0570 Mathematics 2021 Solution Paper 1

Question 1

Given the set of real numbers ℝ, the identity element for multiplication is

A. 1

B. 0

C. -1

D. 2

Show Answer

Answer: 1

Explanation: Step 1: The identity element leaves a number unchanged.
Step 2: For multiplication, a × 1 = a.
Step 3: Therefore, the identity element is 1.

Question 2

The HCF of 21 and 36 is

A. 3

B. 42

C. 84

D. 6

Show Answer

Answer: 3

Explanation: Step 1: Factors of 21 → 1, 3, 7, 21.
Step 2: Factors of 36 → 1, 2, 3, 4, 6, 9, 12, 18, 36.
Step 3: Common factors → 1, 3.
Step 4: Highest common factor = 3.

Question 3

Simplifying 2/3 + 1/4 ÷ 3/5 gives

A. 5/7

B. 11/20

C. 13/12

D. 55/36

Show Answer

Answer: 13/12

Explanation: Step 1: Do division first → (1/4) ÷ (3/5) = (1/4) × (5/3) = 5/12.
Step 2: Add to 2/3 → 2/3 = 8/12.
Step 3: 8/12 + 5/12 = 13/12.

Question 4

Subtracting 1.9705 from 5 gives

A. 6.9705

B. 3.0295

C. -3.0295

D. 4.9705

Show Answer

Answer: 3.0295

Explanation: Step 1: Write subtraction → 5.0000 − 1.9705.
Step 2: Perform subtraction digit by digit.
Step 3: Result = 3.0295.

Question 5

A fraction equivalent to 4/5 is

A. 9/10

B. 8/10

C. 16/25

D. 24/30

Show Answer

Answer: 8/10

Explanation: Step 1: Multiply numerator and denominator by same number.
Step 2: 4/5 × 2/2 = 8/10.
Step 3: Equivalent fractions represent the same value.

Question 6

0.00746 correct to 2 decimal places is

A. 0.0075

B. 0.0074

C. 0.01

D. 0.007

Show Answer

Answer: 0.01

Explanation: Step 1: 2 decimal places means 2 digits after decimal.
Step 2: 0.00746 ≈ 0.01 because digits beyond second decimal round up.
Step 3: Final answer = 0.01.

Question 7

Expressing (3 × 10⁻⁵)(2 × 10⁷) in standard form gives

A. 6.0 × 10²

B. 6.0 × 10⁻²

C. 6.0 × 10⁻²

D. 6.0 × 10³⁵

Show Answer

Answer: 6.0 × 10²

Explanation: Step 1: Multiply numbers → 3 × 2 = 6.
Step 2: Add powers → 10⁻⁵ × 10⁷ = 10².
Step 3: Final = 6 × 10².

Question 8

Given that 1 Euro = 650 FCFA, 65000 FCFA in Euros is

A. 65650

B. 42200

C. 100

D. 64350

Show Answer

Answer: 100

Explanation: Step 1: 1 Euro = 650 FCFA.
Step 2: 65000 ÷ 650 = 100.
Step 3: Answer = 100 Euros.

Question 9

In a test of 75 marks, a candidate scored 60 marks. The percentage score is

A. 30

B. 60

C. 80

D. 90

Show Answer

Answer: 80

Explanation: Step 1: Percentage = (score ÷ total) × 100.
Step 2: (60 ÷ 75) × 100 = 0.8 × 100.
Step 3: = 80%.

Question 10

There are 180 girls in a mixed school. If the ratio of girls to boys is 4:3, the total number of students is

A. 135

B. 180

C. 420

D. 315

Show Answer

Answer: 315

Explanation: Step 1: Ratio girls:boys = 4:3.
Step 2: 4 parts = 180 → 1 part = 45.
Step 3: Boys = 3 × 45 = 135.
Step 4: Total = 180 + 135 = 315.

Question 11

Two sets S = {a, b} and T = {c, d}. The relation that best describes S and T is

A. S ⊂ T

B. S ∩ T = ∅

C. T ⊂ S

D. S ∪ T = ∅

Show Answer

Answer: S ∩ T = ∅

Explanation: Step 1: List elements → S = {a, b}, T = {c, d}.
Step 2: Compare elements → none are common.
Step 3: Intersection means common elements.
Step 4: Since none exist → intersection is empty set (∅).

Question 12

The shaded portion in the Venn diagram represents

A. P ∩ Q′

B. P ∪ Q′

C. P ∩ Q

D. P ∪ Q

Show Answer

Answer: P ∩ Q′

Explanation: Step 1: Observe shaded region is inside P only.
Step 2: It excludes the overlap with Q.
Step 3: Q′ means “not in Q”.
Step 4: So region = elements in P but not in Q → P ∩ Q′.

Question 13

The truth value of p ∧ q is true when

A. p is true and q is false

B. p is true and q is true

C. p is false and q is true

D. p is false and q is false

Show Answer

Answer: p is true and q is true

Explanation: Step 1: Symbol ∧ means AND.
Step 2: AND is true only when both statements are true.
Step 3: Any false statement makes it false.
Step 4: Therefore both must be true.

Question 14

The image of −3 under f: x → 7 − x is

A. 4

B. -10

C. 10

D. -21

Show Answer

Answer: 10

Explanation: Step 1: Substitute x = −3 into function.
Step 2: f(-3) = 7 − (−3).
Step 3: Subtracting a negative = addition.
Step 4: 7 + 3 = 10.

Question 15

Given P × Q = {(m,3),(m,5),(n,3),(n,5)}, the set Q is

A. {m,n}

B. {3,5}

C. {m,5}

D. {n,5}

Show Answer

Answer: {3,5}

Explanation: Step 1: Cartesian product pairs elements from P and Q.
Step 2: First elements (m,n) belong to P.
Step 3: Second elements belong to Q.
Step 4: From pairs → Q = {3,5}.

Question 16

If f(x) = x + 2 and g(x) = x − 4, find f(g(x))

A. 4x + 3

B. 4x − 3

C. x − 2

D. 4 − 3x

Show Answer

Answer: x − 2

Explanation: Step 1: Start with g(x) = x − 4.
Step 2: Substitute into f(x).
Step 3: f(g(x)) = f(x − 4).
Step 4: Replace x → (x − 4) in f(x) = x + 2.
Step 5: (x − 4) + 2 = x − 2.

Question 17

A triangle in which no two sides are equal is

A. Scalene

B. Isosceles

C. Equilateral

D. Right angle

Show Answer

Answer: Scalene

Explanation: Step 1: Scalene → all sides different.
Step 2: Isosceles → two equal sides.
Step 3: Equilateral → all equal sides.
Step 4: Therefore correct answer = scalene.

Question 18

The number of edges of a square-based pyramid is

A. 4

B. 5

C. 8

D. 12

Show Answer

Answer: 8

Explanation: Step 1: Square base has 4 edges.
Step 2: Pyramid has 4 slant edges.
Step 3: Total edges = 4 + 4 = 8.

Question 19

The quadrilateral which has four lines of symmetry is

A. Square

B. Rectangle

C. Rhombus

D. Kite

Show Answer

Answer: Square

Explanation: Step 1: Square has vertical, horizontal, and 2 diagonal lines.
Step 2: Total = 4 lines of symmetry.
Step 3: Others have fewer.

Question 20

In figure 2, O is the centre. The angle marked x is

A. 42°

B. 138°

C. 90°

D. 48°

Show Answer

Answer: 42°

Explanation: Step 1: Angle at centre is twice angle at circumference.
Step 2: Given central angle = 42°.
Step 3: Corresponding angle = 42°.

Question 21

Find angle b in figure

A. 54°

B. 126°

C. 136°

D. 36°

Show Answer

Answer: 126°

Explanation: Step 1: Straight line angles sum to 180°.
Step 2: Given angle = 54°.
Step 3: b = 180 − 54.
Step 4: b = 126°.

Question 22

The diameter of a circle is 14 cm. The area is

A. 28π

B. 198π

C. 89π

D. 49π

Show Answer

Answer: 49π

Explanation: Step 1: Radius = diameter ÷ 2 = 7.
Step 2: Area = πr².
Step 3: = π × 7² = 49π.

Question 23

The total surface area of a cube of side 8 cm is

A. 64

B. 384

C. 512

D. 24

Show Answer

Answer: 384

Explanation: Step 1: Area of one face = 8 × 8 = 64.
Step 2: Cube has 6 faces.
Step 3: Total = 6 × 64 = 384.

Question 24

The area of a square is 144 cm². The side length is

A. 12

B. 14

C. 16

D. 24

Show Answer

Answer: 12

Explanation: Step 1: Area = side².
Step 2: side = √144.
Step 3: side = 12.

Question 25

The coordinates where 2y = x + 3 cuts the y-axis

A. (0,1.5)

B. (-3,0)

C. (0,3)

D. (3,0)

Show Answer

Answer: (0,1.5)

Explanation: Step 1: On y-axis, x = 0.
Step 2: Substitute → 2y = 3.
Step 3: y = 3/2 = 1.5.

Question 26

Two lines are parallel if

A. Product = -1

B. Product = 1

C. Same gradient

D. Sum = -1

Show Answer

Answer: Same gradient

Explanation: Parallel lines have equal slopes (gradients).

Question 27

Midpoint of (-6,7) and (2,1)

A. (-2,4)

B. (-2,-4)

C. (-4,4)

D. (4,4)

Show Answer

Answer: (-2,4)

Explanation: Step 1: Use midpoint formula → ((x₁+x₂)/2, (y₁+y₂)/2).
Step 2: (-6+2)/2 = -2, (7+1)/2 = 4.
Step 3: Midpoint = (-2,4).

Question 28

Equation of line with gradient 2 through (4,3)

A. y + 2x + 11 = 0

B. y + 2x + 5 = 0

C. y − 2x − 11 = 0

D. y − 2x + 5 = 0

Show Answer

Answer: y − 2x + 5 = 0

Explanation: Step 1: Use y − y₁ = m(x − x₁).
Step 2: y − 3 = 2(x − 4).
Step 3: y − 3 = 2x − 8.
Step 4: y = 2x − 5.
Step 5: Rearranged → y − 2x + 5 = 0.

Question 29

The number of arcs in the network is

A. 6

B. 5

C. 4

D. 3

Show Answer

Answer: 5

Explanation: Step 1: Count all connecting lines between nodes.
Step 2: Total arcs = 5.

Question 30

Given y + 4x − 8, the coefficient of y is

A. 0

B. -8

C. 1

D. -1

Show Answer

Answer: 1

Explanation: Step 1: Coefficient = number multiplying the variable.
Step 2: y = 1 × y.
Step 3: So coefficient is 1.

Question 31

Given that p = 3 and q = 2, find p² − 2q

A. 5

B. 9

C. 12

D. 13

Show Answer

Answer: 5

Explanation: Step 1: Substitute p = 3 and q = 2.
Step 2: Calculate p² = 3² = 9.
Step 3: Calculate 2q = 2 × 2 = 4.
Step 4: Subtract: 9 − 4 = 5.

Question 32

The solution of the inequality 9 − x > 2x is

A. x < 3

B. x > 3

C. x ≤ 3

D. x ≥ 3

Show Answer

Answer: x < 3

Explanation: Step 1: Start with 9 − x > 2x.
Step 2: Add x to both sides → 9 > 3x.
Step 3: Divide both sides by 3 → 3 > x.
Step 4: Rewrite → x < 3.

Question 33

Simplify (x/2) ÷ (3/x)

A. x²/6

B. 6/x²

C. x/6

D. 3x/2

Show Answer

Answer: x²/6

Explanation: Step 1: Division of fractions = multiply by reciprocal.
(x/2) ÷ (3/x) = (x/2) × (x/3).
Step 2: Multiply numerators → x × x = x².
Step 3: Multiply denominators → 2 × 3 = 6.
Step 4: Final answer = x²/6.

Question 34

Given that 3ˣ = 3⁻⁵, the value of x is

A. 81

B. -81

C. -5

D. 5

Show Answer

Answer: -5

Explanation: Step 1: Bases are equal (3).
Step 2: Therefore, exponents must be equal.
Step 3: So x = −5.

Question 35

Given v² = u² − 4as, express s in terms of v, u and a

A. (v² − u²)/4a

B. (u² − v²)/4a

C. 4a/(u² − v²)

D. (u² − v²)/a

Show Answer

Answer: (u² − v²)/4a

Explanation: Step 1: Start with v² = u² − 4as.
Step 2: Rearrange → 4as = u² − v².
Step 3: Divide both sides by 4a.
Step 4: s = (u² − v²) / 4a.

Question 36

The next term in the sequence 3, 5, 8, 12, ... is

A. 25

B. 33

C. 20

D. 17

Show Answer

Answer: 17

Explanation: Step 1: Differences: 5−3=2, 8−5=3, 12−8=4.
Step 2: Pattern increases by 1.
Step 3: Next difference = 5.
Step 4: 12 + 5 = 17.

Question 37

The sum of first n terms is Sₙ = n² − 2n. The first term is

A. 0

B. 1

C. -1

D. 3

Show Answer

Answer: -1

Explanation: Step 1: First term = S₁.
Step 2: Substitute n=1 → S₁ = 1² − 2(1).
Step 3: = 1 − 2 = −1.

Question 38

The remainder when x² + 2x − 1 is divided by x − 1 is

A. 2

B. -2

C. 0

D. -4

Show Answer

Answer: 2

Explanation: Step 1: Use remainder theorem → substitute x=1.
Step 2: 1² + 2(1) − 1 = 1 + 2 − 1 = 2.
Step 3: Remainder = 2.

Question 39

The sum of interior angles of a right-angled triangle is

A. 90°

B. 180°

C. 270°

D. 360°

Show Answer

Answer: 180°

Explanation: All triangles have interior angles that sum to 180°.

Question 40

Given sin θ = 1/2 where θ is acute, find θ

A. 90°

B. 60°

C. 45°

D. 30°

Show Answer

Answer: 30°

Explanation: Step 1: Recall sin30° = 1/2.
Step 2: Since θ is acute, θ = 30°.

Question 41

The bearing of Q from P is 050°. Find bearing of P from Q

A. 050°

B. 230°

C. 130°

D. 140°

Show Answer

Answer: 230°

Explanation: Step 1: Reverse bearing differs by 180°.
Step 2: 050° + 180° = 230°.

Question 42

The unit vector parallel to 3i + 4j is

A. −3i − 4j

B. 4i + 3j

C. (3/5)i + (4/5)j

D. (−3/5)i − (4/5)j

Show Answer

Answer: (3/5)i + (4/5)j

Explanation: Step 1: Magnitude = √(3² + 4²) = 5.
Step 2: Divide each component by 5.
Step 3: Result = (3/5)i + (4/5)j.

Question 43

Find vector AB given diagram

A. a − 3b

B. 3b + a

C. −a − 3b

D. a + 3b

Show Answer

Answer: a + 3b

Explanation: Use vector addition following direction shown in diagram.

Question 44

The order of matrix (2 0 11 / 4 13 5) is

A. 2×3

B. 3×2

C. 2×2

D. 2×4

Show Answer

Answer: 2×3

Explanation: Step 1: Count rows = 2.
Step 2: Count columns = 3.
Step 3: Order = 2 × 3.

Question 45

The 2×2 identity matrix is

A. (1 0 / 0 1)

B. (0 1 / 1 0)

C. (1 1 / 1 1)

D. (0 0 / 0 0)

Show Answer

Answer: (1 0 / 0 1)

Explanation: Identity matrix has 1s on diagonal and 0 elsewhere.

Question 46

Transformation maps (3,2) to Q′

A. (8,12)

B. (7,10)

C. (4,0)

D. (1,-2)

Show Answer

Answer: (7,10)

Explanation: Multiply matrix by vector step by step.

Question 47

The mode in the distribution is

A. 1

B. 3

C. 8

D. 12

Show Answer

Answer: 8

Explanation: Mode is the value with highest frequency.

Question 48

The median of 42, 32, 36, 39, 47, 46, 43 is

A. 32

B. 42

C. 36

D. 46

Show Answer

Answer: 42

Explanation: Step 1: Arrange → 32,36,39,42,43,46,47.
Step 2: Middle value = 42.

Question 49

The mean mark from table is

A. 2

B. 3

C. 5

D. 6

Show Answer

Answer: 2

Explanation: Step 1: Multiply marks by frequency.
Step 2: Add totals.
Step 3: Divide by total frequency.

Question 50

Probability of picking a club from deck is

A. 1/3

B. 1/4

C. 1/52

D. 3/4

Show Answer

Answer: 1/4

Explanation: Step 1: There are 4 suits in a deck.
Step 2: Clubs = 13 cards.
Step 3: Total cards = 52.
Step 4: Probability = 13/52 = 1/4.