Question

If p : ax + b = c, a = 0 then equation is linear and \[q:ax+b=c,a\ne b=c=0\], then equation has a solution x = 0. Which of the following options hold?
(a)Both p and q are true
(b)p is true and q is false.
(c)p is false and q is true.
(d)Both p and q are false.

Answer 1

Hint: We have two statements p and q and we have to check if p is a linear equation and q has a solution x = 0. First, we take statement p, put a = 0 and check if it’s linear i.e. there should be one variable apart from constants. Similarly in q, we put b and c as zero and check if it has x = 0 as its solution.

Complete step-by-step answer:
We have been given two statements p and q. We need to check if they are true or not. Let us first take statement p.
Statement p: ax + b = c, a = 0.
Let us put a = 0, in p as given in the question.
\[\begin{align}
  & \Rightarrow ax+b=c \\
 & \Rightarrow 0\times x+b=c\Rightarrow b=c \\
\end{align}\]
It is not linear as b and c are constants linear equations are equations of \[{{1}^{st}}\] order, which consist of at least one variable. But we got b = c, which are both constants and there are no variables.
Thus statement p is false and it’s not linear.
Now let us take statement q : ax + b =c.
It is said \[a\ne b\] and c = 0.
Now let us put b = 0 and c = 0 in the statement q.
\[\Rightarrow ax+b=c\]
\[\Rightarrow ax+0=0\Rightarrow ax=0\] i.e. x = 0.
Now this is a linear equation, which has one variable x. Thus this equation has a solution x = 0. Thus statement q is true.
\[\therefore \] p is false and q is true.
i.e. Option (c) is the correct answer.

Note: Remember that in a linear equation there should for this given statement p and q, for them to be linear there should be the variable ‘x’ in them, or else they are not a linear equation. It is important that students consider the variable term as x and a, b, c as constants to solve this question properly.