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Given:

 The ratio of the radius of a sphere to the radius of a hemisphere is 4 ∶ 5

Formula:

The surface area of the sphere = 4πr²

The curved Surface area of the hemisphere = 2πr²

Calculation:

The radius of sphere: radius of hemisphere = 4 : 5

 Now, the surface area of the sphere: Curved surface area of the hemisphere 

= 4π(R)² : 2πr², where R is the radius of the sphere and r is the radius of the hemisphere

= (2R² : r²)

= (2 × 4² : 5²)

= (2 × 16 : 25)

= 32 : 25

Now, the percentage by which the surface area of a sphere is more than the curved surface area of the hemisphere 

= $\frac{(32-25)}{25}×100$

= ($\frac{7}{25}$) × 100

= 28 %

Hence, the percentage required is 28%.

Shortcut Trick

For mean proportional x between a and b, x = √(a × b).

x = √(12.8 × 64.8) = √(1.28 × 6.48 × 100) / 10 is not needed, simply write:

x = √(128/10 × 648/10) = √(64 × 2 × 324 × 2) / 10 = (8 × 18 × 2) / 10 = 28.8

∴ The correct answer is 28.8.

Alternate Method

Given: First number (a) = 12.8, Second number (b) = 64.8

Formula Used: Mean Proportional (x) = √(a × b)

⇒ x² = 12.8 × 64.8

⇒ x² = (128/10) × (648/10)

⇒ x² = (64 × 2 × 324 × 2) / 100

⇒ x = √((64 × 324 × 4) / 100)

⇒ x = (8 × 18 × 2) / 10

⇒ x = 288 / 10 = 28.8

∴ The correct answer is 28.8.

Additional Information

Third Proportional

The third proportional of two numbers a and b is defined as the value c such that a : b :: b : c, which implies c = b²/a.

Fourth Proportional

The fourth proportional of three numbers a, b, and c is the value d such that a : b :: c : d, which implies d = (b × c) / a.

Invertendo and Alternendo

If a : b = c : d, then Invertendo gives b : a = d : c and Alternendo gives a : c = b : d.

Shortcut Trick

Direct formula for Area of Trapezium = 1/2 × (Sum of parallel sides) × Height

Sum of parallel sides = (2 × Area) ÷ Height

Sum = (2 × 480) ÷ 15 = 960 ÷ 15 = 64 m

Other side = 64 − 20 = 44 m

∴ The correct answer is 44 m.

Alternate Method

Given:

Area (A) = 480 m²

Height (h) = 15 m

Parallel side 1 (a) = 20 m

Formula Used:

Area = 1/2 × (a + b) × h

Calculations:

⇒ 480 = 1/2 × (20 + b) × 15

⇒ 480 × 2 = (20 + b) × 15

⇒ 960 = (20 + b) × 15

⇒ 960 ÷ 15 = 20 + b

⇒ 64 = 20 + b

⇒ b = 64 − 20

⇒ b = 44 m

∴ The correct answer is 44 m.

Additional Information

Properties of a Trapezium

A quadrilateral with at least one pair of parallel sides. The non-parallel sides are called legs.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal in length and base angles are equal.

Median of a Trapezium

The segment connecting the midpoints of the legs. Length = 1/2 × (Sum of parallel sides).

Given:

X is directly proportional to the square of Y. When X is 12, then Y is 2

Calculation:

X : Y²

So,

12 : 2²

⇒ 12 : 4

⇒ 3 : 1

So, X : Y2 =  3 : 1

So, when Y = 3  

⇒X : Y2 = X : 9

X : Y2 = X : 9 = 3 : 1

⇒ X = 27

∴ The required answer is 27.

Given:

Calculations:

Cost of production by company A= 90 

Cost of production by company E = 36

Total cost of production by A and E together = 90 + 36 = 126

So, Total cost required=  126 × 5,000 × 8,00,000/360 = 1,400,000,000 or 140 crores

Hence, The Required value is 140 crores.

Given:

Area of the sector of a circle = 72π cm² and radius = 36 cm

Concept used:

If L is the length of a sector of a circle and r be the radius, then,

Area of the sector = 1/2 × L × r

Calculation:

According to the question,

⇒ 1/2 × L × 36 = 72π

⇒ L =  (72π × 2)/36

⇒ L = 4π 

∴ The length of the corresponding arc of the sector is 4π cm

Shortcut Trick

Let Cost Price (CP) be 100 units.

Marked Price (MP) = 100 + 20% of 100 = 120 units.

Selling Price (SP) = 120 − 5% of 120 = 120 − 6 = 114 units.

Given 114 units = ₹2,280 ⇒ 1 unit = 2,280 ÷ 114 = ₹20.

Cost Price (CP) = 100 units = 100 × 20 = ₹2,000.

∴ The correct answer is ₹2,000.

Alternate Method

Given: Discount = 5%, Markup = 20%, SP = ₹2,280.

Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100

⇒ SP = CP × (120 ÷ 100) × (95 ÷ 100)

⇒ 2,280 = CP × 1.2 × 0.95

⇒ 2,280 = CP × 1.14

⇒ CP = 2,280 ÷ 1.14 = 2,000

∴ The correct answer is ₹2,000.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on the label. Discount is always calculated on the MP to arrive at the Selling Price (SP).

Effective Profit Percentage

The net profit can be found using the formula: Markup% − Discount% − (Markup × Discount) ÷ 100. Here, 20 − 5 − (20 × 5) ÷ 100 = 14% profit.

Let the fifth number be x, then

Average of first four number is = 3x

According to the question

4 × 3x + x = 85.8 × 5

⇒ 12x + x = 429

⇒ 13x = 429

⇒ x = 429/13

∴ x = 33

Given

The numbers are  20, 24, 23, 19, P and Q.

The average = 25

Formula

Average = Total sum of value/No.of values

Solution

The total sum of all numbers = 20 + 24 + 23 + 19 + P + Q

⇒ 86 + P + Q

The total sum of all numbers = 25 × 6 = 150

Equating both,

⇒ 86 + P + Q = 150

⇒ P + Q = 150 - 86

⇒ P + Q = 64

The average of P and Q = 64/2

⇒ 32

The correct option is 2.

Given:

The area of the base of the cylinder = 616 m²

The height of the cylinder (h) = 8 m

Formula used:

Value of π = 22/7

The area of the base of the cylinder = π r ²

The curved or lateral surface area of the cylinder = 2 π r h

where r = radius, h = height

Calculation:

The area of the base of the cylinder = 616 m²

⇒ π r ²  = 616

⇒ (22/7) × r² = 616

⇒ r² = 196

⇒ r = 14 m

The radius of the cylinder = 14 m

The curved surface area (C.S.A.) of the cylinder = 2 π r h

⇒ 2 π r h = 2 × (22/7) × 14 × 8

= 2 × 44 × 8

= 704 m²

∴ The curved surface area of the cylinder is 704 m²

Additional Information