The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then the area of the park (in sq. m) is:

Given that, l = 2b [Here l = length and b = breadth]

Decrease in length = Half of the 10 cm = 10/2 = 5 cm
Increase in breadth = Half of the 10 cm = 10/2 = 5 cm
Increase in the area = (70 + 5) = 75 sq cm

According to the question,
(l - 5) (b + 5) = lb + 75
? (2b - 5) (b + 5) = 2b² + 75 [since l = 2b]
? 5b - 25 = 75
? 5b = 100
? b = 100/ 5 = 20
? l = 2b = 2 x 20 = 40 cm

let the two-digit number be 10p + q
Now according to the question
p + q = 10
and (p +10q) + 36 = (q + 10p)
? -9q + 9p = 36
? -q + p = 4 ....(ii)
on adding eq (i) and (ii) we get
2p = 14 ? p =7
? p = 7 and q = 4

? Required number is the 73
So, the number is a multiple of prime number.

Let the original number be 10p + q.
So from question,
q = 2p + 1 ...(i)
and (10q + p) - (10p + q ) = (10p + q) - 1
? 9q - 9p = 10p + q -1
? 19p - 8q = 1...(ii)
Putting the value of (i) in equation (ii) we get
19p - 8(2p + 1) = 1
? 19p - 16p - 8 = 1
? 3p = 9
? p = 3
So, q = 2 x 3 + 1 = 7
? Original number = 10 x 3 + 7 = 37

Sum of cubes of first n natural numbers = [ n(n+1) ]²
Given n = 15

? Required sum = [ ( 15 x 16 ) / 2 ] ²
= (15 x 8 )²
= 120²
= 14400

Average cost =( w₁x₁ + w₂x₂ + w₃x₃ ) / (w₁ +w₂ +w₃)

= ( 1200 x 30 +1500 x 25 + 2000 x 32) / ( 1200 + 1500 + 2000 )
= 137500 / 4700
= ? 29.25 per kg

We know that, the radius of a circle inscribed in a equilateral triangle = a/[2?3]
Where, a be the length of the side of an equilateral triangle.

Given that, area of a circle inscribed in an equilateral tringle = 154 cm²
? ?(a/2?3)² = 154
? (a/2?3)² = 154 x (7/22) = (7)a²
? a = 42?3 cm

Perimeter of an equilateral triangle = 3a
= 3(14?3)
= 42?3 cm

6² -1 = 35, which is divisible by both 5 and 7.

LCM of 16, 18 and 20 = 720
? Required number = 720k + 4
Where, k is a natural number to be divisible by 7, (720 k + 4) will be a multiple of 7.
Smallest value of k = 4
? Required number = 720 x 4 + 4 = 2884

If A Score 100 then B scores 80 and C score 72.
? B scores 80 while C scores 72
? B scores 100 while C scores 72 * 100 / 80 = 90 points
Hence, in 100 points B can give (100 ? 90) = 10 points