Section B (5 marks)
Instructions: Each of the following mathematical statements is either true or false. If the statement is true or false, write the question number on your answer sheet and write TRUE if the statement is true or FALSE if the statement is false.
1. The region in a circle bounded by two radii and an arc is a sector.
2. A cylinder has a base area of 75 cm² and height of 10 cm. The quantity of water when the cylinder is half full is 375 cm³.
3. Two points P(1, -1) and Q(5, 7) are on a straight line. Given that M is the midpoint of PQ, then the coordinates of M are (-2, -4).
4. Given the equation of the lines L₁: 2x + y = 8 and L₂: 6y - mx = 3, the value of m when L₁ and L₂ are perpendicular is -3.
5. Two angles α and θ are such that α + θ = 90°. The angles α and θ are complementary.
Section C (10 marks). Show all steps in your working.
Exercise 1
The functions f and g are defined as f: x → 2x + 3, g: x → mx - 2, where x ∈ R, the set of real numbers and m is a constant. Find:
a) f(-5/2) (1 mark)
b) f⁻¹(13) (1 mark)
c) fg(x) in terms of m (1 mark)
d) The value of m for which fg(x) = gf(x) (1 mark)
Exercise 2
Mr. Ngede shares a certain amount of money among his three children Azoh, Che, and Ngong. Two-fifths of the money was given to Che, one-half of the remainder to Azoh, and the balance of 150,000 FCFA to Ngong.
PROBLEM (15 MARKS) Show all the steps in your working.
Sixty students were asked to harvest oranges in a school farm. The number of oranges harvested per student was recorded in the frequency distribution table below:
| Number of oranges harvested | Number of students |
|-----------------------------|--------------------|
| 3 - 5 | 1 |
| 6 | 4 |
| 7 | 1 |
| 8 | 2 |
| 9 | 1 |
| 10 | 2 |
| 12 | 13 |
| 13 | 2 |
i) State the mode (1 mark)
ii) Find the mean number of oranges harvested to one decimal place (4 marks)
iii) Calculate the median (3 marks)
iv) Draw a grouped frequency table for the classes: 3 - 5, 6 - 8, 9 - 11, and 12 - 14 (4 marks)
v) Represent the grouped data on a histogram (2 marks)
vii) Repeat is chosen at random. Find the probability that the student picked 13 oranges (2 marks)
Find the fraction of the money given to:
a) Both Azoh and Ngong
b) Ngong
c) Calculate the amount:
i. That was shared
ii. Given to Azoh
iii. Given to Che
e) Find the percentage of the amount given to Azoh
Answer (2)
Let's go through each statement one by one:
The region in a circle bounded by two radii and an arc is a sector.
TRUE. A sector of a circle is indeed the area enclosed between two radii and the arc on the circle.
A cylinder has a base area of $75 \text{ cm}^2 and height of $10 \text{ cm} . The quantity of water when the cylinder is half full is 375 cm 3 .
TRUE. The volume of the cylinder is calculated by: Volume = Base Area × Height = 75 cm 2 × 10 cm = 750 cm 3 . When it's half full, the volume is 2 750 cm 3 = 375 cm 3 .
Two points P ( 1 , − 1 ) and Q ( 5 , 7 ) are on a straight line. Given that M is the midpoint of PQ , then the coordinates of M are ( − 2 , − 4 ) .
FALSE. To find the midpoint M , use the formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) = ( 2 1 + 5 , 2 − 1 + 7 ) = ( 3 , 3 ) .
Given the equation of the lines L 1 : 2 x + y = 8 and L 2 : 6 y − m x = 3 , the value of m when L 1 and L 2 are perpendicular is − 3 .
TRUE. For two lines to be perpendicular, the product of their slopes must be − 1 . Rewriting L 1 in slope-intercept form gives y = − 2 x + 8 , so the slope of L 1 is − 2 . For L 2 , rewriting gives y = 6 m x + 2 1 making its slope 6 m . Solving − 2 × 6 m = − 1 gives m = − 3 .
Two angles α and θ are such that 9 0 ∘ − θ [ t e x ] . T h e an g l es [ / t e x ] α [ t e x ] an d [ / t e x ] θ are complementary.
FALSE. The statement is not complete. While complementary angles sum to 9 0 ∘ , this statement is incomplete and incorrect as it stands without further context.
In summary, the truth values of the statements are: 1) TRUE, 2) TRUE, 3) FALSE, 4) TRUE, 5) TRUE. Each statement has been explained and the logic behind the true or false categorization is provided. It's important to review definitions and calculate correctly when verifying such statements.
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