Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
If P(E) = 0.05, what is the probability of ‘not E’?
In a retail market, fruit vendors were selling mangoes kept in packing boxes. These
boxes contained varying number of mangoes. The following was the distribution of
mangoes according to the number of boxes.
Number of mangoes 50 - 52 53 - 55 56 - 58 59 - 61 62 - 64
Number of boxes 15 110 135 115 25
Find the mean number of mangoes kept in a packing box. Which method of finding
the mean did you choose?
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:
57.19
A hemispherical depression is cut out from one face of a cubical wooden block such
that the diameter l of the hemisphere is equal to the edge of the cube. Determine the
surface area of the remaining solid.
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:
( )
21 244 l π +
In a circle of radius \(21\,\mathrm{cm}\), an arc subtends an angle of 60° at the centre. Find:
(i) the length of the arc (ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord
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:
(i) \(22\text{ cm}\)(ii) \(231\text{ cm}^2\)(iii) \(\left(231-\frac{441\sqrt3}{4}\right)\text{ cm}^2\)
From a point Q, the length of the tangent to a circle is \(24\,\mathrm{cm}\) and the distance of Q from
the centre is \(25\,\mathrm{cm}\). The radius of the circle is
(A) \(7\,\mathrm{cm}\) (B) \(12\,\mathrm{cm}\)
(C) \(15\,\mathrm{cm}\) (D) \(24.5\,\mathrm{cm}\)
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:
A
A kite is flying at a height of \(60\,\mathrm{m}\) above the ground. The string attached to the kite is
temporarily tied to a point on the ground. The inclination of the string with the ground
is 60°. Find the length of the string, assuming that there is no slack in the string.
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:
40 \(3\,\mathrm{m}\)
Given sec θ = 13 ,
12 calculate all other trigonometric ratios.
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:
5 12 5 12 13 , ,, ,sin cos = tan cot cosec =13 13 12 5 5θ = θ θ = θ = θ
Four seats in a classroom grid are at (A\(3,4\)), (B\(6,7\)), (C\(9,4\)), and (D\(6,1\)). Use distances to decide whether \(ABCD\) is a square.
Using the distance formula,
The diagonals are
Thus all four sides are equal and the diagonals are equal. Therefore \(ABCD\) is a square.
\(ABCD\) is a square.
E and F are points on the sides PQ and PR
respectively of a Δ PQR. For each of the following
cases, state whether EF || QR :
(i) PE = \(3.9\,\mathrm{cm}\), EQ = \(3\,\mathrm{cm}\), PF = \(3.6\,\mathrm{cm}\) and FR = \(2.4\,\mathrm{cm}\)
(ii) PE = \(4\,\mathrm{cm}\), QE = \(4.5\,\mathrm{cm}\), PF = \(8\,\mathrm{cm}\) and RF = \(9\,\mathrm{cm}\)
(iii) PQ = \(1.28\,\mathrm{cm}\), PR = \(2.56\,\mathrm{cm}\), PE = \(0.18\,\mathrm{cm}\) and PF = \(0.36\,\mathrm{cm}\)
Formula or theorem used: Mark the corresponding sides and angles before applying the Basic Proportionality Theorem or an AAA, SAS, or SSS similarity test.
Working: State the theorem or similarity criterion, write equal ratios in corresponding order, substitute the given lengths, and end with the required equality or length.
After substitution and simplification:
(i) No (ii) Yes (iiii) Yes
Fill in the blanks in the following table, given that a is the first term, d the common
difference and an the nth term of the AP:
adna n
(i) 738 . . .
(ii) – 18 . . . 10 0
(iii) . . . – 3 18 – 5
(iv) – 18.9 2.5 . . . 3.6
(v) 3.5 0 105 . . .
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) an = 28 (ii) d = 2 (iii) a = 46 (iv) n = 10 (v) an = 3.5