Sehemu A
1.
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Express the following:
Round off:
(a)
0.1136 in a simple fraction.
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(b)
0.00070482 to 3 significant figures.
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(c)
Three fifths of the pupils in a certain school come from the city centre. What is the percentage of pupils who do not come from the city centre?
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Boresha ili kuona majibu ya mfano.
2.
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Rationalize the denominator of
(a)
Write the expression $27^n \times 9^n \times 3$ as a single exponent.
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(b)
$\frac{w^2+1}{\sqrt{2}-5}$
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(c)
By using $\log 2 = 0.3010$ and $\log 3 = 0.4771$, find $\log 72$.
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Boresha ili kuona majibu ya mfano.
3.
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A certain school has 260 students. Out of these students, 130 study Physics, 150 study Chemistry and 40 study both Physics and Chemistry. By using formula, find the number of students who study:
In a sample of 35 animal keepers, 18 keep goats, 20 keep cows and 3 keep both goats and cows. By using a Venn diagram, find the probability of getting a person who keeps goats only.
(a)
Physics only.
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(b)
neither Physics nor Chemistry.
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(c)
By using a Venn diagram, find the probability of getting a person who keeps goats only.
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Boresha ili kuona majibu ya mfano.
4.
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In the following figure, AB = 8 cm, $\angle ABD = 150^{\circ}$. If the area of triangle ABC is 22.6 cm$^2$, find the length of BC.
Find the perimeter of a regular pentagon inscribed in a circle whose radius is 10m.
(a)
In the following figure, AB = 8 cm, $\angle ABD = 150^{\circ}$. If the area of triangle ABC is 22.6 cm$^2$, find the length of BC.
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(b)
Find the perimeter of a regular pentagon inscribed in a circle whose radius is 10m.
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Boresha ili kuona majibu ya mfano.
5.
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(a)
If the exchange rate between Tanzanian shillings and the US dollar is 2,300 Tanzanian shillings to 1 US dollar, convert 27,600,000 Tanzanian shillings into US dollars.
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(b)
The height h of a cylinder is directly proportional to its volume V and inversely proportional to the square of its radius r. Given that, h = 3 cm when V = 900 cm³ and r = 10 cm, find h when V = 1,200 cm³ and r = 5 cm.
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Boresha ili kuona majibu ya mfano.
6.
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(a)
Given that a = (2)i - j, b = -i + 3j and c = 3i - 4j. Determine the magnitude of a + b + c in the form $\sqrt{n}$.
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(b)
The mid-point of a line segment is (3, 5). If one of the end points is (6, 7), find the other end point.
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Boresha ili kuona majibu ya mfano.
7.
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(a)
In making a concrete wall, a builder decides to mix gravel, sand and cement in the ratio of their minimum amounts of 500 kg, 600 kg and 200 kg, respectively. If the concrete wall of mass of 39,000 kg is to be made, how many kilograms of cement will be needed?
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(b)
Find the net profit using the following information:
Opening stock: 2,000/=
Sales: 30,000/=
Purchases: 13,000/=
Expenses: 6,500/=
Closing stock: 500/=
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Boresha ili kuona majibu ya mfano.
8.
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(a)
Joshua deposited Tshs 10,000 in a bank which pays 5% interest compounded annually. How much money did he accumulate after 3 years?
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(b)
If 3, x-1 and 27 are three consecutive terms of a geometric progression, find the possible values of x.
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Boresha ili kuona majibu ya mfano.
9.
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(a)
Find the largest angle of a triangle whose sides are 7 cm, 6 cm and 8 cm.
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(b)
In triangle ABC, angle ABC is a right angle. If CA = 6 cm and AB = 3 cm, find the length of BC and tan C.
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Boresha ili kuona majibu ya mfano.
10.
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(a)
If $x \neq 0$, express the equation $\frac{2}{x} + x + 10 = 0$ in the quadratic form $ax^2 + bx + c = 0$.
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(b)
Solve for x in the quadratic equation obtained in part (a).
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Boresha ili kuona majibu ya mfano.
Sehemu B
11.
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In a Biology examination done by 100 students in a certain school, the marks scored were grouped in the following distribution table:
| Marks | 65-69 | 70-74 | 75-79 | 80 - 84 | 85-89 | 90-94 |
|-----------|-------|-------|-------|---------|-------|-------|
| Number of Students | 10 | 12 | 55 | 10 | 5 | |
| Marks | Number of Students |
|---|
| 65-69 | 10 |
| 70-74 | 12 |
| 75-79 | 55 |
| 80 - 84 | 10 |
| 85-89 | 5 |
| 90-94 | |
(a)
How many students scored marks ranging from 75 to 79?
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(b)
Find the median of the distribution to the nearest whole number.
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Boresha ili kuona majibu ya mfano.
12.
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Given a closed rectangular box ABCDHGFE as shown in the following figure;
[Diagram showing a rectangular box with dimensions labeled. AG is a diagonal.]
(i)
(ii)
(a)
Calculate the distance in kilometres along the circle of latitude between the places located at X(60° N, 30° E) and Y(60°N, 40°E). Use the substitution $\frac{2\pi R}{360°} = 111.6$, where R is the radius of the Earth.
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(b)
Calculate the length of AG, leaving your answer in surd form.
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(c)
If the faces of the sides are to be painted with colour on the outer surfaces, what is the total area that is to be painted?
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13.
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(a)
If
$$ \begin{pmatrix} 2 & a \\ 3 & -1 \end{pmatrix} $$
is a singular matrix, then find the value of a.
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(b)
By using the matrix method, solve the following system of simultaneous equations:
$5x+3y=9$
$10x+7y=11$
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14.
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(a)
If $f(x) = x+2$ and $g(x) = 3-2x$, find the product of $f^{-1}(x)$ and $g^{-1}(x)$.
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(b)
A tailor has 150m$^2$ of cotton material and 90m$^2$ of wool material. He wants to make two types of clothes. A suit requires 1 m$^2$ of cotton and 2m$^2$ of wool while a gown requires 3 m$^2$ of cotton and 1 m$^2$ of wool. If a suit is sold at Tsh 40,000 and a gown at Tsh 60,000; formulate the objective function and the linear inequalities representing this information.
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Boresha ili kuona majibu ya mfano.