Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Evaluate lim(r→1) πr².
Simplify the expression before substitution and preserve one-sided information.
Substitution into the original expression verifies the stated domain and final value.
For P(x_1,y_1) and Q(x_2,y_2), obtain PQ when the segment is (i) vertical(ii) horizontal.
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.
Expand (2x−3)⁶.
Use the coefficients 1,6,15,20,15,6,1.
How many four-letter codes can be made from the first ten English letters if no letter repeats?
Choose and arrange four distinct letters from ten.
Express i^(−39) in the form a+ib.
Use i⁴=1 and write the negative exponent through its reciprocal.
A wheel completes 360 revolutions per minute. Determine the angle, in radians, turned through in one second.
First convert revolutions per minute to revolutions per second.
For \(G=\{7,8\}\) and \(H=\{5,4,2\}\), write
In a Cartesian product, the first coordinate comes from the first set.
\(G\times H=\{(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)\}\).
\(H\times G=\{(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)\}\).
Write these sets in roster form:
The six rosters are shown in the solution, with repeated letters written once.
Why can biological processes decompose some substances but not others?
Microorganisms release specific enzymes, and an enzyme acts only on substances whose chemical structure it can recognise and break. Natural organic materials such as food waste, paper and cotton can therefore be converted into simpler substances by decomposers and are called biodegradable. Many synthetic materials, including common plastics, have structures for which decomposers possess no suitable enzymes. They remain for long periods and are called non-biodegradable.
Biodegradability depends on whether decomposers have enzymes capable of breaking the substance into simpler products.
List important properties of magnetic field lines.