Oxidation state of Mn in MnSO₄ = +2
Oxidation state of Mn in MnO₂ = +4
The increase in oxidation state is 2. Therefore, the equivalent weight of MnSO₄ is half of its molecular weight when it is converted to MnO₂ .

All the given statements are correct.
a. Actinoids show greater number of oxidation states than lanthanoids. This is because the 5f orbitals extend further from the nucleus than the 4f orbitals.
b. The chemistry of actinoids elements is not so smooth as that of lanthanoids.
c. Most of the cations of actinoids and lanthanoids are coloured.

Lanthanum oxide is added to the glass of optical instruments to raise the refractive index of glass and to increase the stability.
Neodymium and praseodymium oxides are used for making coloured glasses for goggles.

In the lanthanide series, there is a regular decrease in the atomic as well as ionic radii with the increase in the atomic number. This is referred to as the lanthanide contraction.

Although the atomic radii do show some irregularities in the trend, the ionic radii of the trivalent ions (M³⁺) decrease with the increase in the atomic number, i.e. from La⁺³ (106 pm) to Lu³⁺ (86 pm).

The ionization energy values increase across the period due to increase in the effective nuclear charge.
However, even though Ni has a higher atomic number than cobalt, it has a lower ionization energy.
This may probably be, because Co with a slightly higher atomic mass has a slightly stronger pull on the valence electrons.

Cr⁺³ is the most stable oxidation state of Cr as the trivalent cation of chromium has a stable electronic configuration of half filled t₂g orbitals.
In this state, chromium is neither oxidizing nor reducing.

Mn²⁺ has a stable configuration of half filled d- orbitals and Mn³⁺ is readily reduced to Mn²⁺.

Therefore, Mn³⁺ is a good oxidizing agent.

The other statements are wrong as Cr⁺³ is more stable than Cr⁺² and as a result Cr⁺² is a strong reducing agent.

K₂Cr₂O₇ + H₂SO₄ + 4H₂O₂ → 2CrO₅ (chromic peroxide) + K₂SO₄ + 5H₂O
The blue colour due to chromic peroxide fades away gradually due to decomposition of CrO₅ to Cr³⁺ ions and oxygen.