GATE2011 GG: GA-6

Question

The number of solutions for the following system of inequalities is

• $X_1≥ 0$
• $X_2 ≥ 0$
• $X_1+ X_2 ≤ 10$
• $2X_1+ 2X_2 ≥ 22$

• A. $0$
• B. infinite
• C. $1$
• D. $2$

Answer 1

$X_1 ≥ 0\quad \to (1)$
$X_2 ≥ 0\quad \to (2)$
$X_1 + X_2 ≤ 10\quad \to (3)$
$2X_1 + 2X_2 ≥ 22 \quad \to(4)$

Now the equation $(4)$ can be written as
$X_1 + X_2 ≥ 11 \quad \to (5)$

Now, equations $(3)$ and $(5)$ cannot hold true together since $X_1 ≥ 0$ and $X_2 ≥ 0.$
Hence, system of inequalities can never be satisfied.

Answer A. $0$

Comments

Although answer is trivial. But in complex question in such category could be solved by drawing appropriate graph.

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may you draw pls ?

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I think the number of solutions remains independent of the values of X1 and X2 because of the inequality equations (3) and (5). Even if X1, X2 belong to all real numbers, the number of solutions still remain 0.

Answer 2

Equations can be written in $AX=B$ form.
Rank of augmented matrix $AB$ is $3$ and rank of $A$ is $2$ so the system of non-homogeneous equations is inconsistent.
So, no solution exists.

Option A.