# GATE Papers >> CSE >> 2019 >> Question No 41

Question No. 41

Which one of the following languages over $\sum\;=\;\{a,\;b\}$ is NOT context-free?

##### Answer : (C) $\{wa^nw^Rb^n\vert w\in\{a,\;b\}\ast,\;n\geq0\}$

Solution of Question No 41 of GATE 2019 CSE Paper

Option (C) is not CFL. Since when we insert 'w' in stack and then $\mathrm a^{\mathrm n}/\mathrm n\geq0$ then wR come so, now we have to match wR (reverse of w) with 'w' and we have to check whether actually it is reverse of 'w' or not. But in the top of the stack we see only a so, here matching of w and wR is not possible because in between w and wR we get a

ex:  let w = ab, let n = 2 then

wR = ba

So, string = ab aa ba bb

w  a2  w  b2

So, for that if we insert w then a2 as show below

New, we have to match wR = ba with 'w' but in the stop of stack we have w so, PDA is not panible for thin language. Hence, it is not regular

But for option (A) we can give PDA because first w come then an come there bn come so, for bn will pop an and then in stack we get w and then wR come so, for that we can do matching so, PDA in possible so, it in CFL

For option (B) we can give NPDA IN option (D) I can be either n, 3n or 5n

$\mathrm{So},\;\mathrm L=\mathrm a^{\mathrm n}\;\mathrm b^{\mathrm n}\cup\mathrm a^{\mathrm n}\;\mathrm b^{3\mathrm n}\cup\mathrm a^{\mathrm n}\;\mathrm b^{5\mathrm n}$

Since, whenever we can write language in union, we can give NPDA for that language

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