1
JEE Main 2023 (Online) 10th April Morning Shift
Numerical
+4
-1

Let a, b, c be three distinct positive real numbers such that $${(2a)^{{{\log }_e}a}} = {(bc)^{{{\log }_e}b}}$$ and $${b^{{{\log }_e}2}} = {a^{{{\log }_e}c}}$$.

Then, 6a + 5bc is equal to ___________.

2
JEE Main 2023 (Online) 25th January Morning Shift
Numerical
+4
-1

Let $$S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}$$. Then the maximum value of $$\beta$$ for which the equation $${x^2} - 2{\left( {\sum\limits_{\alpha \in s} \alpha } \right)^2}x + \sum\limits_{\alpha \in s} {{{(\alpha + 1)}^2}\beta = 0}$$ has real roots, is ____________.

3
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
The number of solutions of the equation

$${\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0$$, x > 0, is :
4
JEE Main 2021 (Online) 26th February Morning Shift
Numerical
+4
-1
The number of solutions of the equation log4(x $$-$$ 1) = log2(x $$-$$ 3) is _________.
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