Basic Mathematics Examination

JIHUDUMIE SCHOOL

BASIC MATHEMATICS EXAMINATION

Time: 3 Hours

Instructions:

Answer all 14 questions.
Each question carries equal weight.
Show clearly all the working steps.

1. (Numbers/Fractions, Decimals, Percentages/Approximations)

(a) Evaluate the following expression and express your final answer in standard form (scientific notation):

\frac{0.0084 \times 1.23}{0.246 \times 0.0002}

(b) In a certain mixed secondary school, 40% of the students are boys. If there are 120 more girls than boys, find the total number of students in that school.

2. (Exponents/Radicals/Logarithms)

(a) Rationalize the denominator and simplify the following expression completely:

\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}

(b) Solve for \( x \) in the following logarithmic equation:

\log_2(x + 3) + \log_2(x - 3) = 4

3. (Sets/Probability)

(a) In a class of 50 students, 30 like Basketball, 25 like Football, and 10 like neither of the two sports. Use a Venn diagram to find how many students like both sports.

(b) A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random from the bag one after the other without replacement. Find the probability that both balls drawn are of the same colour.

4. (Co-ordinate geometry/Vectors)

(a) Find the equation of the perpendicular bisector of the line segment joining the points \( A(-2, 5) \) and \( B(4, 1) \). Express your answer in the form \( ax + by + c = 0 \).

(b) Given the vectors \( \mathbf{u} = 3\mathbf{i} - 4\mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} + y\mathbf{j} \). If vector \( \mathbf{u} \) is perpendicular to vector \( \mathbf{v} \), calculate the value of \( y \).

5. (Geometry/Perimeters and areas/Congruence and similarity)

(a) Two similar triangles have areas of \( 16 \text{ cm}^2 \) and \( 36 \text{ cm}^2 \). If the length of the base of the smaller triangle is \( 6 \text{ cm} \), find the length of the base of the larger triangle.

(b) The length of an arc of a circle is \( 11 \text{ cm} \) and it subtends an angle of \( 60^\circ \) at the centre of the circle. Calculate the radius of the circle and the area of the sector formed. (Use \( \pi = \frac{22}{7} \))

6. (Units/Rates and variation)

(a) A train is travelling at a speed of \( 72 \text{ km/h} \). Convert this speed into meters per second (\( \text{m/s} \)).

(b) The variable \( P \) varies directly as the square of \( Q \) and inversely as the square root of \( R \). If \( P = 18 \) when \( Q = 3 \) and \( R = 4 \), find the exact value of \( P \) when \( Q = 5 \) and \( R = 25 \).

7. (Ratios, profit and loss/Accounts)

(a) Ali, Juma, and Rose shared a business profit of TZS 1,200,000 in the ratio 2 : 3 : 5 respectively. How much more money did Rose get compared to Ali?

(b) A businessman bought an article for TZS 50,000. He marked its price such that after allowing a discount of 10% to his customers, he still made a profit of 20% on the cost price. Calculate the marked price of the article.

8. (Sequences and series)

(a) The sum of the first \( n \) terms of an Arithmetic Progression (A.P.) is given by the formula \( S_n = 2n^2 + 3n \). Find the \( n \)-th term of the sequence and state its common difference.

(b) A Geometric Progression (G.P.) has a first term of 5 and a common ratio of 2. Find the least number of terms required for the sum of the progression to exceed 1000.

9. (Trigonometry and Pythagoras theorem)

(a) Without using mathematical tables or a calculator, evaluate the following:

\frac{\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ}{\tan 45^\circ}

(b) An observer standing on top of a vertical building \( 50 \text{ m} \) high sees a car parked on the level ground at an angle of depression of \( 30^\circ \). How far is the car from the base of the building?

10. (Algebra/Quadratic equations)

(a) Make \( x \) the subject of the formula:

y = \sqrt{\frac{x - a}{x + b}}

(b) The length of a rectangular garden is \( 4 \text{ m} \) more than its width. If the total area of the garden is \( 96 \text{ m}^2 \), formulate a quadratic equation and solve it to find the dimensions (length and width) of the garden.

11. (Statistics/Circles)

(a) The following are the marks out of 100 obtained by 10 students in a mathematics test: 45, 55, 60, 45, 70, 65, 50, 45, 80, 75. Calculate the mean, mode, and median of these marks.

(b) In a circle with centre \( O \), a chord \( AB \) of length \( 16 \text{ cm} \) is at a perpendicular distance of \( 6 \text{ cm} \) from the centre \( O \). Find the radius of the circle.

12. (Three dimensional figures/The Earth as a sphere)

(a) A right circular cone has a base radius of \( 5 \text{ cm} \) and a slant height of \( 13 \text{ cm} \). Calculate its total surface area and its volume. (Leave your answers in terms of \( \pi \))

(b) Two places \( P(40^\circ\text{N}, 30^\circ\text{W}) \) and \( Q(40^\circ\text{N}, 50^\circ\text{E}) \) are located on the Earth's surface. Calculate the distance between \( P \) and \( Q \) in kilometers measured along the parallel of latitude. (Take the radius of the Earth \( R = 6370 \text{ km} \) and \( \pi = 3.14 \))

13. (Matrices and transformations)

(a) Find the values of \( x \) and \( y \) given the following matrix equation:

\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 17 \end{pmatrix}

(b) A triangle with vertices \( A(1, 2) \), \( B(3, 2) \), and \( C(2, 4) \) is mapped onto triangle \( A'B'C' \) by a reflection in the line \( y = x \). Find the coordinates of the image vertices \( A' \), \( B' \), and \( C' \).

14. (Linear Programming/Functions/Relations)

(a) Given the functions \( f(x) = 2x - 3 \) and \( g(x) = x^2 + 1 \), find the composite function \( g(f(x)) \) and hence evaluate \( g(f(2)) \).

(b) A tailor makes two types of shirts: Type A and Type B. Type A requires 2 meters of cotton and 1 meter of silk. Type B requires 1 meter of cotton and 2 meters of silk. He has 10 meters of cotton and 8 meters of silk available in his shop. If the profit on a Type A shirt is TZS 4000 and on a Type B shirt is TZS 5000, write down all the inequalities that represent this information and state the objective function for maximizing profit.