CAT (QA) PYQs

For some constant real numbers $p, k$ and $a$ consider the following system of linear equations in $x$ and $y$:

$px - 4y = 2$

$3x + ky = a$

A necessary condition for the system to have no solution for $(x, y)$ is

In an examination, there were $75$ questions. $3$ marks were awarded for each correct answer, $1$ mark was deducted for each wrong answer and $1$ mark was awarded for each unattempted question. Rayan scored a total of $97$ marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is

John gets Rs $57$ per hour of regular work and Rs $114$ per hour of overtime work. He works altogether $172$ hours and his income from overtime hours is $15\%$ of his income from regular hours. Then, for how many hours did he work overtime?