CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.
Question 5 : Let \(a, b, c\) be non-zero real numbers such that \(b^2 \lt 4 a c\), and \(f(x)=a x^2+b x+c\). If the set \(S\) consists of al integers \(m\) such that \(f(m)\lt0\), then the set \(S\) must necessarily be
We are given the definition of the function, \(f(x)=a x^2+b x+c\), and been told that \(b^2<4 a c\). This means that f(x) has imaginary roots and therefore no real roots at all.
If a function has no real roots, the graph of the function can never touch the x-axis, because touching the x-axis means, for some real value x, the value of f(x) is 0. In other words, it means that the function has roots for some real value x.
When we graph this function quadratic function f(x), we get a parabola, that should never touch the x-axis. Such a parabola should be completely above the x-axis or completely below the x-axis.
If it is completely above the x-axis:
It means that the value of f(x) is always positive for any value of x.
The set of values of x that satisfies the condition that f(x) < 0 is an empty set.
If it is completely below the x-axis:
It means that the value of f(x) is always negative for any value of x.
The set of values of x that satisfies the condition that f(x) < 0 is the set of all real numbers.
Since set S contains all the integers ‘m’ that satisfy the above conditions, Set S is either an empty set or the set of all integers.
Choice B is the correct answer.
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