Live practice

Use \(1+\cot^2A=\cosec^2A\). Since \(A\) is acute, all ratios are positive.

Also, \(\tan A=1/\cot A\). Then \(\cos A=\sin A\cot A\), so

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:

x = 6, y = 3

In \(\triangle AOB\) and \(\triangle COD\),

Therefore, \(\triangle AOB\sim\triangle COD\) by AA similarity. The corresponding sides are proportional:

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:

100

(i) (P(\text{defective})=\frac4{20}=\frac15).

(ii) After one good bulb is removed, (19) bulbs remain, of which (15) are good.

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:

x = 8, y = 7

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:

Not correct. Correct answer is 346.51 cm3.

The line from a vertex to \(O\) bisects the angle between the two tangents from that vertex. Hence, in \(\triangle AOB\),

Similarly,

Adding and using \(\angle A+\angle B+\angle C+\angle D=360^\circ\),

From \(\sec^2A=1+\tan^2A\),

Also \(\cos A=1/\sec A\). Hence,

Taking reciprocals gives

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:

(3, – 10)