Live practice

Total coins (=100+50+20+10=180).

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:

Mode = 36.8 years, Mean = 35.37 years. Maximum number of patients admitted in the
hospital are of the age 36.8 years (approx.), while on an average the age of a patient
admitted to the hospital is 35.37 years.

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:

π cm3

The ribs form eight equal sectors, each with central angle \(360^\cir\frac{c}{8}=45^\circ\). Therefore,

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:

\(3\,\mathrm{cm}\)

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:

20 \(3\,\mathrm{m}\), 20m, 60m

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:

12 5 12, ,sin P = cos P = tan P =13 13 5

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:

3x + y – 5 = 0

Since \(DE\parallel BC\), the Basic Proportionality Theorem gives

As \(D\) is the midpoint of \(AB\), \(AD=DB\). Therefore,

Hence \(E\) is the midpoint of \(AC\).

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:

No