Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Solve 3(x−1)≤2(x−3).
Expand both sides and isolate x.
Let \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{5,6\}\), \(D=\{5,6,7,8\}\). Verify (i) \(A\times(B\cap C)=(A\times B)\cap(A\times C)\)(ii) \(A\times C\subset B\times D\).
(i) \(B\cap C=\varnothing\), so the left side is \(\varnothing\). The second coordinates in \(A\times B\) lie in \(B\), while those in \(A\times C\) lie in disjoint \(C\); hence their intersection is also \(\varnothing\).
(ii) Since \(A\subset B\) and \(C\subset D\), every \((a,c)\in A\times C\) has \(a\in B,c\in D\). Thus \(A\times C\subset B\times D\).
Both statements are verified.
Which are null sets:
No odd natural number is divisible by 2, so (i) is empty. The even-prime set is \(\{2\}\), so (ii) is not empty. No natural number satisfies \(x<5\) and \(x>7\), and distinct parallel lines have no common point. Thus (iii) and (iv) are empty.
Null sets: (i), (iii), (iv). Part (ii) equals \(\{2\}\).
Suggest any two actions an individual can take to reduce the waste-disposal problem.
These measures reduce the amount reaching a landfill and allow each waste stream to be treated correctly.
Segregate and compost/recycle waste, and reduce single-use products by reusing durable items.
Choose the correct description of the magnetic field inside a long current-carrying solenoid.
A long solenoid produces nearly parallel, equally spaced field lines in its central interior. The field is therefore strong and approximately uniform, with the same magnitude at all interior points away from the ends.
It is the same at all interior points - option (D).
State the factors on which resistance of a uniform conductor depends.
\(R=\rho \frac{l}{A}\). Resistance increases with length \(l\), decreases with cross-sectional area \(A\), and depends on material through resistivity \(\rho\). Resistivity also changes with temperature.
Length, cross-sectional area, material/resistivity and temperature.
What is the normal least distance of distinct vision for a young adult?
A normal young eye can focus comfortably down to approximately \(25\,\mathrm{cm}\).
\(25\,\mathrm{cm}\) - option (C).
A light ray travels obliquely from air into water. State and explain its direction of bending.
Water is optically denser than air, so light slows on entry. The ray bends towards the normal; hence the angle of refraction is smaller than the angle of incidence.
It bends towards the normal because its speed decreases in water.
A tall violet-flowered pea crossed with a short white-flowered plant produces all violet offspring, but about half are short. Determine the tall parent's genotype.
Let tall T dominate short t, and violet W dominate white w. The short white parent is ttww. About half the offspring are short, so the tall parent must supply t in half its gametes: it is Tt. All offspring are violet when crossed with ww, so the tall parent must always supply W: it is WW. Hence the genotype is TtWW.
TtWW - option (C).
Why must DNA be copied during reproduction?
The new individual needs the complete genetic instructions required to build and maintain its cells. DNA replication supplies each daughter cell or gamete lineage with hereditary information. Without copying, offspring could not retain the organisation characteristic of the species.
Every offspring requires a complete hereditary instruction set.