New Curriculum First Term Mathematics examination for SS 1

Class: SS 1
Term: First Term
Subject: General Mathematics
Time: 2 hours
Total Marks: 70

SECTION A: OBJECTIVES (40 Marks)
Instruction: Answer all 40 questions. Each question carries 1 mark. Show rough working separately where necessary. Diagrams are not necessarily drawn to scale.

1. The base-four numeral (231.2)₄ represents which decimal quantity?
   (a) 47.25    (b) 44.5    (c) 39.5    (d) 45.5
2. The correct first step in adding (243)₅ and (132)₅ directly in base five is to
   (a) write 5 in the units position and continue    (b) add digits from left to right without carrying    (c) add the unit digits 3 and 2, write 0 and carry 1    (d) subtract 5 from each original digit before adding
3. Simplify (a⁵b⁻²)² ÷ (a³b⁻¹), where a and b are non-zero.
   (a) a¹³/b³    (b) a²/b    (c) a⁷/b⁵    (d) a⁷/b³
4. If 9ˣ = 27ˣ⁻¹, find x.
   (a) −3    (b) 3    (c) 6    (d) 2
5. In the right-angled triangle shown, calculate sin θ at A.
   (a) 3/5    (b) 5/4    (c) 3/4    (d) 4/5
6. Two observation points are 20 m apart on level ground, on the same side of a mast. The angles of elevation are 30° at the farther point and 60° at the nearer point. Calculate the mast’s height.
   (a) 20√3 m    (b) 30 m    (c) 10√3 m    (d) 10 m
7. The index form of log₂(1/32) = −5 is ________.
   (a) 2⁵ = 1/32    (b) (−5)² = 1/32    (c) 2⁻⁵ = 1/32    (d) (1/32)⁻⁵ = 2
8. Given log₁₀2 = 0.3010, calculate log₁₀50 to four decimal places.
   (a) 1.6990    (b) 0.6990    (c) 2.3010    (d) 1.3010
9. An eight-bit code 00101101 is interpreted as an unsigned binary number. Its decimal value is
   (a) 53    (b) 45    (c) 37    (d) 41
10. Calculate (3C)₁₆ + (1F)₁₆ and keep the answer in hexadecimal.
    (a) (6B)₁₆    (b) (5B)₁₆    (c) (5C)₁₆    (d) (4B)₁₆
11. Solve 4²ˣ⁻¹ = 8ˣ⁺¹ by expressing both sides as powers of 2.
    (a) x = −5    (b) x = 3    (c) x = 5    (d) x = 1
12. For an acute angle θ, tan θ = 3/4. The corresponding value of sin θ is
    (a) 5/3    (b) 3/4    (c) 3/5    (d) 4/5
13. From the top of a 25 m tower the angle of depression of a car is 30°. The car’s horizontal distance from the foot of the tower is
    (a) 25 m    (b) 50 m    (c) 25/√3 m    (d) 25√3 m
14. If log₁₀N = −3/2, express N in index form.
    (a) 10⁻³⁄²    (b) 3/20    (c) −10³⁄²    (d) 10³⁄²
15. A calculation requires log₁₀8 but only log₁₀2 = 0.3010 is known. The correct value to four decimal places is
    (a) 2.4080    (b) 0.6020    (c) 0.9030    (d) 1.3010
16. The hexadecimal digit ________ corresponds to the four-bit pattern 1111.
    (a) 10    (b) A    (c) F    (d) E
17. Subtract (3B)₁₆ from (A0)₁₆.
    (a) (55)₁₆    (b) (75)₁₆    (c) (6F)₁₆    (d) (65)₁₆
18. A three-digit numeral (213)ᵦ has decimal value 81. Determine the base b.
    (a) 8    (b) 6    (c) 7    (d) 5
19. Evaluate (10110)₂ − (111)₂ and give the answer in binary.
    (a) (1111)₂    (b) (1011)₂    (c) (1101)₂    (d) (1001)₂
20. In the identity (xᵖ)³ = x¹⁵ for all positive x other than 1, p is ________.
    (a) 5    (b) 18    (c) 45    (d) 12
21. Calculate log₃81 + log₂8.
    (a) 4    (b) 7    (c) 12    (d) 6
22. The solution of 10ˣ = 0.01 is ________.
    (a) x = 2    (b) x = −1    (c) x = 0.1    (d) x = −2
23. The main reason hexadecimal notation shortens a binary code is that
    (a) each hexadecimal digit represents four binary digits    (b) hexadecimal eliminates all positional values    (c) each hexadecimal digit represents two binary digits    (d) every hexadecimal digit is always a decimal digit
24. When converted to base ten, the binary fraction (0.101)₂ becomes ________.
    (a) 0.75    (b) 0.375    (c) 0.625    (d) 0.505
25. The decimal equivalent of hexadecimal (7D)₁₆ is
    (a) 132    (b) 113    (c) 117    (d) 125
26. Multiplying (23)₄ by (12)₄ gives ________ in base four.
    (a) (1020)₄    (b) (1002)₄    (c) (1001)₄    (d) (1012)₄
27. ________ is the exact value of 16³⁄⁴.
    (a) 12    (b) 4    (c) 64    (d) 8
28. A laboratory sample begins with 5 cells and doubles every hour. After how many complete hours will it contain 160 cells?
    (a) 4 hours    (b) 5 hours    (c) 8 hours    (d) 6 hours
29. Evaluate sin 60° × cos 30° + cos 60° × sin 30° without a calculator.
    (a) 1/2    (b) √3/2    (c) 1    (d) 3/4
30. A 10 m ladder leans against a vertical wall with its foot 5 m from the wall. Its angle with the ground is
    (a) 75°    (b) 60°    (c) 30°    (d) 45°
31. Which calculation correctly completes (120)₃ ÷ (10)₃?
    (a) (12)₃    (b) (21)₃    (c) (20)₃    (d) (11)₃
32. Which of the following is NOT generally a law of indices?
    (a) aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a ≠ 0    (b) aᵐ + aⁿ = aᵐ⁺ⁿ    (c) aᵐ × aⁿ = aᵐ⁺ⁿ    (d) (aᵐ)ⁿ = aᵐⁿ
33. To solve 2³ˣ ÷ 2ˣ⁻¹ = 32, the correct equivalent equation is
    (a) 2x + 1 = 5    (b) 4x − 1 = 5    (c) 3x = 5    (d) 2x − 1 = 5
34. Which expression is equal to tan 60° ÷ sin 30°?
    (a) √3    (b) 2/√3    (c) √3/2    (d) 2√3
35. An observer is 12 m from the foot of a tree and measures an elevation of 60°. Find the tree’s height, ignoring the observer’s eye level.
    (a) 12/√3 m    (b) 24 m    (c) 12√3 m    (d) 6√3 m
36. The equation log₁₀(5x) = 2 has the solution
    (a) x = 20    (b) x = 500    (c) x = 10    (d) x = 5
37. The curve shows y = 10ˣ on the axes. At which value of x does the graph reach y = 10?
    (a) 1    (b) 2    (c) −1    (d) 0
38. Which of these is not a valid single-byte hexadecimal value?
    (a) (00)₁₆    (b) (100)₁₆    (c) (A4)₁₆    (d) (FF)₁₆
39. An eight-bit hexadecimal counter advances from (FF)₁₆ by one before considering any storage overflow. The mathematical sum is
    (a) (F10)₁₆    (b) (10F)₁₆    (c) (FF0)₁₆    (d) (100)₁₆
40. All these can represent numbers in base four except ________.
    (a) (304)₄    (b) (233)₄    (c) (102)₄    (d) (301)₄

SECTION B: THEORY (30 Marks)
Instruction: Answer Question 1 and any other TWO questions. Each question carries 10 marks. Show all necessary workings and label diagrams clearly.

Question 1: Number bases and arithmetic
(a) Convert (231.2)₄ to decimal, showing all place-value terms. (3 marks)
(b) Evaluate (243)₅ + (132)₅ without converting to base ten. (3 marks)
(c) Express (3C)₁₆ in decimal and as an 8-bit binary pattern. (4 marks)

Question 2: Indices and exponential modelling
(a) Simplify (a⁵b⁻²)² ÷ (a³b⁻¹), where a,b ≠ 0. (4 marks)
(b) Solve 9ˣ = 27ˣ⁻¹. (3 marks)
(c) A sample starts with 5 cells and doubles each hour. After how many hours does it reach 160? (3 marks)

Question 3: Trigonometry and surveying
(a) State the exact value of tan 60°. (2 marks)
(b) Two points on level ground are 30 m apart. The nearer point has elevation 60° to a mast top; the farther point has elevation 30°. Find the height of the mast. (5 marks)
(c) A 10 m ladder touches a wall with its foot 5 m from the wall. Find its angle with the ground. (3 marks)

Use the angle arrangement shown.

Question 4: Logarithms and hexadecimal arithmetic
(a) Write log₂(1/32) = −5 in index form. (2 marks)
(b) Given log₁₀2=0.3010, determine log₁₀50. (4 marks)
(c) Evaluate (A7)₁₆ + (B9)₁₆, leaving the answer in base sixteen. (4 marks)

Objective Marking Scheme
Question 1 – ✅ (d) 45.5
Question 2 – ✅ (c) add the unit digits 3 and 2, write 0 and carry 1
Question 3 – ✅ (d) a⁷/b³
Question 4 – ✅ (b) 3
Question 5 – ✅ (d) 4/5
Question 6 – ✅ (c) 10√3 m
Question 7 – ✅ (c) 2⁻⁵ = 1/32
Question 8 – ✅ (a) 1.6990
Question 9 – ✅ (b) 45
Question 10 – ✅ (b) (5B)₁₆
Question 11 – ✅ (c) x = 5
Question 12 – ✅ (c) 3/5
Question 13 – ✅ (d) 25√3 m
Question 14 – ✅ (a) 10⁻³⁄²
Question 15 – ✅ (c) 0.9030
Question 16 – ✅ (c) F
Question 17 – ✅ (d) (65)₁₆
Question 18 – ✅ (b) 6
Question 19 – ✅ (a) (1111)₂
Question 20 – ✅ (a) 5
Question 21 – ✅ (b) 7
Question 22 – ✅ (d) x = −2
Question 23 – ✅ (a) each hexadecimal digit represents four binary digits
Question 24 – ✅ (c) 0.625
Question 25 – ✅ (d) 125
Question 26 – ✅ (b) (1002)₄
Question 27 – ✅ (d) 8
Question 28 – ✅ (b) 5 hours
Question 29 – ✅ (c) 1
Question 30 – ✅ (b) 60°
Question 31 – ✅ (a) (12)₃
Question 32 – ✅ (b) aᵐ + aⁿ = aᵐ⁺ⁿ
Question 33 – ✅ (a) 2x + 1 = 5
Question 34 – ✅ (d) 2√3
Question 35 – ✅ (c) 12√3 m
Question 36 – ✅ (a) x = 20
Question 37 – ✅ (a) 1
Question 38 – ✅ (b) (100)₁₆
Question 39 – ✅ (d) (100)₁₆
Question 40 – ✅ (a) (304)₄

Theory Marking Scheme

Question 1: Number bases and arithmetic
(a) 2×4² + 3×4 + 1 + 2×4⁻¹ = 32 + 12 + 1 + 0.5 = 45.5. Award 2 for correct expansion, 1 for result.
(b) Units: 3+2=5, write 0 carry 1; fives: 4+3+1=8, write 3 carry 1; twenty-fives: 2+1+1=4. Sum (430)₅. Award 2 for carries, 1 for result.
(c) 3×16 + 12 = 60₁₀ (2); hexadecimal nibbles 3→0011 and C→1100, giving 00111100₂ (2).

Question 2: Indices and exponential modelling
(a) a¹⁰b⁻⁴ ÷ a³b⁻¹ = a⁷b⁻³ = a⁷/b³. Award 1 for expanding power, 2 for combining exponents, 1 for positive-index form.
(b) 3²ˣ = 3³ˣ⁻³; 2x = 3x−3; x = 3. Award 1 each.
(c) Model 5×2ᵗ = 160 (1); 2ᵗ=32=2⁵ (1); t=5 hours (1).

Question 3: Trigonometry and surveying
(a) tan 60° = √3; give 2 for correct exact value.
(b) Let x be the nearer distance and h the height. h = x√3 and h = (x+30)/√3 (2); 3x=x+30, x=15 (2); h=15√3 m (1).
(c) cos θ = 5/10 = 1/2 (2); θ=60° (1).

Question 4: Logarithms and hexadecimal arithmetic
(a) 2⁻⁵=1/32. Award 2.
(b) 50=100/2 (1); log50 = log100 − log2 (1) = 2−0.3010=1.6990 (2).
(c) Units: 7+9=16, write 0 carry 1 (1); sixteens: A+B+1 = 10+11+1 =22 = 16+6, write 6 carry 1 (2); result (160)₁₆ (1).