GATE Mechanical 2023 | GA | Question: 7

Question

Consider the following inequalities
$$
\begin{aligned}
& p^2-4 q<4 \\
& 3 p+2 q<6
\end{aligned}
$$
where $p$ and $q$ are positive integers.

The value of $(p+q)$ is __________.

• A. $2$
• B. $1$
• C. $3$
• D. $4$

Comments

Answer here!   

Answer 1

key fact: p and q are positive integers.

means: $\boxed{ p \geq 1, \quad q \geq 1 }$         ...............................(1)

given inequalities:

1. $p^{2} - 4q < 4$
2. $3p + 2q < 6$

from the $2$nd inequality, we can see that:

• if $p \geq 2$ then $2$nd inequality will not hold because $3 \times 2 + 2q = 6 + 2q \nless 6$ even if $q=0$ (which itself is not possible)
• if $q \geq 2$ then $2$nd inequality will not hold because $3p + 2 \times 2 = 3p + 4 \nless 6$ (because min. value of $q = 1$ and after putting it $3p+4 = 3 \times 1 + 4 = 7 \nless 6$)

hence: $\boxed{ p < 2, \quad q < 2 }$         ....................................(2)

from (1) and (2),

$\boxed{ p = 1, \quad q = 1 }$

putting these values in equality (1),

$(1)^{2} - 4 \times 1 < 4$

$1 - 4  < 4$

$-3  < 4$

so, $\boxed{ p = 1, \quad q = 1 }$ satisfies both inequalities.

therefore, $p + q = 1 + 1 = 2$

Answer 2

Given P and Q are Positive integers.

Let p , q=1,Then Inequality becomes

i)1-4<4 is True

ii)3+2<6 is True

Then  Ans might be p + q=1+1=2

Lets take p=1,q=2;

i)1-8<4 is True

ii)3+4<6 is False

Let take p=2,q=1;

i)4-4<4 is True

ii}6+2<6 is False

Hence p=q=1 is Right Choice to select

Therefore, answer is option A.

Comments

answer is fine but how 4-4<0?

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@pavansan $0<4$

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haha oops sorry sometimes my eyes will dominate my consciousness

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:)