If \(c \in (1,3)\) satisfies Lagrange’s Mean Value Theorem for \[ f(x)=x^3-2x^2+x-1 \] on \([1,3]\), then find: \[ 9c^2-12c = \ ? \]
\[ \int \frac{3e^x + 5e^{-x}}{1 - 4e^{-x}} \, dx = 3f(x) + \frac{5}{4}g(x) + \frac{53}{4}\log h(x) + c \] Given the above conditions, find: \[ f(1) + g(1) + h(1) \]
\[ \int_{0}^{8} x^{\frac{5}{3}} \left(4 - x^{\frac{2}{3}}\right)^{\frac{3}{2}} \, dx = \ ? \]
The percentage error in the measurement of length when a metal scale calibrated at \(30^{\circ}C\) is used at \(-10^{\circ}C\) is (Coefficient of linear expansion of the metal \(= 12 \times 10^{-6}\ ^{\circ}C^{-1}\)).
The radius of first Bohr orbit of hydrogen atom is \(r_o\)\( \AA\). The wavelength (in \(\AA\)) of electron associated with sixth orbit of same atom is
\[ S_2O_3^{2-}(aq) + OH^{-}(aq) \rightarrow SO_4^{2-}(aq) + H_2O(l) + e^{-} \] After the above half reaction is balanced, which of the following are the coefficients of \(OH^{-}\) and \(SO_4^{2-}\) respectively?
\[ \lim_{x \rightarrow \frac{2}{3}} \frac{\sin\left(\pi \cos^2(3x-2)\right)} {9x^2-12x+4} = \ ? \]
If \(A\) and \(B\) are the domain and range of the real valued function, \[ f(x)=\dfrac{|x|}{\sqrt{1-|x|}} \] then \(A \cup B =\ ?\)
