IPM Indore 2023


For the following questions answer them individually

Question 31

If the harmonic mean of the roots of the equation $$(5 + \sqrt{2})x^{2} โˆ’ bx + 8 + 2\sqrt{5} = 0$$ is 4 then the value of b is

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Question 32

Let [x] denote the greatest integer not exceeding x and {x} = x -[x].
If n is a natural number, then the sum of all values of x satisfying the equation 2[x] = x + n{x} is

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Question 33

If $$\frac{a+b}{b+c} = \frac{c+d}{d+a}$$, which of the following statements is always true?

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Question 34

The set of all real values of x satisfying the inequality $$\frac{x^{2}(x + 1)}{(x - 1)(2x + 1)^{3}}> 0$$ is

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Question 35

In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot represent the scores of the five players?

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Question 36

If $$\cos \alpha + \cos \beta = 1$$ ,then the maximum value of $$\sin \alpha โˆ’ \sin \beta$$ is

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Question 37

A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The remainder when P(x) is divided by (x - 1) (x - 2) is

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Question 38

In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is

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Question 39

Let p be a positive integer such that the unit digit of $$p^{3}$$ is 4. What are the possible unit digits of $$(p+3)^{3}$$?

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Question 40

If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4?

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