Live practice

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:

3m, 2 \(3\,\mathrm{m}\)

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:

7 3,cos A = tan A =4 7

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:

No

The ratio of every side of the rhombus to the corresponding side of the square is

Equal side ratios alone are insufficient. A square has four right angles, whereas the displayed rhombus has acute and obtuse angles. Since corresponding angles are not equal, the quadrilaterals are not similar.

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) a = 3, d = – 2 (ii) a = – 5, d = 4
(iii) 1 4,
3 3a d= = (iv) a = 0.6, d = 1.1

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) – 2, 5 (ii) 3– 2, 2
(iii) 5 , 2
2
− −
(iv) 1 1,
4 4 (v) 1 1,
10 10

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) Consistent (ii) Inconsistent (iii) Consistent
(iv) Consistent (v) Consistent

If the sum and product of the zeroes are (S) and (P), a required quadratic polynomial is (x^2-Sx+P). A non-zero constant multiple represents the same zeroes.

Every probability lies in ([0,1]). Since (-1.5<0), it cannot be a probability. Hence the answer is (B).

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:

75.9