1
GATE CE 2005
MCQ (Single Correct Answer)
+1
-0.3
Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}$$ will be
A
$$2 \times 2$$
B
$$3 \times 3$$
C
$$\,4 \times 3\,$$
D
$$\,3 \times 4\,$$
2
GATE CE 2005
MCQ (Single Correct Answer)
+1
-0.3
Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,{X_i}} \right)$$ be an eigen value and its corresponding eigen vector for real matrix $$A$$. Let $${{\rm I}_{nxn}}$$ be unit matrix. Which one of the following statement is not correct?
A
For a homogeneous $$nxn$$ system of linear equations $$\left( {A - \lambda {\rm I}} \right)X = 0,$$ having a non trivial solution, the rank of $$\left( {A - \lambda {\rm I}} \right)$$ is less then $$n.$$
B
For matrix $${A^m},$$ $$m$$ being a positive integer, $$\left( {{\lambda _i}^m,\,{X_i}^m} \right)$$ will be eigen pair for all $$i.$$
C
If $${A^T} = {A^{ - 1}}$$ then $$\left| {{\lambda _i}} \right| = 1$$ for all $$i.$$
D
If $${A^T} = A$$ then $${{\lambda _i}}$$ are real for all $$i.$$
3
GATE CE 2005
MCQ (Single Correct Answer)
+2
-0.6
Value of the integral $$\,\,\oint {xydy - {y^2}dx,\,\,} $$ where, $$c$$ is the square cut from the first quadrant by the line $$x=1$$ and $$y=1$$ will be (Use Green's theorem to change the line integral into double integral)
A
$$1/2$$
B
$$1$$
C
$$3/2$$
D
$$5/3$$
4
GATE CE 2005
MCQ (Single Correct Answer)
+2
-0.6
The line integral $$\int {\,\,V.dr\,\,} $$ of the vector function $$V\left( r \right) = 2xyz\widehat i + {x^2}z\widehat j + {x^2}y\widehat k\,\,$$ from the origin to the point $$P(1,1,1)$$
A
is $$1$$
B
is zero
C
is $$-1$$
D
cannot be determined without specifying the path
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