1
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\mathrm{R}$$ be a relation on $$\mathrm{N} \times \mathbb{N}$$ defined by $$(a, b) ~\mathrm{R}~(c, d)$$ if and only if $$a d(b-c)=b c(a-d)$$. Then $$\mathrm{R}$$ is

A
symmetric and transitive but not reflexive
B
reflexive and symmetric but not transitive
C
transitive but neither reflexive nor symmetric
D
symmetric but neither reflexive nor transitive
2
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If the domain of the function $$f(x)=\frac{[x]}{1+x^{2}}$$, where $$[x]$$ is greatest integer $$\leq x$$, is $$[2,6)$$, then its range is
A
$$\left(\frac{5}{37}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$$
B
$$\left(\frac{5}{37}, \frac{2}{5}\right]$$
C
$$\left(\frac{5}{26}, \frac{2}{5}\right]$$
D
$$\left(\frac{5}{26}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$$
3
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

A wire of length $$20 \mathrm{~m}$$ is to be cut into two pieces. A piece of length $$l_{1}$$ is bent to make a square of area $$A_{1}$$ and the other piece of length $$l_{2}$$ is made into a circle of area $$A_{2}$$. If $$2 A_{1}+3 A_{2}$$ is minimum then $$\left(\pi l_{1}\right): l_{2}$$ is equal to :

A
6 : 1
B
1 : 6
C
4 : 1
D
3 : 1
4
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\alpha \in (0,1)$$ and $$\beta = {\log _e}(1 - \alpha )$$. Let $${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)$$. Then the integral $$\int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} $$ is equal to

A
$$ - \left( {\beta + {P_{50}}\left( \alpha \right)} \right)$$
B
$$\beta - {P_{50}}(\alpha )$$
C
$${P_{50}}(\alpha ) - \beta $$
D
$$\beta + {P_{50}} - (\alpha )$$
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