NECTA CHEMISTRY 1 2025 SOLUTION

NECTA Form 4 Basic Mathematics 2025

Step-by-Step Solutions

SECTION A (60 Marks)

1. Numbers and Approximations

(a) Express 0.1136 as a fraction:
0.1136 =

113610000

Simplify by dividing by 16:
Ans:

71625

(b) Rounding off:
(i) 0.00070482 to 3 sig figures: 0.000705
(ii) 1,233,388 to nearest ten thousands: 1,230,000

(c) Percentage:
Pupils not from city = 100% - (3/5 × 100%) = 100% - 60% = 40%.

2. Exponents & Logarithms

(a) Simplify Exponents:
Expression: 27ⁿ × 9²ⁿ × 3
= (3³)ⁿ × (3²)²ⁿ × 3¹
= 3³ⁿ × 3⁴ⁿ × 3¹
= 3^{(3n + 4n + 1)}
Ans: 3^{7n + 1}

(b) Rationalize:

√2 + 1√2 - √5

×

√2 + √5√2 + √5

Denominator becomes: 2 - 5 = -3.
Ans: -

2 + √10 + √2 + √53

(c) Logarithms:
log 72 = log(2³ × 3²) = 3log2 + 2log3
= 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542
Ans: 1.8572

3. Sets & Probability

(a) Sets: n(P)=130, n(C)=150, n(P∩C)=40, Total=260.
(i) Physics only = 130 - 40 = 90.
(ii) Neither = 260 - (90 + 110 + 40) = 20.

(b) Probability:
n(Goats only) = 18 - 3 = 15.
Probability =

1535

=

37

4. Geometry

(a) Triangle Area:
∠ABC = 180° - 150° = 30°.
Area =

12

ab sinθ
22.6 = 0.5 × 8 × BC × sin(30°)
22.6 = 2 × BC → BC = 11.3 cm.

(b) Pentagon Perimeter:
Side length s = 2r sin(36°) = 2(10)(0.5878) = 11.756 m.
Perimeter = 5 × 11.756 = 58.78 m.

5. Variations

(a) Currency: 27,600,000 ÷ 2,300 = 12,000 USD.

(b) Joint Variation:
Formula: h =

k Vr²

Find k: 3 = k(900)/100 → k =

13

.
Calculate h:

13

×

120025

= 16 cm.

6. Vectors

(a) Magnitude:
Sum = (2-1+3)i + (-1+3-4)j = 4i - 2j.
Magnitude = √4² + (-2)² = √20 = 2√5.

(b) Coordinate:
Midpoint formula. (6+x)/2 = 3 → x=0. (7+y)/2 = 5 → y=3.
Point: (0, 3)

9. Trigonometry

(a) Cosine Rule:
8² = 6² + 7² - 2(6)(7)cosA
cosA = 0.25 → A ≈ 75.5°

(b) Right Triangle:
tan C =

ABBC

. From Pythagoras, BC = √6²-3² = √27 = 3√3.
tan C =

33√3

=

1√3

.

SECTION B (40 Marks)

13. Matrices

(a) Singular Matrix: Determinant is 0.
Det = (-1)(3) - (11)(a) = 0 → -3 - 11a = 0.
a = -

311

(b) Simultaneous Equations:
Matrix equation:

5	3
10	7

x

y

=

9

11

Determinant = 35 - 30 = 5.
Inverse =

15

7	-3
-10	5

Multiply Inverse by B:
x =

15

(63 - 33) = 6.
y =

15

(-90 + 55) = -7.
Solution: x = 6, y = -7

14. Functions

(a) Inverse Product:
f^-1(x) = x - 2
g^-1(x) =

x + 23

Product = (x - 2) ×

x + 23

=

x² - 43