IPM Indore 2023

If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of f(2023) is _________

Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker to complete the task alone would be ______.

If f(n)= 1 + 2 + 3 +∙∙∙+(n+1) and $$\sum_{k=1}^{k=n}\frac{1}{f(k)}$$ then the least value of n for which g(n) exceeds the value $$\frac{99}{100}$$ is ____________.

In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more than the other candidate, then the number of voters in the voting list is_________.

The total number of positive integer solutions of $$21 \leq a + b + c \leq 25$$ is __________.

If three consecutive coefficients in the expansion of $$(x + y)^{n}$$ are in the ratio 1 : 9 : 63, then the value of n is ___________.

The polynomial $$4x^{10} − x^{9} + 3x^{8} − 5x^{7} + cx^{6} + 2x^{5} − x^{4} + x^{3} − 4x^{2} + 6x − 2$$ when divided by $$x - 1$$ leaves a remainder 2. Then the value of $$c + 6$$ is________.

The product of the roots of the equation $$\log_{2}2^{(\log_{2}x)^{2} }− 5\log_{2}x + 6 = 0$$ is ________

The remainder when 1! + 2! + 3! +∙∙∙+95! is divided by 15 is _____________.

Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, then the total time taken, in minutes, to complete the journey, will be______.