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Given:

Radius of metallic cylinder = 3 cm 

Height of metallic cylinder = 5 cm 

Height of cone to be formed = 1 cm 

Radius of cone to be formed = 1 mm or 0.1 cm 

Formulas used:

Volume of cylinder = πR²H 

Volume of cone = 1/3 × πr²h 

Calculation:

Radius and height of the cylinder is R = 3 cm, H = 5 cm respectively. 

Radius and height of the cone is r = 0.1 cm, h = 1 cm respectively. 

As per the question;

Volume of metallic cylinder = No of cones × Volume of a cone 

⇒ π × 3 × 3 × 5 = No of cones × 1/3 × π × 0.1 × 0.1 × 1 

⇒ No of cones =  (3 × 3 × 5 × 3)/(0.1 × 0.1) 

⇒  No of cones = 13500 

∴ The number of cones formed is 13500. 

 Mistake Points

1 cm = 10 mm 

0.1 cm = 1 mm

Given:

A 120m long train crosses a 280m long tunnel in 20 seconds.

Formula Used:

Time = [ L1 + L2] / Relative speed

Where, L1 = length of train; L2 = length of object that train is crossing

Relative Speed (RS) = Speed of train + speed of platform(= 0)  

Calculation:

20 seconds = [120 + 280]/RS

⇒ 20 seconds = [120 + 280]/Speed of train [∵ speed of platform = 0]

⇒ Speed of train = 400/20 = 20 m/s

Now, time taken by train to cross the pole = Distance/Speed

Also, length of pole = 0 and speed of pole = 0

⇒ Time taken by train to cross the pole = Length of train / Speed of train

= 120/20 = 6 seconds

Given:

(Raju + Rajat) together complete the work = 5 days

Raju alone can complete the work = 7 days.

Formula used:

Total work = efficiency × time

Calculation:

Now,

Efficiency of Rajat = (7 - 5) = 2 unit/day

Time taken to complete the work by Rajat = 35/2 = 17.5 days

∴ The correct answer is 17.5 days.

GIVEN:

Ram : Shiv = 9 : 11

difference = 4 years

EXPLANATION:

let Ram's age be 9x 

∴ Shiv's age = 11x

difference = 11x - 9x = 2x

A.T.Q 

2x = 4 years 

x = 2 years 

Ram's age after 4 years = (9 × 2) + 4 = 22 years 

Shiv's age after 4 years = (11 × 2) + 4 = 26 years

Requires ratio = 22/26 = 11 : 13

∴ the ratio will be 11 : 13

2 ratio = 4 years (difference)

after 4 years 2 ratios will increase in ages of both 

∴ required ratio = (9 + 2) :(11 + 2) = 11 : 13

∴ the ratio will be 11 : 13

Given:

CP of 13 books = SP of 17 books

Formula Used:

Loss % = (CP - SP)/CP × 100

Calculation:

CP × 13 = SP × 17

⇒ CP : SP = 17 : 13

Loss % = (CP - SP)/CP × 100

⇒ (17 - 13)/17 × 100

⇒ 400/17

⇒ $23\frac{9}{17}$%

∴ The loss percentage is $23 \frac{9}{17}$ %.

Given:

The ratio of the number of boys to girls in a school is 10 : 11.

The number of girls in the school is 8360.

Formula Used:

Let the number of boys be 10x and the number of girls be 11x.

Total number of students = Number of boys + Number of girls

Calculation:

Number of girls = 11x

11x = 8360

⇒ x = 8360 / 11

⇒ x = 760

Number of boys = 10x = 10 × 760 = 7600

Total number of students = 7600 + 8360 = 15960

The total number of students in the school is 15960.

Given:

Length = 6 breadth

Length of the field = 120 m

Formula used:

Perimeter = 2 × (L + B)

Calculation:

According to question,

Breadth = 20 m, Length = 120 m

Perimeter = 2 × (120 + 20)

⇒ 2 × (140)

⇒ 280 m

∴ The correct answer is 280 m.

Given:

A can fill the tank in 3 hours and B can empty the tank in 6 hours

Concept Used:

If time taken to fill the tank = X days/hours

∴ Part of tank fill in one day/hours = 1/X

Efficiency ∝ 1/Time

Calculation:

A can fill the tank in 3 hours

∴ Part of tank fill in one hours = 1/3

B can empty the tank in 6 hours

∴ Part of tank emptied in one hour = 1/6

Now, if both the pipes are open together

∴ Time taken to fill the tank = 1/3 – 1/6 = 1/6

So, both the pipes will fill the tank in 6 hours

Hence, option (2) is correct

Concept use:

Total Surface Area = 2πr² + 2πrh  and h is the height of Cylinder and r is the Radius of Circle

Explanation:

968 = 2 × 22/7 × r (r + h) = 2 × 22/7 × 7 ( h + 7)

⇒ 968 × 7/22 × 1/2 × 1/7 = h + 7

⇒ 22 = h + 7 

⇒ 22 - 7 = h

⇒ height of Cylinder = 15cm 

Hence, The Height of Cylinder is 15cm.

Concept used:

a^b × a^c = a^{b + c}

If a^b = a^c then b = c

(a^b)^c = a^{b × c}

Calculation:

1252 × (625)-0.25 = 5x

⇒ 5^{3 × 2} × 5^{4 × -0.25} = 5^x

⇒ 5⁶ × 5^-1 = 5^x

⇒ 5^{6 - 1} = 5^x

⇒ 5⁵ = 5x

⇒ x = 5