Get comprehensive solutions to all NCERT Class 9 Maths questions with detailed explanations. Master board exam concepts and build a strong foundation for Class 10 mathematics.
Q1. Rationalize: 1/(√5 + √3)
Answer:
1/(√5 + √3) × (√5 - √3)/(√5 - √3)
= (√5 - √3)/((√5)² - (√3)²)
= (√5 - √3)/(5 - 3) = (√5 - √3)/2
Q2. Express 0.overline{23} as a fraction.
Answer:
Let x = 0.232323...
100x = 23.232323...
100x - x = 23
99x = 23, x = 23/99
Q1. Find the remainder when x³ - 2x² + x + 2 is divided by (x - 1).
Answer:
By remainder theorem, remainder = f(1)
f(1) = 1³ - 2(1)² + 1 + 2 = 1 - 2 + 1 + 2 = 2
Q2. Factorize: x³ + 6x² + 11x + 6
Answer:
f(-1) = -1 + 6 - 11 + 6 = 0
So (x + 1) is a factor
x³ + 6x² + 11x + 6 = (x + 1)(x² + 5x + 6)
= (x + 1)(x + 2)(x + 3)
Q1. Solve: 2x + 3y = 11 and 2x - 4y = -24
Answer:
Subtracting equation 2 from equation 1:
7y = 35, y = 5
Substituting in equation 1:
2x + 15 = 11, 2x = -4, x = -2
Solution: x = -2, y = 5
Q2. The cost of 2 kg apples and 1 kg grapes is ₹160. The cost of 1 kg apples and 2 kg grapes is ₹200. Find the cost per kg of each.
Answer:
Let cost of apples = ₹x per kg, grapes = ₹y per kg
2x + y = 160 ... (1)
x + 2y = 200 ... (2)
Solving: x = ₹40, y = ₹80
Q1. In triangle ABC, if AB = AC and ∠B = 70°, find ∠A and ∠C.
Answer:
Since AB = AC, triangle ABC is isosceles
∠B = ∠C = 70° (base angles are equal)
∠A + ∠B + ∠C = 180°
∠A + 70° + 70° = 180°
∠A = 40°
Q2. State and prove the angle sum property of a triangle.
Answer:
Statement: The sum of angles in a triangle is 180°.
Proof: Draw a line parallel to BC through A. By alternate interior angles, the three angles at A sum to 180°.
Q1. Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.
Answer:
s = (13 + 14 + 15)/2 = 21 cm
Area = √[s(s-a)(s-b)(s-c)]
= √[21 × 8 × 7 × 6]
= √7056 = 84 sq cm
Q2. A triangular park has sides 120 m, 80 m, and 50 m. Find the area and cost of leveling at ₹15 per sq m.
Answer:
s = (120 + 80 + 50)/2 = 125 m
Area = √[125 × 5 × 45 × 75] = 1500 sq m
Cost = 1500 × 15 = ₹22,500
Q1. Find the volume of a sphere with radius 6 cm.
Answer:
Volume = (4/3)πr³ = (4/3) × (22/7) × 6³
= (4/3) × (22/7) × 216 = 905.14 cu cm
Q2. A cone has base radius 5 cm and height 12 cm. Find its curved surface area.
Answer:
Slant height l = √(r² + h²) = √(25 + 144) = 13 cm
CSA = πrl = (22/7) × 5 × 13 = 204.28 sq cm
Q: How many chapters are there in Class 9 Maths?
A: There are 15 chapters in NCERT Class 9 Maths covering all major mathematical concepts for board exams.
Q: Which topics are most important for Class 9 board exams?
A: Polynomials, coordinate geometry, triangles, circles, surface areas and volumes, and statistics are high-weightage topics.
Q: How should I prepare for Class 9 Maths board exam?
A: Focus on theorem proofs, practice numerical problems, understand geometric constructions, and solve previous year papers regularly.
