New Curriculum Second Term Mathematics examination for SS 1

Class: SS 1
Term: Second 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 remainder when 73 + 58 is divided by 9 is
   (a) 8    (b) 6    (c) 5    (d) 4
2. A digital clock reads 9:00 a.m. What time will it show 29 hours later?
   (a) 1:00 p.m. the next day    (b) 2:00 a.m. the next day    (c) 2:00 p.m. the next day    (d) 9:00 a.m. two days later
3. Let A = {x ∈ ℤ : −2 ≤ x ≤ 3}. The number of elements in A is
   (a) 7    (b) 6    (c) 5    (d) 4
4. ________ correctly describes A = {0,1,2} relative to B = {0,1,2,3}.
   (a) A and B are equal    (b) B is a proper subset of A    (c) A ∩ B is empty    (d) A is a proper subset of B
5. The Venn diagram represents 50 pupils who study Mathematics (M) or Physics (P). How many study neither subject?
   (a) 10    (b) 11    (c) 12    (d) 17
6. Suppose p is false and q is true. Which compound statement has the truth value true?
   (a) q ∧ ¬p is false    (b) p ∨ q    (c) p ∧ q    (d) p ↔ q
7. For p = false and q = true, evaluate ¬(p ∧ q).
   (a) neither true nor false    (b) equal to p    (c) false    (d) true
8. The main reason p → q is false when p is true and q is false is that
   (a) both condition and conclusion are false    (b) the implication requires p and q always to have the same truth value    (c) the stated condition holds but its conclusion does not    (d) the condition is false and the conclusion is true
9. The formula y = √(x + 4) is rearranged to make x the subject. The result is
   (a) x = y − 4    (b) x = y² − 4    (c) x = y² + 4    (d) x = √y − 4
10. A quantity y varies directly as x². If y = 27 when x = 3, determine y when x = 5.
    (a) 45    (b) 125    (c) 60    (d) 75
11. Which assertion about the empty set is correct?
    (a) It contains exactly one element.    (b) It is a subset of every set.    (c) It cannot be a subset of a nonempty set.    (d) It is the universal set in every problem.
12. For the same survey in the diagram, the number studying exactly one of the two subjects is
    (a) 29    (b) 23    (c) 40    (d) 39
13. All the following are mathematical statements with definite truth values except ________.
    (a) Three is less than ten.    (b) Close the classroom door.    (c) 7 is a prime number.    (d) 9 is an even number.
14. A statement equivalent to ¬(p ∨ q) is ________.
    (a) ¬p ∨ ¬q    (b) ¬p ∧ ¬q    (c) p ↔ q    (d) p ∧ q
15. If p is false and q is false, what is the truth value of p → q?
    (a) unknown    (b) not a statement    (c) false    (d) true
16. If V = πr²h, the correct expression for positive radius r is
    (a) r = √(V/(πh))    (b) r = √(Vh/π)    (c) r = V/(πh)    (d) r = V/(2πh)
17. The time y for a task varies inversely as the number x of workers. If five workers take 12 hours, eight workers take
    (a) 19.2 hours    (b) 9 hours    (c) 8 hours    (d) 7.5 hours
18. Evaluate 7 × 11 (mod 6).
    (a) 5    (b) 0    (c) 1    (d) 4
19. A meeting held on Tuesday will be repeated after 103 days. On which weekday will it fall?
    (a) Saturday    (b) Monday    (c) Thursday    (d) Sunday
20. Given A = {2,3,5} and B = {1,3,5,7}, determine A ∩ B.
    (a) {2,7}    (b) {1,2,3,5,7}    (c) {2,3,5}    (d) {3,5}
21. If both p and q are false, which operator produces a true compound statement?
    (a) ¬p ∨ q    (b) p ∨ q    (c) p ↔ ¬q    (d) p ∧ q
22. The biconditional p ↔ q is false when ________.
    (a) both p and q are false    (b) p and q agree    (c) p and q have different truth values    (d) both p and q are true
23. The correct first step in making r the subject of V = πr²h is to
    (a) divide both sides by πh    (b) take the square root of V without rearranging    (c) subtract πh from both sides    (d) multiply both sides by πh
24. Suppose z varies jointly as x and y; z = 24 when x = 2, y = 3. Calculate z when x = 4, y = 5.
    (a) 40    (b) 120    (c) 80    (d) 60
25. In modulo 8 arithmetic, the residue represented by 23 − 47 is
    (a) 0    (b) 5    (c) 3    (d) 7
26. Solve 4x + 3 ≡ 1 (mod 7) for the residue 0 ≤ x < 7.
    (a) 2    (b) 4    (c) 3    (d) 5
27. The set {12,15,18,21} is correctly expressed in set-builder form as
    (a) {x ∈ ℕ : 12 ≤ x ≤ 21, x is divisible by 4}    (b) {x ∈ ℕ : x is even and 12 ≤ x ≤ 21}    (c) {x ∈ ℕ : 12 ≤ x ≤ 21, x is divisible by 3}    (d) {x ∈ ℕ : 12 < x < 21, x is divisible by 3}
28. In the universal set U = {1,2,3,4,5,6}, let A = {2,4,6}. Determine how many elements are not in A.
    (a) 2    (b) 6    (c) 4    (d) 3
29. In a survey, 18 take French, 12 take Arabic and 25 take at least one. How many take both languages?
    (a) 7    (b) 5    (c) 6    (d) 30
30. The statement “It is not raining” is the negation of which proposition?
    (a) It may rain tomorrow.    (b) It is cloudy.    (c) It is sunny.    (d) It is raining.
31. The residue ________ is the least non-negative solution of 5x ≡ 2 (mod 9).
    (a) 4    (b) 7    (c) 5    (d) 2
32. Which of these is not an exact representation of {2,4,6,8}?
    (a) {2,4,6,8}    (b) {x ∈ ℕ : 2 ≤ x ≤ 8 and x is even}    (c) {x ∈ ℕ : x < 8 and x is even}    (d) {8,6,4,2}
33. The set B = {2,4} is a subset of A = {1,2,3,4}. Which statement is NOT true?
    (a) B ∪ A = A    (b) B ∩ A = B    (c) A ⊆ B    (d) B ⊆ A
34. Given U = {1,2,3,4,5,6,7,8}, A = {2,4,6,8} and B = {3,6}, find A′ ∩ B.
    (a) {3,6}    (b) {3}    (c) ∅    (d) {6}
35. If p is true and q is false, then the conjunction p ∧ q is
    (a) undetermined    (b) true    (c) false    (d) equivalent to p
36. Examine the truth table shown. The final column represents which operation?
    (a) p ∧ q    (b) p → q    (c) p ↔ q    (d) p ∨ q
37. Under what condition will “If I revise, then I pass” be logically false?
    (a) The student passes without revising.    (b) The student revises and passes.    (c) The student neither revises nor passes.    (d) The student revises but does not pass.
38. A scientist uses E = mc². The expression needed to calculate m is
    (a) m = E²/c    (b) m = E/c²    (c) m = Ec²    (d) m = √(E/c)
39. For partial variation y = ax + b, measurements give y = 11 at x = 3 and y = 17 at x = 5. The value of y at x = 7 is
    (a) 25    (b) 23    (c) 29    (d) 21
40. If 17 ≡ 5 (mod n), which listed value may be n?
    (a) 6    (b) 5    (c) 7    (d) 8

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: Sets, Venn diagrams and modular arithmetic
(a) Of 70 students, 42 study Mathematics, 36 study Physics and 19 study both. Find the number studying at least one. (3 marks)
(b) Using the same data, find the number studying neither subject. (3 marks)
(c) Calculate (24+49) mod 8, stating the non-negative remainder. (4 marks)
Use a correctly labelled Venn diagram for your working.

Question 2: Modular equations and weekday applications
(a) Calculate 7×11 (mod 6). (2 marks)
(b) Solve 4x+3 ≡ 1 (mod 7), giving 0 ≤ x < 7. (4 marks)
(c) A deadline is set for Tuesday, 103 days from now. On which weekday will it occur? (4 marks)

Question 3: Logic and truth-value analysis
(a) Construct the complete four-row truth table for p → q. (4 marks)
(b) Write a logically equivalent expression for ¬(p ∨ q). (3 marks)
(c) State precisely when the biconditional p ↔ q is false. (3 marks)

Question 4: Change of subject and variations
(a) Make r the subject of V=πr²h for r>0. (2 marks)
(b) y varies directly as x². If y=27 when x=3, find y when x=5. (4 marks)
(c) For partial variation y=ax+b, the pairs (3,11) and (5,17) are observed. Find y at x=7. (4 marks)

Objective Marking Scheme
Question 1 – ✅ (c) 5
Question 2 – ✅ (c) 2:00 p.m. the next day
Question 3 – ✅ (b) 6
Question 4 – ✅ (d) A is a proper subset of B
Question 5 – ✅ (a) 10
Question 6 – ✅ (b) p ∨ q
Question 7 – ✅ (d) true
Question 8 – ✅ (c) the stated condition holds but its conclusion does not
Question 9 – ✅ (b) x = y² − 4
Question 10 – ✅ (d) 75
Question 11 – ✅ (b) It is a subset of every set.
Question 12 – ✅ (a) 29
Question 13 – ✅ (b) Close the classroom door.
Question 14 – ✅ (b) ¬p ∧ ¬q
Question 15 – ✅ (d) true
Question 16 – ✅ (a) r = √(V/(πh))
Question 17 – ✅ (d) 7.5 hours
Question 18 – ✅ (a) 5
Question 19 – ✅ (d) Sunday
Question 20 – ✅ (d) {3,5}
Question 21 – ✅ (a) ¬p ∨ q
Question 22 – ✅ (c) p and q have different truth values
Question 23 – ✅ (a) divide both sides by πh
Question 24 – ✅ (c) 80
Question 25 – ✅ (a) 0
Question 26 – ✅ (c) 3
Question 27 – ✅ (c) {x ∈ ℕ : 12 ≤ x ≤ 21, x is divisible by 3}
Question 28 – ✅ (d) 3
Question 29 – ✅ (b) 5
Question 30 – ✅ (d) It is raining.
Question 31 – ✅ (a) 4
Question 32 – ✅ (c) {x ∈ ℕ : x < 8 and x is even}
Question 33 – ✅ (c) A ⊆ B
Question 34 – ✅ (b) {3}
Question 35 – ✅ (c) false
Question 36 – ✅ (a) p ∧ q
Question 37 – ✅ (d) The student revises but does not pass.
Question 38 – ✅ (b) m = E/c²
Question 39 – ✅ (b) 23
Question 40 – ✅ (a) 6

Theory Marking Scheme

Question 1: Sets, Venn diagrams and modular arithmetic
(a) n(M∪P)=42+36−19=59. Formula 2, result 1.
(b) 70−59=11. Method 2, answer 1.
(c) 24+49=73 (1); 73=8×9+1 (2); remainder 1 (1).

Question 2: Modular equations and weekday applications
(a) 77 ≡ 5 (mod 6), award 2.
(b) 4x ≡ −2 ≡ 5 (mod 7) (2); inverse of 4 is 2 modulo 7, hence x ≡ 10 ≡ 3 (mod 7) (2).
(c) 103=14×7+5 (2); move five weekdays from Tuesday: Sunday (2).

Question 3: Logic and truth-value analysis
(a) Rows (T,T)→T, (T,F)→F, (F,T)→T, (F,F)→T. Award 1 per correct row.
(b) ¬p ∧ ¬q; award 3, or 1 for recognising De Morgan rule with correct operation only.
(c) When truth values differ: (T,F) or (F,T). Award 1 for principle and 1 for each row.

Question 4: Change of subject and variations
(a) r²=V/(πh) (1), r=√[V/(πh)] (1).
(b) y=kx² (1); 27=9k, k=3 (1); y=3×25=75 (2).
(c) 11=3a+b and 17=5a+b (1); subtract: a=3 (1); b=2 (1); y(7)=21+2=23 (1).