Class 11

Math

JEE Main Questions

Binomial Theorem

Find the coefficient of $x_{5}$ in the expansioin of the product $(1+2x)_{6}(1−x)_{7}˙$

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Simplify: $x_{5}+10x_{4}a+40x_{3}a_{2}+80x_{2}a_{3}+80xa_{4}+32a_{5}˙$

Using binomial theorem, evaluate : $(96)_{3}$

Find the condition for which the formula $(a+b)_{m}a_{m}+ma_{m−1}b+1×2m(m−1) a_{m−2}b_{2}+$ holds.

There are two bags each of which contains $n$ balls. A man has to select an equal number of balls from both the bags. Prove that the number of ways in which a man can choose at least one ball from each bag $is_{2n}C_{n}−1.$

If $a_{1},a_{2},a_{3},a_{4}$ be the coefficient of four consecutive terms in the expansion of $(1+x)_{n},$ then prove that: $a_{1}+a_{2}a_{1} +a_{3}+a_{4}a_{3} =a_{2}+a_{3}2a_{2} ˙$

Show that the middle term in the expansion of $(1+x)_{2n}$is $n!1.3.5.2n−1˙ ˙ $$2nx_{n}2nx_{n}$, where n is a positive integer.

Find the value of \displaystyle{\left\lbrace{3}^{{{2003}}}/{28}\right\rbrace},{w}{h}{e}{r}{e}{\left\lbrace{\dot}\right\rbrace} denotes the fractional part.

If $(1+x)_{n}=r=0∑n C_{r}x_{r},$ then prove that $C_{1}+2c_{2}+3C_{1}++nC_{n}=n2_{n−1}˙$ .