If n is even, f(n) = n (n + 1)
So, f (2) = 2(2+1) = 2 (3) = 6
If n is odd, f(n) = n + 3
f (1) = 1 + 3 = 4
It is given that, 8 x f(m+1) - f(m) =2
So, m can either be even or odd
Case-1: If m were even and m+1 odd
So, 8 x f(m+1) - f(m) =2
8(m + 4) - m (m + 1) = 2
(m-10) (m+3) = 0
m = 10 or -3
m = 10, since m is positive
Case-2: If m were even and m+1 odd
8 x f(m+1) - f(m) =2
8 (m +1) (m + 2) - (m + 3) = 2
Now, let us substitute m = 1 which is the minimum possible value
8 (1 + 1) (1 + 2) - (1 + 3) 3
Case 2 does not work
If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be
Now, let us observe the second pattern.
In 2021, the value increases to 2(2p + 3) + 3
On expanding, (2 x 2) +(2 x 3) + 3 = 2 x 2 + 3(1 + 2)
In 2022, the value increases to 2[2(2p+3) +3]+3
On expanding, 2((2 x 2p) + (2 x 3) + 3) + 3
= (2 x 2 x 2 x p) + (2x 2 x 3) + (2 x 3) + 3
A person invested a total amount of Rs 15 lakh. A part of it was invested in a fixed deposit earning 6% annual interest, and the remaining amount was invested in two other deposits in the ratio 2 : 1, earning annual interest at the rates of 4% and 3%, respectively. If the total annual interest income is Rs 76000 then the amount (in Rs lakh) invested in the fixed deposit was [TITA]
Let Rs x be invested in FD.
So, Remaining amount = Rs. 15 Lakhs - x, which is invested in the ratio 2 : 1
Rewriting Rs. 76000 as Rs. 0.76 Lakhs,
Multiply both sides by 100,
165 + 18x -11x = 228
7x = 63
x = 9 (or) Rs. 9 Lakhs
The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is
Let the two numbers be a and b
From the question, we know
a x b = 616
a - b = 6, ab = 616
616 can be rewritten as 22 x 28, where 28 - 22= 6 using trial and error
So, a + b = 28 + 22 = 50
On selling a pen at 5% loss and a book at 15% gain, Karim gains Rs. 7. If he sells the pen at 5% gain and the book at 10% gain, he gains Rs. 13. What is the cost price of the book in Rupees?
Two cars travel the same distance starting at 10:00 am and 11:00 am, respectively, on the same day. They reach their common destination at the same point of time. If the first car travelled for at least 6 hours, then the highest possible value of the percentage by which the speed of the second car could exceed that of the first car is
The minimum time travelled by Car 1 = 6 hours.
Minimum time travelled by Car 2 = 5 hours
Now, the time taken can be of any value → (6,5) (7,6) … (1000,999)
However, since we want to calculate the highest possible percentage by which speed of car 2 could exceed car 1, we consider the lowest time taken - 6 hours and 5 hours, respectively.
Let the distance between the start and finish be 30 Kms (LCM of 6 and 5)
So, Car 1 travels at 5 Kmph and Car 2 travels at 6 Kmph
Percentage increase in speed by Car 2
At their usual efficiency levels, A and B together finish a task in 12 days. If A had worked half as efficiently as she usually does, and B had worked thrice as efficiently as he usually does, the task would have been completed in 9 days. How many days would A take to finish the task if she works alone at her usual efficiency?
We know that A and B together would finish a task in 12 days
Now, if A works in half the efficiency and B works thrice as efficiently, the work gets completed in 9 days.
Multiply (1) by 3
Solving (2) and (3) we get,
A = 18 days
2cos (x (x + 1)) = 2cos (0(0 + 1)) = 2cos (0) = 2 x = 0 is a valid solution
Any value other than this wont work.
Therefore, only x = 0 works and there is only one real solution
The income of Amala is 20% more than that of Bimala and 20% less than that of Kamala. If Kamala’s income goes down by 4% and Bimala’s goes up by 10%, then the percentage by which Kamala’s income would exceed Bimala’s is nearest to
Let Bimala’s income be B. Amala’s income is 20% more than B.
So, A = 1.2 B
Similarly, Amala’s income is 20% less than that of Kamala.
A= 0.8K
Now, Bimala’s new income goes up by 10% = 1.1B
Kamala’s income goes down by 4%
4% (1.5B) = 0.06B
Therefore, Kamala’s new income is = 1.5B - 0.06B = 1.44B
So, K = 1.44B and B = 1.1 B
In a race of three horses, the first beat the second by 11 metres and the third by 90 metres. If the second beat the third by 80 metres, what was the length, in metres, of the racecourse? [TITA]
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