KILIMANJARO FORM FOUR MOCK EXAMINATION - BASIC MATHEMATICS (with comprehensive solution)

FORM FOUR MOCK EXAMINATION - BASIC MATHEMATICS

Time: 2 Hours 30 Minutes
Total Marks: 100

---

SECTION A (60 MARKS)

Answer ALL questions in this section.

Question 1 (6 Marks)

Given the number 0.000375:

(i) Write it correct to 2 significant figures.
Explanation: Significant figures are the digits that give meaning to a number's precision. We count the first two non-zero digits and round accordingly.
Solution:

• First non-zero digits: 3 and 7

• Next digit is 5, so we round 7 up to 8
  ✅ Answer: 0.00038

(ii) Write it correct to 4 decimal places.
Explanation: Decimal places count digits after the decimal point.
Solution:

• First four digits after the decimal: 0.0003

• Next digit is 7, so we round 3 up to 4
  ✅ Answer: 0.0004

(iii) Express it in standard form.
Explanation: Standard form is written as A × 10ⁿ, where 1 ≤ A < 10.
Solution:

• Move the decimal point after the first non-zero digit (3.75)

• We moved 4 places to the right → exponent is -4
  ✅ Answer: 3.75 × 10⁻⁴

---

Question 2 (6 Marks)

Given the set {1, 3, 4, 5, 6, 8, 10, 15, 17, 21, 27, ...}, identify:

(i) Prime numbers
✅ Answer: 3, 5, 17 (Numbers with only two factors: 1 and themselves)

(ii) Multiples of 3
✅ Answer: 3, 6, 15, 21, 27 (Numbers divisible by 3)

(iii) Factors of 60
✅ Answer: 1, 3, 4, 5, 6, 10, 15 (Numbers that divide 60 without a remainder)

---

Question 3 (4 Marks)

If m = 0.27 and n = 0.027, find n/m in simplest form.
Solution:

• n/m = 0.027 ÷ 0.27

• Multiply numerator & denominator by 1000 → 27/270

• Simplify by dividing by 27 → 1/10
  ✅ Answer: 1/10

---

Question 4 (6 Marks)

Find the equation of the line passing through (2, 3) and the intersection of x + y = 4 and 2x − y = 8.
Solution:

1. Find intersection point:

   • Add equations: 3x = 12 → x = 4

   • Substitute: 4 + y = 4 → y = 0

   • Intersection at (4, 0)

2. Find slope (m):

   • m = (0 − 3)/(4 − 2) = -3/2

3. Equation:

   • Using point-slope form: y − 3 = (-3/2)(x − 2)

   • Simplify: 3x + 2y = 12
     ✅ Answer: 3x + 2y = 12

---

Question 5 (5 Marks)

Three brothers visit their grandfather every 5, 7, and 12 days. If they start together on September 15, when will they next meet? *(Assume 30-day months)*
Solution:

• LCM of 5, 7, 12 = 420 days

• 420 days = 14 months

• September 15 + 14 months = November 15 (next year)
  ✅ Answer: November 15

---

Question 6 (5 Marks)

Find the final image of (3, 1) after reflection over x=0 (y-axis), then y=−2.
Solution:

1. Reflect over x=0: (−3, 1)

2. Reflect over y=−2:

   • Distance from y=−2 to (1) = 3 units

   • New y-coordinate: −2 − 3 = −5
     ✅ Answer: (−3, −5)

---

Question 7 (4 Marks)

If √3 / (p + √4) = m + n√p, find m, n, p.
Note: Question seems flawed—requires additional constraints.

---

Question 8 (8 Marks)

In a class:

• 50 take Geography

• 48 take Biology

• 36 take both

(a) Total students?
✅ Answer: 62 *(14 Geo-only + 36 Both + 12 Bio-only)*

(b) Geography-only students?
✅ Answer: 14 (50 total − 36 taking both)

---

Question 9 (10 Marks)

(a) Roza & Juma share 64,000 Tsh in a 3:5 ratio. Find the difference.
Solution:

• Total parts = 8

• 1 part = 8,000 Tsh

• Roza: 24,000 Tsh, Juma: 40,000 Tsh
  ✅ Difference: 16,000 Tsh

(b) Define accounting terms:

1. Cash balance: Money available (cash + bank).

2. Balance sheet: Snapshot of assets, liabilities, equity.

3. Trading account: Shows gross profit from sales.

4. Profit & loss account: Summarizes income & expenses.

---

Question 10 (6 Marks)

(a) Masudi saves 2,000,000 Tsh in Year 1, increasing by 400,000 Tsh yearly. Total after 22 years?
Solution:

• Arithmetic series sum: S₂₂ = 11 × [4,000,000 + 8,400,000] = 136,400,000 Tsh
  ✅ Answer: 136,400,000 Tsh

(b) Find the 10th term of 5, 15, 45, ... (Geometric sequence)
Solution:

• Common ratio (r) = 3

• aₙ = 5 × 3⁽ⁿ⁻¹⁾

• a₁₀ = 5 × 3⁹
  ✅ Answer: 5 × 3⁹

SECTION B

Question 11: Statistics (10 Marks)

Marks (%) scored by 36 candidates in a History test:
43, 70, 50, 45, 64, 62, 50, 53, 46, 62, 65, 83, 59, 54, 58, 64, 55, 52, 53, 54, 52, 59, 48, 54, 55, 48, 40, 58, 64, 40, 71, 74, 55, 70, 72, 48

(a) Construct a frequency distribution table (5 marks)
✅ Solution:

Class Interval	Tally	Frequency (f)
40 - 44	III	3
45 - 49	IIII II	6
50 - 54	IIII IIII I	11
55 - 59	IIII II	7
60 - 64	IIII I	6
65 - 69	I	1
70 - 74	IIII I	5
80 - 84	I	1

(b) Calculate the mean using assumed mean (A = 62%) (3 marks)
✅ Working:

Class Interval	f	x (Midpoint)	d = x - 62	fd
40 - 44	3	42	-20	-60
45 - 49	6	47	-15	-90
50 - 54	11	52	-10	-110
55 - 59	7	57	-5	-35
60 - 64	6	62	0	0
65 - 69	1	67	+5	+5
70 - 74	5	72	+10	+50
80 - 84	1	82	+20	+20
Total	40			-220

$$\text{Mean}=A+\frac{\sum fd}{\sum f}=62+\frac{-220}{40}=62-5.5=56.5\mathrm{\%}$$

✅ Answer: 56.5%

(c) Find the mode and median (2 marks)

• Mode: Class with highest frequency → 50 - 54%

• Median:

  • Total frequency = 40 → Median position = 20th term

  • Cumulative frequency reaches 20 in class 50 - 54%
    ✅ Answers:

  • Mode = 50 - 54%

  • Median class = 50 - 54%

---

Question 12: Trigonometry & Geometry (10 Marks)

(a) Distance between Towns A (76°S, 131°W) and B (36°N, 131°W) (4 marks)
✅ Solution:

• Both towns on same longitude (131°W).

• Latitude difference = 76° + 36° = 112°

• Distance = $\frac{112\mathrm{°}}{360\mathrm{°}}\times 2\pi \times 6400\text{ km}$

• ≈ 12,462.22 km

(b) Given $\sin A=\frac{3}{5}$, find:
(i) $\cos A$ (2 marks)
✅ Solution:

$${\sin }^{2}A+{\cos }^{2}A=1⟹\cos A=\sqrt{1-{\left(\frac{3}{5}\right)}^2}=\frac{4}{5}$$

(ii) $\frac{\tan A-\sin A}{1-2\cos A}$ (4 marks)
✅ Solution:

$$\tan A=\frac{\sin A}{\cos A}=\frac{3\mathrm{/}5}{4\mathrm{/}5}=\frac{3}{4}$$$$\frac{\frac{3}{4}-\frac{3}{5}}{1-2\times \frac{4}{5}}=\frac{\frac{3}{20}}{-\frac{3}{5}}=-\frac{1}{4}$$

---

Question 13: Algebra & Trigonometry (10 Marks)

(a) Given $f(x)=x^2-3x+2$, find:
(i) Inverse $f^{-1}(x)$ (3 marks)
✅ Solution:

$$y=x^2-3x+2⟹x=y^2-3y+2$$$$f^{-1}(x)=\frac{3\pm \sqrt{1+4x}}{2}$$

(ii) Turning point (2 marks)
✅ Solution:

$$x=\frac{-b}{2a}=\frac{3}{2}=1.5$$$$f(1.5)=(1.5{)}^2-3(1.5)+2=-0.25$$

✅ Answer: (1.5, -0.25)

(iii) Range of $f(x)$ (1 mark)
✅ Answer: $y\ge -0.25$

(b) Height of a tree with angle of elevation 37° from 30m away (4 marks)
✅ Solution:

$$\tan 37\mathrm{°}=\frac{\text{Height}}{30}⟹\text{Height}=30\times 0.7536\approx 22.61\text{m}$$

---

Question 14: Matrices & Business Math (10 Marks)

(a) Grocery sales problem:

• Cake = 1,500 Tsh | Soda = 500 Tsh

• Total items sold = 225 | Total revenue = 255,500 Tsh

(i) Simultaneous equations: (2 marks)

$$c+s=225$$$$1500c+500s=255500$$

(ii) Matrix form: (2 marks)

$$\left[\begin{array}{cc}1& 1\\ 1500& 500\end{array}\right]\left[\begin{array}{c}c\\ s\end{array}\right]=\left[\begin{array}{c}225\\ 255500\end{array}\right]$$

(iii) Inverse matrix: (3 marks)

$$A^{-1}=\frac{1}{-1000}\left[\begin{array}{cc}500& -1\\ -1500& 1\end{array}\right]=\left[\begin{array}{cc}-0.5& 0.001\\ 1.5& -0.001\end{array}\right]$$

(iv) Solve for cakes (c) and sodas (s): (3 marks)

$$\left[\begin{array}{c}c\\ s\end{array}\right]=A^{-1}\left[\begin{array}{c}225\\ 255500\end{array}\right]=\left[\begin{array}{c}112\\ 113\end{array}\right]$$

✅ Answer: 112 cakes, 113 sodas

---

Final Tips:

✔ Double-check calculations for matrix operations.
✔ Label all answers clearly (e.g., "Mean = 56.5%").
✔ Show ALL steps to earn partial marks.

CLICK HERE TO OPEN BASIC MATHEMATICS