Live practice

In \(\triangle PQR\), \(\angle PQR=\angle PRQ\). Sides opposite equal angles are equal; therefore,

Using \(1\) in the given proportion,

Because \(P\) lies on \(QT\) and \(S\) lies on \(QR\),

Equations \(2\) and \(3\) give two proportional sides including an equal angle. Hence \(\triangle PQS\sim\triangle TQR\) by SAS similarity.

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:

128

There are 90 equally likely outcomes. The two-digit labels run from 10 to 90, giving \(90-10+1=81\) outcomes. The perfect squares are \(1,4,9,16,25,36,49,64,81\), giving 9 outcomes. There are \(\frac{90}{5}=18\) multiples of 5.

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:

Median age = 35.76 years

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.

Write each reason beside the corresponding equality. The conclusion follows only after the stated theorem or congruence/similarity criterion has been established.

After substitution and simplification:

(i) B (ii) C (iii) D (iv) D

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:

\(\left(-\frac27,-\frac{20}{7}\right)\)

The first pair of angles is given:

Since \(S\) lies on \(PR\) and \(T\) lies on \(QR\), rays (RS,RP) are collinear and rays (RT,RQ) are collinear. Therefore,

Two corresponding angle pairs are equal. Hence \(\triangle RPQ\sim\triangle RTS\) by AA similarity.

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:

60

The six faces are (A,B,C,D,E,A). Thus (A) appears twice and (D) once.

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:

Median length = \(146.75\,\mathrm{mm}\)