Class 8 Maths — NCERT Exercise Solutions

(i) 2x + 5 = 17 → 2x = 12 → x = 6

(ii) (3x − 4)/2 = 7 → 3x − 4 = 14 → 3x = 18 → x = 6

(iii) 5x − 3 = 3x + 5 → 5x − 3x = 5 + 3 → 2x = 8 → x = 4

Let the three consecutive even numbers be x, x+2, x+4

Equation: x + (x+2) + (x+4) = 54

3x + 6 = 54 → 3x = 48 → x = 16

The three numbers are: 16, 18, 20

Verification: 16 + 18 + 20 = 54 ✓

Let Rahul's present age = 3x and Riya's present age = 4x

Five years ago: Rahul = 3x − 5, Riya = 4x − 5

Given ratio: (3x−5)/(4x−5) = 2/3

3(3x−5) = 2(4x−5) → 9x − 15 = 8x − 10 → x = 5

Rahul's present age = 3 × 5 = 15 years

Riya's present age = 4 × 5 = 20 years

Let the smaller number = x, so the larger number = x + 12

x + (x+12) = 48 → 2x + 12 = 48 → 2x = 36 → x = 18

The two numbers are 18 and 30.

The sum of measures of all angles of a convex quadrilateral = 360°.

Yes, this property holds for a non-convex (concave) quadrilateral too. Any quadrilateral can be divided into two triangles by drawing a diagonal. The sum of angles of each triangle = 180°, so total = 180° + 180° = 360°.

Sum of all angles of a quadrilateral = 360°

Fourth angle = 360° − (75° + 90° + 110°) = 360° − 275° = 85°

Properties of a Parallelogram:

• Opposite sides are equal and parallel.
• Opposite angles are equal.
• Adjacent angles are supplementary (sum = 180°).
• Diagonals bisect each other.

Rhombus vs Parallelogram: A rhombus is a special parallelogram where all four sides are equal. Additionally, diagonals of a rhombus bisect each other at right angles (90°), which is not true for all parallelograms. Every rhombus is a parallelogram, but not vice versa.

Adjacent angles of a parallelogram are supplementary: 3x + 2x = 180°

5x = 180° → x = 36°

The four angles are: 108°, 72°, 108°, 72°

(3x = 108°, 2x = 72°, and opposite angles are equal)

Arranged data: 2, 3, 4, 5, 7, 7, 7, 8, 9 (n = 9)

Mean = Sum/n = (2+3+4+5+7+7+7+8+9)/9 = 52/9 ≈ 5.78

Median = Middle value = 5th term = 7

Mode = Most frequent value = 7 (appears 3 times)

Sample space = {1, 2, 3, 4, 5, 6}, Total outcomes = 6

(i) Prime numbers: {2, 3, 5} → Favourable = 3

P(prime) = 3/6 = 1/2

(ii) Greater than 4: {5, 6} → Favourable = 2

P(>4) = 2/6 = 1/3

(iii) Number 6: {6} → Favourable = 1

P(6) = 1/6

Sum of 5 numbers = 5 × 18 = 90

Sum of remaining 4 numbers = 4 × 16 = 64

Excluded number = 90 − 64 = 26

Prime factorisation: 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹⁰

√1024 = 2⁵ = 32

Prime factorisation: 180 = 2² × 3² × 5

5 is unpaired. To make it a perfect square, multiply by 5.

New number = 180 × 5 = 900

√900 = √(2² × 3² × 5²) = 2 × 3 × 5 = 30

The unit digit of a square depends only on the unit digit of the original number:

• (i) 81: Unit digit = 1 → 1² = 1 → Unit digit of 81² = 1
• (ii) 272: Unit digit = 2 → 2² = 4 → Unit digit of 272² = 4
• (iii) 799: Unit digit = 9 → 9² = 81 → Unit digit = 1
• (iv) 3853: Unit digit = 3 → 3² = 9 → Unit digit = 9

(i) 343 = 7 × 7 × 7 = 7³ → ∛343 = 7

(ii) −2744: ∛(−2744) = −∛2744. 2744 = 2³ × 7³ = 14³ → −14

(iii) 46656 = 2⁶ × 3⁶ = (2²×3²)³ = 36³ → ∛46656 = 36

Prime factorisation: 392 = 2³ × 7²

7² is an incomplete triplet. Need one more 7 to complete the triplet 7³.

Multiply by 7 → 392 × 7 = 2744 = 2³ × 7³ = 14³ ✓

392 is not a perfect cube. Smallest multiplier = 7.

CP = ₹14,400, SP = ₹16,200

Profit = SP − CP = 16200 − 14400 = ₹1800

Profit% = (Profit/CP) × 100 = (1800/14400) × 100 = 12.5%

P = ₹8000, R = 10%, n = 2 years

Formula: A = P(1 + R/100)ⁿ

A = 8000 × (1 + 10/100)² = 8000 × (11/10)² = 8000 × 121/100 = ₹9680

CI = A − P = 9680 − 8000 = ₹1680

% Increase: (100/500) × 100 = 20%

% Decrease: (100/600) × 100 = 16.67%

Note: % increase and % decrease are NOT equal because the base changes.

(i) (ab−bc) + (bc−ca) + (ca−ab)

= ab − bc + bc − ca + ca − ab = 0

(ii) (2x²+3xy−y²) + (−4x²+xy+3y²)

= (2−4)x² + (3+1)xy + (−1+3)y² = −2x² + 4xy + 2y²

(i) 103² = (100+3)² = 100² + 2×100×3 + 3² = 10000 + 600 + 9 = 10609

(ii) 98² = (100−2)² = 100² − 2×100×2 + 2² = 10000 − 400 + 4 = 9604

(i) (2x+3)(x−5) = 2x² − 10x + 3x − 15 = 2x² − 7x − 15

(ii) (a+b+c)(a−b−c) = a(a−b−c) + b(a−b−c) + c(a−b−c)

= a² − ab − ac + ab − b² − bc + ac − bc − c²

= a² − b² − c² − 2bc

Area of trapezium = ½ × (sum of parallel sides) × height

= ½ × (25 + 15) × 12 = ½ × 40 × 12 = 240 cm²

r = 7 cm, h = 20 cm, π = 22/7

Volume = πr²h = 22/7 × 7 × 7 × 20 = 22 × 7 × 20 = 3080 cm³

TSA = 2πr(h + r) = 2 × 22/7 × 7 × (20 + 7) = 44 × 27 = 1188 cm²

l = 15 cm, b = 12 cm, h = 10 cm

Volume = l × b × h = 15 × 12 × 10 = 1800 cm³

LSA = 2h(l + b) = 2 × 10 × (15+12) = 20 × 27 = 540 cm²

TSA = 2(lb + bh + hl) = 2(180 + 120 + 150) = 2 × 450 = 900 cm²

(i) 2³ × 3⁴ × 4/3 = 2³ × 3⁴ × 2²/3 = 2⁵ × 3³ = 32 × 27 = 864

(ii) (2³)² = 2^(3×2) = 2⁶ = 64

(iii) 3⁷/3² = 3^(7−2) = 3⁵ = 243

Standard form: A × 10ⁿ where 1 ≤ A < 10

(i) 5960000 = 5.96 × 10⁶

(ii) 0.000023 = 2.3 × 10⁻⁵

(iii) 302000000 = 3.02 × 10⁸

This is an inverse proportion (fewer hours → more workers needed).

Workers × Hours = constant

15 × 48 = x × 30

x = (15 × 48)/30 = 720/30 = 24 workers

This is direct proportion (more time → more distance).

Speed = constant: 60/1.5 = x/5

x = (60 × 5)/1.5 = 300/1.5 = 200 km

(i) 12x²y − 18xy² = 6xy(2x − 3y) — common factor = 6xy

= 6xy(2x − 3y)

(ii) x² + 7x + 12. Need two numbers with product 12 and sum 7: 3 and 4.

= (x + 3)(x + 4)

(iii) x² − 5x − 24. Need product −24 and sum −5: −8 and 3.

= (x − 8)(x + 3)

(i) a² − 25 = a² − 5² = (a+5)(a−5) [using a²−b² = (a+b)(a−b)]

(ii) 4x²−12x+9 = (2x)² − 2×2x×3 + 3² = (2x−3)²

(iii) 16m²−(4m−1)² = (4m)²−(4m−1)²

= (4m+4m−1)(4m−4m+1) = (8m−1)(1) = (8m−1)

Plot the four points: A(3,5), B(−1,5), C(−1,−2), D(3,−2) on the coordinate plane.

AB is horizontal (y=5), CD is horizontal (y=−2), BC is vertical (x=−1), AD is vertical (x=3).

All angles are 90°. AB = CD = 4 units, BC = AD = 7 units.

The figure formed is a Rectangle.

Table:

• s = 1 cm → p = 4 cm
• s = 2 cm → p = 8 cm
• s = 3 cm → p = 12 cm
• s = 4 cm → p = 16 cm

Direct proportion: Yes, because p/s = 4 (constant) for all values. The graph is a straight line passing through the origin.

For s = 6.5 cm: p = 4 × 6.5 = 26 cm

O = (0,0), A = (4,0), B = (4,3)

In rectangle OABC, OC must be parallel to AB and OC = AB = 3 units vertical. OA is horizontal.

C lies directly above O at x = 0: C = (0, 3)

Dimensions: length OA = 4 units, width OC = 3 units. Area = 12 sq. units.