Live practice

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:

8.1 plants. We have used direct method because numerical values of xi and fi are small.

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:

160 cm2

Formula or theorem used: Use \(C=2\pi r\), \(A=\pi r^2\), sector area \(=\frac{\theta}{360^\circ}\pi r^2\), and arc length \(=\frac{\theta}{360^\circ}2\pi r\).

Working: Split the figure into sectors, segments, triangles, or rectangles; calculate each labelled area and combine them with the correct addition or subtraction.

After substitution and simplification:

2132 cm7

Formula or theorem used: Use that a tangent is perpendicular to the radius at the point of contact and that tangents drawn from an external point are equal.

Working: Name the radii and tangents, state the theorem, establish congruent right triangles when required, and then obtain the requested angle or length.

After substitution and simplification:

Infinitely many

The rope is the hypotenuse. If \(h\) is the height,

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:

(i) 7 24,sin A = cos A =25 25
(ii) 24 7,sin C = cos C =25 25

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:

(i) 2 2 (ii) 4 2 (iii) 22 2 +a b

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.

After substitution and simplification:

(i) Similar (ii) Similar (iii) Equilateral
(iv) Equal, 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:

(i) Yes: 15, 23, 31, … has common difference 8. (ii) No. (iii) Yes: 150, 200, 250, … is an AP. (iv) No.

Formula or theorem used: Write the equation in the form \(ax^2+bx+c=0\). Factorise where possible; otherwise use \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).

Working: Display the standard form, the factorisation or discriminant calculation, both roots, and reject any value that is impossible in the stated context.

After substitution and simplification:

(i) Yes (ii) Yes (iii) No (iv) Yes
(v) Yes (vi) No (vii) No (viii) Yes