Which of the following statements are true or false? Give reason

Question

Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.
a) The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.
b) The greatest interger function is continuous on $\mathbf{R}$ .
c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .
d) Every integrable function is monotonic.
e) $a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

09 Jan 2026

Answer :

Word Count : 144

a) True. The set of solutions to an equation involving real numbers is generally infinite unless the equation is specifically constrained to have a finite number of solutions. For example, for the equation (x + 1 _______ ______ ____ _______ __________ _________ _______ ___ ________ __________.
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