Live practice

There are (36) ordered outcomes. On each throw, (5) of the (6) faces are not (5).

Using the complementary event,

Since \(AB=AC\), the base angles of the isosceles triangle are equal:

As \(D\) lies on \(BC\) and (E,B,C) are collinear,

Also, \(AD\perp BC\) and \(EF\perp AC\), so

By \(2\), \(3\), and AA similarity, \(\triangle ABD\sim\triangle ECF\).

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:

10

\(i\) The equally likely ordered outcomes are (HH,HT,TH,TT), not the three verbal descriptions. Therefore,

The three descriptions are not equally likely, so the \(\frac{1}{3}\) claim is incorrect.

\(ii\) A fair die has six equally likely outcomes. Three—1, 3, and 5—are odd. Hence,

The second argument reaches the correct probability.

Let the common ratio be \(k\). Then

Since \(AD\) and \(PM\) are medians, \(BD=B\frac{C}{2}\) and \(QM=Q\frac{R}{2}\). By Apollonius' theorem,

Substitute \(1\) and the midpoint relations in \(2\):

Therefore \(AC^2=k^2PR^2\), so \(AC=kPR\). Thus \[\begin{aligned}A\frac{B}{P}Q &= B\frac{C}{Q}R\\&= A\frac{C}{P}R\end{aligned}\], and the triangles are similar by SSS.

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:

(i) 245 (ii) –180 (iii) 5505 (iv) 33
20

Since \(D\) lies on \(BC\), rays \(CD\) and \(CB\) coincide. Therefore,

It is also given that \(\angle ADC=\angle BAC\). Hence \(\triangle ADC\sim\triangle BAC\) by AA similarity.

Using corresponding sides in the correct order,

Cross-multiplication gives

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:

(i) 1046 1
2 (ii) 286 (iii) – 8930

Formula or theorem used: Mark the corresponding sides and angles before applying the Basic Proportionality Theorem or an AAA, SAS, or SSS similarity test.

Working: State the theorem or similarity criterion, write equal ratios in corresponding order, substitute the given lengths, and end with the required equality or length.

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:

Produce AD to a point E such that AD = DE and produce PM to a point N such that
PM = MN. Join EC and NR.

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:

(i) n = 16, Sn = 440 (ii) d = 7 ,
3 S13 = 273 (iii) a = 4, S12 = 246
(iv) d = –1, a10 = 8 (v) a = 9
35 85,
3 3− = a (vi) n = 5, an = 34
(vii) n = 6, d = 54
5 (viii) n = 7, a = – 8 (ix) d = 6
(x) a = 4