Live practice

Form the required totals explicitly and divide by the full number of observations.

The variance is nonnegative and the reported standard deviation is its positive square root.

Simplify the expression before substitution and preserve one-sided information.

Substitution into the original expression verifies the stated domain and final value.

Use rectangular-coordinate definitions and keep corresponding coordinates aligned.

The final relation satisfies the original coordinate or distance condition, completing the verification.

Translate every condition into an equation and retain exact values throughout the conic calculation.

The last line satisfies the original conditions and supplies every requested part.

Use the sequence definition or geometric-progression formula, keeping the first-term offset explicit.

The displayed substitutions cover every requested part and verify the result against the original data.

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

A difference is odd precisely when one number is odd and the other even. Odd members of \(A\) pair with even members 4 and 6 of \(B\); the even member 2 pairs with odd member 9.

\(R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}\).

Plan the first substituent by its directing effect. Install an ortho/para director before nitration for para products; install a meta director first for meta product; use Friedel–Crafts acylation for acetophenone.

An anion with a lone pair attacks an electron-deficient centre; a positively charged acylium ion accepts ring electrons.

A rise in oxidation number identifies oxidation; a fall identifies reduction. The oxidised reactant is the reducing agent, and the reduced reactant is the oxidising agent.

(a) Hydroquinone, (mathrm{C_6H_6O_2}), is oxidised to quinone, (mathrm{C_6H_4O_2}). Silver(I) in AgBr is reduced to Ag. Therefore hydroquinone is the reducing agent and AgBr is the oxidising agent.

(b) Carbon in HCHO changes from 0 to +2 in (mathrm{HCOO^-}), so HCHO is oxidised and is the reducing agent. Ag(I) in (mathrm{[Ag(NH_3)_2]^+}) changes to Ag(0), so the silver complex is reduced and is the oxidising agent.

(c) Carbon in HCHO changes from 0 to +2 in (mathrm{HCOO^-}), so HCHO is oxidised and is the reducing agent. Cu(II) changes to Cu(I) in (mathrm{Cu_2O}), so (mathrm{Cu^{2+}}) is reduced and is the oxidising agent.

(d) Nitrogen changes from -2 in (mathrm{N_2H_4}) to 0 in (mathrm{N_2}), so hydrazine is oxidised and is the reducing agent. Oxygen changes from -1 in (mathrm{H_2O_2}) to -2 in water, so hydrogen peroxide is reduced and is the oxidising agent.

(e) Pb(0) changes to Pb(II) in (mathrm{PbSO_4}), so Pb is oxidised and is the reducing agent. Pb(IV) in (mathrm{PbO_2}) changes to Pb(II), so (mathrm{PbO_2}) is reduced and is the oxidising agent.