Mensuration
- A plane divides a right circular cone into two parts of equal volume. If the plane is parallel to the base, then the ratio, in which the height of the cone is divided, is
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OA' = h units
AA' = H units
AB = R units
A'B' = r units.
A'B'|| AB
∠OA'B' = ∠OAB
∠OB'A' = ∠OBA
∴ ∆OAB ~ ∆OA;B;∴ AO' = A'B' OA AB ⇒ h = r H + h R
According to the question,1 πr²h = 1 πR²(H + h) - 1 πr²h 3 3 3 ⇒ 2 πr²h = 2 πR²(H + h) 3 3 ⇒ 2 = r² = (H + h) R² h ⇒ (H + h)³ = 2 h³ ⇒ H + h = ³√2 h ⇒ H + 1 = ³√2 h ⇒ H = ³√2 - 1 h 1 ⇒ h = 1 : ³√2 - 1 H Correct Option: D
OA' = h units
AA' = H units
AB = R units
A'B' = r units.
A'B'|| AB
∠OA'B' = ∠OAB
∠OB'A' = ∠OBA
∴ ∆OAB ~ ∆OA;B;∴ AO' = A'B' OA AB ⇒ h = r H + h R
According to the question,1 πr²h = 1 πR²(H + h) - 1 πr²h 3 3 3 ⇒ 2 πr²h = 2 πR²(H + h) 3 3 ⇒ 2 = r² = (H + h) R² h ⇒ (H + h)³ = 2 h³ ⇒ H + h = ³√2 h ⇒ H + 1 = ³√2 h ⇒ H = ³√2 - 1 h 1 ⇒ h = 1 : ³√2 - 1 H
- The radii of two solid iron spheres are 1 cm and 6 cm respectively. A hollow sphere is made by melting the two spheres. If the external radius of the hollow sphere is 9 cm, then its thickness (in cm) is
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Volume of sphere = 4 πr³ 3 ∴ Total volume of both spheres = 4 π(r1³ + r2³) 3 = 4 π(1³ + 6³) 3 = 4 π(1 + 216) 3 = 
4π × 217 
cu.metre. 3
If the internal radius of hollow sphere = r cm, then∴ Volume of the iron of this sphere = 4 π (9³ - r³) cu.cm. 3
According to the question,4 π (9³ - r³) = 4π × 217 3 3
⇒ 729 – r³ = 217
⇒ r³ = 729 – 217 = 512
⇒ r³ = (8)³
⇒ r = 8 cm
∴ Required thickness = 9 – r = 9 – 8 = 1 cm.Correct Option: D
Volume of sphere = 4 πr³ 3 ∴ Total volume of both spheres = 4 π(r1³ + r2³) 3 = 4 π(1³ + 6³) 3 = 4 π(1 + 216) 3 = 4π × 217 cu.metre. 3
If the internal radius of hollow sphere = r cm, then∴ Volume of the iron of this sphere = 4 π (9³ - r³) cu.cm. 3
According to the question,4 π (9³ - r³) = 4π × 217 3 3
⇒ 729 – r³ = 217
⇒ r³ = 729 – 217 = 512
⇒ r³ = (8)³
⇒ r = 8 cm
∴ Required thickness = 9 – r = 9 – 8 = 1 cm.
- The base of a right prism is a trapezium whose lengths of two parallel sides are 10 cm and 6 cm and distance between them is 5 cm. If the height of the prism is 8 cm, its volume is
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Area of the base of prism
= 1 (10 + 6) × 5 2 = 1 × 16 × 5 = 40 sq.cm. 2
∴ Volume of prism = Area of base × height
= 40 × 8 = 320 cu. cm.Correct Option: A
Area of the base of prism
= 1 (10 + 6) × 5 2 = 1 × 16 × 5 = 40 sq.cm. 2
∴ Volume of prism = Area of base × height
= 40 × 8 = 320 cu. cm.
- The radius of a hemispherical bowl is 6 cm. The capacity of the bowl is (Take π = 22/7)
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Capacity of bowl = 2 π r³ 3 = 2 × 22 × 6 × 6 × 6 cu.cm. 3 7 = 3168 = 452.57 cu. cm. 7 Correct Option: D
Capacity of bowl = 2 π r³ 3 = 2 × 22 × 6 × 6 × 6 cu.cm. 3 7 = 3168 = 452.57 cu. cm. 7
- Length of each edge of a regular tetrahedron is 1 cm. Its volume is :
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Volume of regular tetrahedron = a³ cu. cm. 6√2 = 1 = √2 cu. cm. 6√2 6√2 × √2 = √2 cu. cm. 12 Correct Option: D
Volume of regular tetrahedron = a³ cu. cm. 6√2 = 1 = √2 cu. cm. 6√2 6√2 × √2 = √2 cu. cm. 12
