1
GATE CSE 2005
MCQ (Single Correct Answer)
+2
-0.6
Consider the following system of equations in three real variables $$x1, x2$$ and $$x3$$ :
$$2x1 - x2 + 3x3 = 1$$
$$3x1 + 2x2 + 5x3 = 2$$
$$- x1 + 4x2 + x3 = 3$$
This system of equations has
A
no solution
B
a unique solution
C
more than one but a finite number of solutions
D
an infinite number of solutions
2
GATE CSE 2004
MCQ (Single Correct Answer)
+2
-0.6
How many solutions does the following system of linear equations have?

- x + 5y = - 1
x - y = 2
x + 3y = 3
A
infinitely many
B
two distinct solutions
C
unique
D
None
3
GATE CSE 2004
MCQ (Single Correct Answer)
+2
-0.6
In an M$$\times$$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is
A
$$\le a + b$$
B
$$\le \max \left\{ {a,\,b} \right\}$$
C
$$\le$$ $$\min \left\{ {M - a,\,N - b} \right\}$$
D
$$\le \min \left\{ {a,\,b} \right\}$$
4
GATE CSE 2004
MCQ (Single Correct Answer)
+2
-0.6
If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$
and $${X^2} - X + 1 = 0$$
($${\rm I}$$ is the identity matrix and $$O$$ is the zero matrix), then the inverse of $$X$$ is
A
$$\left[ {\matrix{ {1 - a} & { - 1} \cr {{a^2}} & a \cr } } \right]$$
B
$$\left[ {\matrix{ {1 - a} & { - 1} \cr {{a^2} - a + 1} & a \cr } } \right]$$
C
$$\left[ {\matrix{ { - a} & 1 \cr { - {a^2} + a - 1} & {a - 1} \cr } } \right]$$
D
$$\left[ {\matrix{ {{a^2} - a + 1} & a \cr 1 & {1 - a} \cr } } \right]$$
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