A certain sum of money at simple interest amounts to Rs. 1012 in 2 ¹/ ₂ years and to Rs. 1067.20 in 4 years. The rate of interest per annum is?

Let the present ages of father be 3x and daughter be x .
so the 4 year ago father's age and daughter's age was
( 3x - 4 ) and ( x - 4 )
? ( 3x - 4 / x -4 ) = 4 / 1
? x = 12 year and 3x = 36 year

Total number of cases is 17.
? Number divisible by 3 are 3, are 3, 6, 9, 12, 15 (These are 5 in number )
Number divisible by 7 are 7, 14. (These are 2 in number )
There are two favourable number of cases
Total no. of favourable number = 5 + 2
Required probability = 7/17.

Total ways = 100
Squares of following no's lie between 1 and 100,
1², 2², 3² , 4², 5² , 6², 7², 8² , 9² , 10²
(Which are 10 in numbers.)
So. Required probability = 10/100 = 1/10

Required number of ways = ²⁵C₃
= (25 x 24 x 23) / (1 x 2 x 3) = 2300

Let p(N) = 8 (N⁵ - N³ + N)
= 4 x 2 x N (N⁴ - N² + 1)
and q(N) = 28 (N⁶ + 1)
= 7 x 4 [( N²)³ + (1)³]
= 4 x 7 (N² + 1) ( N⁴ - N² + 1)
? HCF of p(N) and q (N) = 4 (N⁴ - N² + 1)

Using the formula, A = p ( 1 + R/100 )ⁿ
? 1352 = p(1 + 4/100 )²
? 1352 = p(1.04 )²
? p = 1352/(1.04)² = ? 1250

Let third number = y
Then, second number = 2y
and first number =4y

? ( y + 2y + 4y) / 3 = 42
? 7y = 42 x 3
? y = 18

so, (largest) - (smallest) = ( 4y - y)
= 3y
=54

length of tunnel + length of train = distances covered by train in 1 min
? length of tunnel + 700 = 72/60 km = 1.2 km = 1200 m
? length of tunnel = 1200 - 700 = 500 m

Let the required number be N,
Then, N/6 = N - 40
? N + 240 = 6N
? N = 48.

? P(1 + r/100)^t = A
? 390625(1 + 4/100)^t = 456976
? (1 + 4/100)^t = 456976/390625 =(26/25)⁴
? (26/25)^t = (26/25)⁴
? t = 4
? The required time is 4 years.