If the maximum value of the term independent of t in the expansion of (t2x\(^{\frac{1}{5}}\)+\(\frac{(1−x)^{\frac{1}{10}}}{t}\)),x≥0 is K, then 8 K is equal to__________.
Correct Answer: 6006
Solution and Explanation
General Term
=15Cr(\(t^2x^{\frac{1}{5}}\))15−r(\(\frac{(1−x)^{\frac{1}{10}}}{t}\))r
for term independent on t
2(15 – r) – r = 0
⇒ r = 10
∴ T11 = 15C10x(1 – x)
Maximum value of x (1 – x) occur at
x=\(\frac{1}{2}\)
i.e.,(15(1−x))max=\(\frac{1}{4}\)
⇒K=15C10×\(\frac{1}{4}\)
⇒8K=2(15C10)=6006
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Concepts Used:
Inequalities
In mathematics, inequality is a relationship that compares two numbers or other mathematical expressions in a non-equal fashion. It is most commonly used to compare the size of two numbers on a number line.
Specifically, a linear inequality is a mathematical inequality that integrates a linear function. One of the symbols of inequality is observed in a linear inequality: In graph form, it represents data that is not equal.
Some of the linear inequality symbols are given below:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
- = equal to
Inequalities can be demonstrated as questions that are solved using alike procedures to equations, or as statements of fact in the form of theorems. It is used to contrast numbers and find the range or ranges of values that pleases a variable's criteria.