Live practice

Figures are similar when they have the same shape: corresponding angles are equal and corresponding lengths are proportional.

Any two circles are similar because one is an enlargement or reduction of the other. Any two squares are similar because all corresponding angles are \(90^\circ\) and the ratios of corresponding sides are equal.

A square and a rectangle whose adjacent sides are unequal are not similar because their side ratios differ. An equilateral triangle and a scalene triangle are not similar because their corresponding angles are not equal.

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) 10, 20, 30, 40 (ii) – 2, – 2, – 2, – 2 (iii) 4, 1, – 2, – 5
(iv) 1 1,–1, 0,2 2− (v) – 1.25, – 1. 50, – 1.75, – 2.0

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) 2x2 + x – 528 = 0, where x is breadth (in metres) of the plot.
(ii) x2 + x – 306 = 0, where x is the smaller integer.
(iii) x2 + 32x – 273 = 0, where x (in years) is the present age of Rohan.
(iv) u2 – 8u – 1280 = 0, where u (in km/h) is the speed of the train.

Formula or theorem used: Translate the information into two linear equations. Use the method named in the exercise, keeping equivalent equations aligned.

Working: Eliminate or substitute one variable, obtain the other variable, substitute back, and verify the ordered pair in both original equations.

After substitution and simplification:

(i) Intersect at a point (ii) Coincident (iii) Parallel

(i) (x^2-2x-8=(x-4)(x+2)). Thus (alpha=4,eta=-2). Hence (alpha+eta=2=-\frac{-2}{1}) and (alphaeta=-8=\frac{-8}{1}).

(ii) (4s^2-4s+1=(2s-1)^2). Thus (alpha=eta=\frac12). Hence (alpha+eta=1=-\frac{-4}{4}) and (alphaeta=\frac14=\frac14).

(iii) (6x^2-7x-3=(3x+1)(2x-3)). Thus (alpha=-\frac13,eta=\frac32). Therefore (alpha+eta=\frac76=-\frac{-7}{6}) and (alphaeta=-\frac12=\frac{-3}{6}).

(iv) (4u^2+8u=4u(u+2)). Thus (alpha=0,eta=-2). Their sum is (-2=-\frac84), and their product is (0=\frac04).

(v) (t^2-15=(t-\sqrt{15})(t+\sqrt{15})). Thus the zeroes are (sqrt{15}) and (-\sqrt{15}); their sum is (0) and product is (-15), agreeing with (-\(\frac{b}{a}\)) and (\(\frac{c}{a}\)).

(vi) (3x^2-x-4=(3x-4)(x+1)). Thus (alpha=\frac43,eta=-1). Hence (alpha+eta=\frac13=-\frac{-1}{3}) and (alphaeta=-\frac43=\frac{-4}{3}).

A fair coin has sample space (S=\{H,T\}). Since the two outcomes have equal chance, (P(H)=P(T)=\frac12); neither team is favoured.

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:

f = 20

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:

214.5 cm2

In 5 minutes the hand turns through

The swept region is a sector of radius \(14\,\mathrm{cm}\):

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:

D