Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Differentiate 4x−2.
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
Calculate the area enclosed by y=x, x+y=0, and x=k.
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.
If a,b,c,d form a geometric progression, prove that aⁿ+bⁿ, bⁿ+cⁿ and cⁿ+dⁿ also form a geometric progression.
Translate the conditions into a sequence equation, then retain every admissible real solution.
The final value satisfies the original recurrence, sum, product or financial condition.
Complete every requested part of miscellaneous exercise, question 1: 2 cos cos + cos + cos =0 13 13 13 13
Proceed from the given expression and keep every transformation explicit.
The last line satisfies the required domain and establishes the result.
The ionization constant of dimethylamine is 5.4 × 10–4. Determine its degree of ionization in its 0.02M solution. What percentage of dimethylamine is ionized if the solution is also 0.1M in NaOH?
For the weak base alone use α≈√(Kb/C). Added NaOH suppresses ionisation, so α≈Kb/[OH⁻].
A 150 pm photon ejects an inner electron at 1.5 × 10⁷ \(\mathrm{m\,s}\)⁻¹. Calculate the electron's binding energy.
Energy conservation gives photon energy = binding energy + electron kinetic energy.
A hyperbola has vertices (±2,0) and foci (±3,0). Find its equation.
Translate every condition into an equation and retain exact values throughout the hyperbola calculation.
The last line satisfies the original conditions and supplies every requested part.
Differentiate (ax+b)ⁿ.
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
A hyperbola has vertices (0,±5) and foci (0,±8). Find its equation.
Translate every condition into an equation and retain exact values throughout the hyperbola calculation.
The last line satisfies the original conditions and supplies every requested part.
Choose p so that 3x+y−2=0, px+2y−3=0, and 2x−y−3=0 are concurrent.
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.