PW Arjuna JEE 2027 Basic Mathematics Practice Sheet Flashcards

Vocabulary flashcards covering key logarithmic equations, absolute value inequalities, and algebraic simplifications from the JEE Arjuna 2027 Basic Mathematics replica sheet.

Classification of $3^{\sqrt{\log_3(2)}} - 2^{\sqrt{\log_2(3)}}$

An irrational number (JEE Main 2023).

Root Set $S$ of $3^{(3^x - 1)} + 2 = |3^{3^x} - 1| + |3^{3^x} - 2|$

A singleton set containing exactly one real root (JEE Main 2020).

Number of Elements in Set $\{x \in \mathbb{R} : (|x| - 3)|x + 4| = 6\}$

$2$ elements (JEE Main 2021).

Sum of Roots of Equation $x + 1 - 2 \log_4(3 + 2^x) + 2 \log_{16}(10 - 2^{-x}) = 0$

$\log_2(11)$ (JEE Main 2021).

Number of Real Roots of Equation $5^x + |2^x - 1| = 2^x(2^x - 2)$

$1$ real root (JEE Main 2019).

Product of Positive Real Solutions to $x^{\frac{\log_5(x) + 16}{3}} = 5^{-16}$

$1$ (JEE Main 2022).

Logarithmic Inequality Base Dependence

Statement-1 ($\log_e(x) > \log_e(y) \implies 0 < y < x$) is true, whereas Statement-2 ($\log_a(x) > \log_a(y) \implies 0 < y < x$) is false unless $a > 1$.

Nested Radical Expression $\log_p \left( \sqrt{p \sqrt{p \dots p}} \right)$

Independent of $n$, but dependent on $p$ when evaluated for $n$ radical signs.

Magnitude of Logarithmic Base Conversion $|\log_a(b)|$

$1$, given $a = \sqrt{3 + 2\sqrt{2}}$ and $b = \sqrt{3 - 2\sqrt{2}}$.

Number of Solutions to Logarithmic Equation $\log_4(x - 1) = \log_2(x - 3)$

$1$ solution.

Absolute Inequality Solution Metric $|a| + |b| + |c| + |d|$ for $(|x - 1| - 3)(|x + 2| - 5) < 0$

$16$, where the complete solution set is $(a, b) \cup (c, d)$.

Exponential Expression Reduction $12^{\frac{1 - a - b}{2(1 - b)}}$

$2$, given $60^a = 3$ and $60^b = 5$.