Live practice

Because \(D\) and \(E\) are midpoints,

Hence,

The two sides \(AB\) and \(AC\) are divided in the same ratio. By the converse of the Basic Proportionality Theorem, the joining segment is parallel to the third side:

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:

178

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:

Yes. \(40\,\mathrm{m}\), \(20\,\mathrm{m}\)

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) x = 19
5 , y = 6
5 (ii) x = 2, y = 1 (iii) x = 9
13 , y = 5
13−
(iv) x = 2, y = –3

The sample space is \({1,2,3,4,5,6,7,8}\), containing 8 outcomes.

The odd outcomes are \(1,3,5,7\), so (P\(\text{odd}\)=\(\frac{4}{8}\)=\(\frac{1}{2}\)). Six outcomes \(3,4,5,6,7,8\) exceed 2, so (P(>2)=\(\frac{6}{8}\)=\(\frac{3}{4}\)). Every label is below 9, so (P(

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:

Modal monthly expenditure = ₹ 1847.83, Mean monthly expenditure = ₹ 2662.5.

The radius is \(1.4\) cm and the cylinder length is (5-2\(1.4\)=2.2) cm. The two hemispheres form a sphere.

For 45 sweets, syrup volume is

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:

189.97 km2

Let the contact points on (AB,BC,CD,DA) be (P,Q,R,S). Tangents from the same external point are equal:

Now

Replacing each segment by its equal tangent gives

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:

( )7 3 \(1\,\mathrm{m}\)+