Q. Given: log₄ z = B log₂/₃z, for all z > 0. Find the value of constant B.

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Q. Solve for x the equation 2ˣ⁺³ = 5ˣ⁺².

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Q. Solve for x: log₂(x²-3x)=log₂(5x-15).

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Q. Find the value of x: 3 x² aˡᵒᵍᵃˣ = 348?

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Q. Determine the logarithmic function of ln(1+5x)⁻⁵.

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Q. Solve the logarithmic function of ln(\(\frac{1+5x}{1+3x}\)).

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Q. Computation of the discrete logarithm is the basis of the cryptographic system _______

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Q. From 1, 2, 3, …, 320 one number is selected at random. Find the probability that it is either a multiple of 7 or a multiple of 3.

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Q. An integer from 300 through 780, inclusive is to be chosen at random. Find the probability that the number is chosen will have 1 as at least one digit.

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Q. A card is drawn randomly from a standard deck of cards. Determine the probability that the card drawn is a queen or a heart.

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Q. There are 9 letters having different colors (red, orange, yellow, green, blue, indigo, violet) and 4 boxes each of different shapes (tetrahedron, cube, polyhedron, dodecahedron). How many ways are there to place these 9 letters into the 4 boxes such that each box contains at least 1 letter?

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Q. The sum of all integers from 1 to 520 that are multiples of 4 or 5?

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Q. In class, students want to join sports. 15 people will join football, 24 people will join basketball, and 7 people will join both. How many people are there in the class?

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Q. In a renowned software development company of 240 computer programmers 102 employees are proficient in Java, 86 in C#, 126 in Python, 41 in C# and Java, 37 in Java and Python, 23 in C# and Python, and just 10 programmers are proficient in all three languages. How many computer programmers are there those are not proficient in any of these three languages?

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Q. The numbers between 1 and 520, including both, are divisible by 2 or 6 is _______

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Q. At a software company, skilled workers have been hired for a project. Out of 75 candidates, 48 of them were software engineer; 35 of them were hardware engineer; 42 of them were network engineer; 18 of them had skills in all three jobs and all of them had skills in at least one of these jobs. How many candidates were hired who were skilled in exactly 2 jobs?

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Q. There are 70 patients admitted in a hospital in which 29 are diagnosed with typhoid, 32 with malaria, and 14 with both typhoid and malaria. Find the number of patients diagnosed with typhoid or malaria or both.

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Q. Find the sequence generated by 1/1−x²−x⁴. (Assume that 1, 1, 2, 3, 5, 8,… has generating function 1/1−x−x².)

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Q. Suppose G is the generating function for the sequence 4, 7, 10, 13, 16, 19,…, the find a generating function (in terms of G) for the sequence of differences between terms.

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Q. What is the generating function for the sequence with closed formula aₙ=4(7ⁿ)+6(−2)ⁿ?

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