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.

Subtract arrangements with the four I’s treated as one block from all arrangements.

Let the third score be x and translate ‘at least 60’ into an inequality.

Proceed from the given expression and keep every transformation explicit.

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

(i) Every real input is allowed, and every real \(y\) occurs by taking \(x=-y\). Domain and range are \(\mathbb R\).

(ii) \(9-x^2\ge0\Rightarrow -3\le x\le3\). The square root is non-negative, with minimum 0 at \(x=\pm3\) and maximum 3 at \(x=0\).

1. \(X\cup Y=\{1,2,3,5\}\).
2. \(A\cup B=\{a,b,c,e,i,o,u\}\).
3. \(A\cup B=\{1,2,3,4,5,6,9,12,15,\ldots\}\).
4. \(A\cup B=\{2,3,4,5,6,7,8,9\}\).
5. \[\begin{aligned}A\cup\varnothing &= A\\&= \{1,2,3\}\end{aligned}\].

It occurs when live and neutral wires come into direct or very low-resistance contact, often because insulation fails or wiring is faulty. Resistance collapses and a dangerously large current flows.

By Joule's law:

The object must be between P and F, so its distance is greater than 0 but less than \(15\,\mathrm{cm}\). One incident ray parallel to the axis reflects through F; another directed towards the pole reflects symmetrically. Their backward extensions meet behind the mirror, forming a virtual, erect and enlarged image.

In a unicellular organism, cell division itself creates new individuals, so fission or budding is sufficient. In a multicellular organism, ordinary cell division usually serves growth and repair because cells are specialised and organised into tissues. Reproduction requires specialised organs, germ cells or propagules and coordinated development of a whole body.