Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Solve the compound inequality 2≤3x−4≤5.
Add 4 to all three members and then divide by 3.
Find the domain of \(f(x)=\dfrac{x^2+2x+1}{x^2-8x+12}\).
The denominator must be non-zero:
Thus \(x e2,6\).
\(\mathbb R-\{2,6\}\).
Let \(A=\mathbb N\), \(B\) be even natural numbers, \(C\) odd natural numbers, and \(D\) prime numbers. Find \(A\cap B\), \(A\cap C\), \(A\cap D\), \(B\cap C\), \(B\cap D\), and \(C\cap D\).
Because \(B,C,D\subset A\), the first three intersections equal \(B,C,D\). No natural number is both even and odd, so \(B\cap C=\varnothing\). The only even prime is 2, while every other prime is odd.
\(B,C,D,\varnothing,\{2\}\), and the set of odd primes, respectively.
Two identical wires are connected first in series and then in parallel across the same voltage. Find heat ratio, series:parallel.
Equivalent resistances are \(2R\) and \(\frac{R}{2}\). Since \(H=\frac{V^2t}{R_{eq}}\), ratio is \([1/(2R)]/[\frac{2}{R}]=\frac{1}{4}\).
1:4 - option (C).
Interpret the statement that a plane mirror has magnification +1.
\(|m|=1\) means image height equals object height. The positive sign means the image is erect. For a plane mirror it is also virtual and as far behind the mirror as the object is in front.
The virtual upright image is the same size as the object.
State the role of saliva in digestion.
Saliva moistens and lubricates food, helping formation and swallowing of a bolus. Salivary amylase begins chemical digestion by hydrolysing cooked starch into simpler sugars such as maltose. Its near-neutral medium supports the enzyme.
Saliva lubricates food and amylase begins starch digestion.
Give
Add bromine water in the absence of strong light, or add cold dilute alkaline potassium permanganate. An unsaturated hydrocarbon decolourises the reagent by addition/oxidation at the multiple bond. A saturated hydrocarbon produces no immediate colour change under these conditions.
Give scientific reasons: (a) platinum, gold and silver are used for jewellery(b) sodium, potassium and lithium are stored under oil(c) aluminium is used for cooking utensils despite being reactive(d) carbonate and sulphide ores are converted to oxides before reduction.
(a) These metals are lustrous, malleable and very unreactive, so ornaments retain their appearance.
(b) Alkali metals react violently with air and moisture; oil excludes both.
(c) Aluminium quickly forms a thin, adherent \(\ce{Al2O3}\) layer that prevents further corrosion. It also conducts heat well.
(d) Metal oxides are generally easier to reduce. Carbonates are calcined and sulphides roasted:
The reasons follow from low reactivity, safe storage, protective oxide formation, and easier reduction of oxides.
Why does dry hydrogen chloride gas fail to display acidic behaviour without water?
Hydrogen chloride shows acidity only after ionising in water:
Without water, hydronium ions cannot form. Therefore dry HCl gas does not affect dry indicators or conduct as an acid.
Water is required for HCl to form \(\ce{H3O+}\) ions.
Among 60 students, 30 chose NCC, 32 chose NSS and 24 chose both. Find probabilities of (i) NCC or NSS(ii) neither(iii) NSS but not NCC.
Define the sample space and count each favourable outcome once.
The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.