Live practice

A \(p^\circ\) sector is the fraction \(\frac{p}{360}\) of the full circle:

This is option D.

The radii to the contact points are perpendicular to the tangents:

The angle sum of quadrilateral \(OAPB\) is \(360^\circ\). Therefore,

The vertical separation from the observer's eye is \(88.2-1.2=87\) m. If the two horizontal distances are \(x_1,x_2\),

Thus the balloon travels \(x_2-x_1=58\sqrt3\) m.

Formula or theorem used: In a right triangle use \(\sin\theta=\frac{P}{H}\), \(\cos\theta=\frac{B}{H}\), and \(\tan\theta=\frac{P}{B}\), together with the standard-angle values and identities.

Working: Write the relevant ratio or identity, substitute the known value, simplify one equality at a time, and state restrictions when a denominator could be zero.

After substitution and simplification:

∠ A = 45°, ∠ B = 15°

Formula or theorem used: Use \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) or the section formula \(\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)\).

Working: Substitute the coordinates before simplifying. For a shape, compare the required side lengths or diagonals and state the conclusion.

After substitution and simplification:

2 : 7

1. The three corresponding angles are equal, so \(\triangle ABC\sim\triangle PQR\) by AA.
2. \[\begin{aligned}\frac{2}{4} &= \frac{2.5}{5}\\&= \frac{3}{6}\\&= \frac{1}{2}\end{aligned}\], so the triangles are similar by SSS, with the order fixed by the matching sides.
3. The side ratios are not all equal: \(\frac{2.7}{4}\ne\frac{3}{6}\). Therefore the triangles are not similar.
4. The two sides enclosing the \(70^\circ\) angle are proportional, \(\frac{2.5}{5}=\frac{5}{10}\). Hence the triangles are similar by SAS.
5. The given equal angle is not included between the corresponding proportional sides; the data do not establish a similarity criterion. The pair is not similar.
6. Two corresponding angles are equal and the third pair is consequently equal. Hence the triangles are similar by AA.

Formula or theorem used: For an AP use \(a_n=a+(n-1)d\) and \(S_n=\frac n2[2a+(n-1)d]\).

Working: Identify \(a\), \(d\), and the required term or sum; substitute them in the appropriate formula and simplify line by line.

After substitution and simplification:

1

Initially there are (5) cards, so (P(Q)=\frac15). After the queen is removed, (4) cards remain: one ace and no queen.

Formula or theorem used: For grouped data use class marks and frequencies systematically. Apply the direct, assumed-mean, or step-deviation formula and show the working table totals.

Working: Write the relevant mean, median, or mode formula, identify the modal/median class, substitute the table totals, and interpret the numerical result in the question's context.

After substitution and simplification:

Mode = 44.7 cars

Formula or theorem used: Write the surface-area or volume formula for every solid before combining them. Hidden joining faces are excluded from exposed surface area.

Working: Identify every radius, height, and slant height; calculate the component measures, add or subtract them as the model requires, and attach cubic or square units.

After substitution and simplification:

\(892.26\,\mathrm{kg}\)