Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of 3.33 h at 25$$^\circ$$C. After 9 h, the fraction of sucrose remaining is f. The value of $${\log _{10}}\left( {{1 \over f}} \right)$$ is ________ $$\times$$ 10^$$-$$2. (Rounded off to the nearest integer)

[Assume : ln 10 = 2.303, ln 2 = 0.693]

The magnitude of the change in oxidising power of the $$MnO_4^ - /M{n^{2 + }}$$ couple is x $$\times$$ 10^$$-$$4 V, if the H⁺ concentration is decreased from 1M to 10^$$-$$4 M at 25$$^\circ$$C. (Assume concentration of $$MnO_4^ - $$ and $$M{n^{2 + }}$$ to be same on change in H⁺ concentration). The value of x is ___________. $$\left[ {Given\,:{{2.303RT} \over F} = 0.059} \right]$$

The total number of amines among the following which can be synthesized by Gabriel synthesis is __________.

Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A²B² $$-$$ B²A²) X = O, where X is a 3 $$\times$$ 1 column matrix of unknown variables and O is a 3 $$\times$$ 1 null matrix, has :