SATHEE: Binomial Theorem 1 Question 22

Binomial Theorem 1 Question 22

24.

The expression ${[x + {(x^{3} - 1)}^{1 / 2}]}^{5} + {[x - {(x^{3} - 1)}^{1 / 2}]}^{5}$ is a polynomial of degree

(1992, 2M)

(a) $5$

(b) $6$

(d) $8$

Show Answer

Answer:

Correct Answer: 24. (c)

Solution:

We know that,

$\begin{matrix} (a + b)^{5} + (a - b)^{5} =^{5} C_{0} a^{5} +^{5} C_{1} a^{4} b +^{5} C_{2} a^{3} b^{2} \\ +^{5} C_{3} a^{2} b^{3} +^{5} C_{4} a b^{4} +^{5} C_{5} b^{5} +^{5} C_{0} a -^{5} C_{1} a^{4} b \\ +^{5} C_{2} a^{3} b^{2} -^{5} C_{3} a^{2} b^{3} +^{5} C_{4} a b^{4} -^{5} C_{5} b^{5} \\ = 2 [a^{5} + 10 a^{3} b^{2} + 5 a b^{4}] \\ ∴ {[x + {(x^{3} - 1)}^{1 / 2}]}^{5} + {[x - {(x^{3} - 1)}^{1 / 2}]}^{5} \\ = 2 [x^{5} + 10 x^{3} (x^{3} - 1) + 5 x {(x^{3} - 1)}^{2}] \end{matrix}$

Therefore, the given expression is a polynomial of degree 7.

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Binomial Theorem 1 Question 22

24.

Answer:

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