Live practice

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.

Proceed from the given expression and keep every transformation explicit.

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

Use pH=−log₁₀[H⁺].

The element symbol already fixes its atomic number; the upper-left number must be mass number A and the lower-left number, when shown, must be atomic number Z.

1. False. Take \(A=\{1\}\), \(B=\{A\}\), \(x=1\). Then \(A\in B\), but \(1\notin B\).
2. False. Take \(A=\{1\}\), \(B=\{1,2\}\), \(C=\{B\}\).
3. True. Every element of \(A\) lies in \(B\), and hence in \(C\).
4. False. Take \(A=\{1\}\), \(B=\{2\}\), \(C=\{1\}\); then \(A\subset C\).
5. False. With \(A=\{1,2\}\), \(B=\{1\}\), choose \(x=2\).
6. True. If \(x\in A\), subsethood would imply \(x\in B\); its contrapositive gives the statement.

(a) Tungsten has very high melting point, adequate resistivity and can be drawn into thin wire. (b) Alloys have high resistivity, high melting point and resist oxidation. (c) Series devices share voltage/current, cannot switch independently and all stop if one fails. (d) \(R=\rho \frac{l}{A}\), so resistance is inversely proportional to area. (e) Copper and aluminium have low resistivity, are ductile and economical enough for low-loss transmission.

Apply the definition or the stated differentiation rules term by term.

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

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

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

Translate every condition into an equation and retain exact values throughout the straight-line calculation.

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