
(a) symmetric (b) skew symmetric (c) orthogonal (d) none of these
(a) symmetric (b) skew symmetric (c) orthogonal (d) none of these
1 M
1(a)(vii)
Which of the following is not an elementary matrix?
(a) \( \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \) (b) \( \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix} \) (c) \( \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \) (d) \( \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \)
(a) \( \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \) (b) \( \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix} \) (c) \( \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \) (d) \( \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \)
1 M
1(b)(i)
If r = xi + yj + zk then curl (r) is
(a) 1 (b) 2 (c) 0 (d) none of these
(a) 1 (b) 2 (c) 0 (d) none of these
1 M
1(b)(ii)
If ϕ = xyz, then the value of |gradϕ| at (1, 2, -1) is
(a) 0 (b) 1 (c) 2 (d) 3
(a) 0 (b) 1 (c) 2 (d) 3
1 M
1(b)(iii)
The set {(0, 0) , (1, 0)} is
(a) linearly independent
(b) linearly dependent
(c) basis of R2
(d) none of these
(a) linearly independent
(b) linearly dependent
(c) basis of R2
(d) none of these
1 M
1(b)(iv)
If eigen values of a 3 × 3 matrix A are -1, 0, 1 the trace ( A ) is
(a) 0 (b) 1 (c) -1 (d) none of these
(a) 0 (b) 1 (c) -1 (d) none of these
1 M
1(b)(v)
Dimension of P3 = {a+bx+cx2+dx3 : a, b, c, d &isin R} is
(a) 1 (b) 2 (c) 3 (d) 4
(a) 1 (b) 2 (c) 3 (d) 4
1 M
1(b)(vi)
If u and v are nonzero orthogonal vectors in R2 with Euclidian inner product then
(a) ||u+v||2 = ||u||2 + ||v||2
(b) ||u+v||2 = 2||u||2 + 2||v||2
(c) ||u+v||2 = ||u||2 + 2||v||2
(d) ||u+v||2 = 2||u||2 + ||v||2
(a) ||u+v||2 = ||u||2 + ||v||2
(b) ||u+v||2 = 2||u||2 + 2||v||2
(c) ||u+v||2 = ||u||2 + 2||v||2
(d) ||u+v||2 = 2||u||2 + ||v||2
1 M
1(b)(vii)
If det A ≠ 0 then
(a) AX = 0 has no solution
(b) AX = 0 has unique solution
(c) AX = 0 has infinitely many solution
(d) none of these
(a) AX = 0 has no solution
(b) AX = 0 has unique solution
(c) AX = 0 has infinitely many solution
(d) none of these
1 M
2(a)
Express (5, -1, 9) as a linear combination of
v1 = (2, 9, 0), v2 = (3, 3, 4), v3 = (1, 2, 1).
v1 = (2, 9, 0), v2 = (3, 3, 4), v3 = (1, 2, 1).
3 M
2(b)
Let u = (u1, u2), v = (v1, v2) ∈ R2. Check whether (u, v) defined as (u, v) = 4u1v1+6u2v2 is an inner product on R2?
4 M
2(c)
Solve
x1 + x2 + 2x3 - 5x4 = 3
2x1 + 5x2 - x3 - 9x4 = -3
2x1 + x2 - x3 + 3x4 = -11
x1 - 3x2 + 2x3 + 7x4 = -5
Using Gauss Jordan method.
x1 + x2 + 2x3 - 5x4 = 3
2x1 + 5x2 - x3 - 9x4 = -3
2x1 + x2 - x3 + 3x4 = -11
x1 - 3x2 + 2x3 + 7x4 = -5
Using Gauss Jordan method.
7 M
3(a)
Find the inverse of the matrix \( A=\begin{bmatrix}
1 & 2 & 3\\
2 & 5 & 3\\
1 & 0 & 8
\end{bmatrix} \)
3 M
3(b)
Find the basis of column space of the matrix \[\begin{bmatrix}
1 & -3 & 4 & -2 & 5 & 4\\
1 & -6 & 9 & -1 & 8 & 2\\
2 & -6 & 9 & -1 & 9 & 7\\
-1 & 3 & -4 & 2 & -5 & -4
\end{bmatrix}\]
Hence, find the rank of the matrix
Hence, find the rank of the matrix
4 M
3(c)
Determine linear transformation T : R2 → R3 such that T(1, 0) = (1, 2, 3) and T(1, 1) = (0, 1, 0). Also find T(2, 3)
7 M
4(a)
Check whether the function T : R2 → R2 given by the formula T(x, y) = (x + 2y, 3x ' y) is linear transformation or not.
3 M
4(b)
Check whether set of following matrices is linearly dependent? \[\left \{ \begin{bmatrix}
1 & 1\\
1 & 1
\end{bmatrix},\begin{bmatrix}
2 & 3\\
1 & 2
\end{bmatrix},\begin{bmatrix}
3 & 1\\
2 & 1
\end{bmatrix}, \begin{bmatrix}
2 & 2\\
1 & 1
\end{bmatrix}\right \}\].
4 M
4(c)
Show that the following set is basis for P3.
{1 + 4x ' 2x2, 2x + x2, -3 + x + x2, 5 ' 2x ' 3x3
{1 + 4x ' 2x2, 2x + x2, -3 + x + x2, 5 ' 2x ' 3x3
7 M
5(a)
Find eigen values of \( A=\begin{bmatrix}
-5 & 4 & 34\\
0 & 0 & 4\\
0 & 0 & 4
\end{bmatrix}. \) Is A invertible?
3 M
5(b)
State why the following set are not vector space
(i) V = R2 with the operation
(x1, y1) + (x2, y2) = (x1 + y1 + 1, x2 + y2 + 1)
k(x, y) = (kx, ky)
(ii) V = { p ∈ P2 : p(0) = 1} with the usual operation.
(i) V = R2 with the operation
(x1, y1) + (x2, y2) = (x1 + y1 + 1, x2 + y2 + 1)
k(x, y) = (kx, ky)
(ii) V = { p ∈ P2 : p(0) = 1} with the usual operation.
4 M
5(c)
Find eigenvalues and basis for eigenspace for the matrix \[A=\begin{bmatrix}
3 & -1 & 0\\
-1 & 2 & -1\\
0 & -1 & 3
\end{bmatrix}.\]
7 M
6(a)
Find curl F, if F = (y2 cos x + z3)i + (2y sin x ' 4)j + 3xz2k. Whether F is irrotational?
3 M
6(b)
Find the directional derivative of f(x, y, z) = x3 - xy2 - z at (1, 1, 0) in the direction of 2i ' 3j + 6k
4 M
6(c)
For which value of ' a ' will the following system have
(i) No solution?, (ii) Unique solution? (iii) Infinitely many solution.
x + 2y ' 3z = 4
3x ' y + 5z = 2
4x + y + (a2 - 14)z = a + 2
(i) No solution?, (ii) Unique solution? (iii) Infinitely many solution.
x + 2y ' 3z = 4
3x ' y + 5z = 2
4x + y + (a2 - 14)z = a + 2
7 M
7(a)
Find the unit normal to the surface z2 = 4(x2 + y2) at a point (1, 0, 2).
3 M
7(b)
If F = (2xy + z3)i + x2j + 3xz2k. Show that \( \int _C F.dr \) is independent of path of integration. Hence find the integral when C is many path joining (1, -2, 1) and (3, 1, 4)
4 M
7(c)
Verify Green's theorem for the function F = (x + y)i + 2xyj and C is the rectangle in the xy ' plane bounded by x = 0, y = 0, x = a, y = b.
7 M
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