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A solid sphere of uniform density and radius \( R \) cools radiatively with a time constant \( \tau \) as per the following expression: \[ T(t) = T_s + (T_h - T_s)e^{-t/\tau}, \] where \( T_h \) is the initial temperature of the object and \( T_s \) is the surrounding temperature. Then, Enumerate Test qqq Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} x^2 - 1, & x \leq 0 \\ 2x + 1, & x > 0 \end{cases} \] Then the range of \( f \circ f \) is: \[ .
\] A \([0, \infty)\) \(.
\) B \([1, \infty)\) C \([2, \infty)\) D \([1, 2] \cup [5, \infty)\) rrrrr Let \(f(x)=\int_{0}^{t} t\left(t^{2}-9 t+20\right) d t, 1 \leq x \leq 5\). If the range of \(f\) is \([\alpha, \beta]\), then \(4(\alpha+\beta)\) equals:
(1) 253
(2) 154
(3) 125
(4) 157 If \(A\) and \(B\) are two non-zero \(n \times n\) matrices such that \(A^{2}+B=A^{2} B\), then
(1) \(A B=I\)
(2) \(A^{2} B=I\)
(3) \(A^{2}=I\) or \(B=I\)
(4) \(A^{2} B=B A^{2}\) 21 August 2025 In which one of the following sets all species show disproportionation reaction?
(1) \(\mathrm{ClO}_{4}^{-}, \mathrm{MnO}_{4}^{-}, \mathrm{ClO}_{2}^{-}\)and \(\mathrm{F}_{2}\)
(2) \(\mathrm{MnO}_{4}^{2-}, \mathrm{ClO}_{2}^{-}, \mathrm{Cl}_{2}\) and \(\mathrm{Mn}^{3+}\)
(3) \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}, \mathrm{MnO}_{4}^{-}, \mathrm{ClO}_{2}^{-}\)and \(\mathrm{Cl}_{2}\)
(4) \(\mathrm{ClO}_{2}^{-}, \mathrm{F}_{2}, \mathrm{MnO}_{4}^{-}\)and \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) Identify correct \(\mathrm{A}, \mathrm{B}\) and C in the reaction sequence given below : \[ \text{conc. \(\mathrm{HNO}_{3}\)} \xrightarrow[\Delta]{\text { conc. } \mathrm{H}_{2} \mathrm{SO}_{4}} \mathrm{~A} \xrightarrow[\text { Anhyd. } \mathrm{AlCl}_{3}]{\mathrm{Cl}_{2}} \mathrm{~B} \xrightarrow{\mathrm{Fe} / \mathrm{HCl}} \mathrm{C} \] The \(\mathrm{Eu}^{2+}\) ion is a strong reducing agent in spite of its ground state electronic configuration (outermost) : [Atomic number of \(\mathrm{Eu}=63\) ]
(1) \(4 f^{6} 6 s^{2}\)
(2) \(4 f^{7}\)
(3) \(4 f^{7} 6 s^{2}\)
(4) \(4 f^{6}\)