Live practice

\(i\) Let \(x\) be girls and \(y\) be boys.

The two graphs intersect at (\(7,3\)). Therefore there are 7 girls and 3 boys.

\(ii\) Let \(x\) be the price of one pencil and \(y\) the price of one pen.

Multiplying the first equation by 7 and the second by 5 gives \(35x+49y=350\) and \(35x+25y=230\). Hence \(24y=120\), so \(y=5\). Substitution gives \(x=3\). The graphs therefore meet at (\(3,5\)).

A zero of (p\(x\)) is the \(x\)-coordinate of a point at which its graph meets or touches the \(x\)-axis. Count each distinct contact point once.

Equally likely means that no outcome is favoured by the conditions. Starting a car and scoring a basket depend on the machine or the player's skill, so (i) and (ii) are not equally likely. A random true/false guess and the two stated birth outcomes are treated as equally likely here. Hence (iii) and (iv) .

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:

₹ 545.20

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:

572 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:

277 cm8 3.
2154 cm3

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:

(i) One (ii) Secant (iii) Two (iv) Point of contact

Formula or theorem used: Draw and label a right triangle for each line of sight. Use \(\tan\theta=\frac{\text{vertical height}}{\text{horizontal distance}}\).

Working: Define the unknown height or distance, write a tangent equation for each observation, solve the equations, and include the observer's height when the question requires it.

After substitution and simplification:

8 \(3\,\mathrm{m}\)

By Pythagoras' theorem,

Therefore,

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:

39; \(39\,\mathrm{km}\)