Assertion (A) : In the expansion of $(x + x^{- 2})^{n}$ the coefficient of eighth term and nineteenth term are equal then $n = 25$ .

Reason (R) : Middle term in the expansion of $(x + a)^{n}$ has greatest coefficient where n is an even integer.

Both A and R are individually true and R is the correct explanation of A.

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Both A and R are individually true and R is not correct explanation of A.

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A is true but R is false

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A is false but R is true

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Solution

The correct option is C Both A and R are individually true and R is not correct explanation of A.
It is given that coefficients of $T_{8} = T_{19}$
$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$
Applying to the above question we get
$^{n} C_{7} =^{n} C_{18}$
Now $^{n} C_{r} =^{n} C_{n - r}$
Using the formula we get
$n - 7 = 18$
$n = 25$
It is also true that the middle term has the greatest coefficient in the expansion of $(x + a)^{n}$ since in pascal's triangle the middle term has the largest value. However this is not the correct explanation of the assertion Hence the correct answer is option B.

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Similar questions

DIRECTIONS (Qs. 52 AND 53): The following item consist of two statements: one labelled as the Assertion a) and the other as Reason (R). You are to examine these two statements carefully and select the answers to these items using the codes given below:

a) Both A and R are individually true but R is the correct explanation of A

b) Both A and R are individually true but R is not the correct explanation of A

c) A is true but R is false

d) Both A and R are false

53. Assertion a): Saluva Narasimha put an end to the old dynasty and assumed the royal title.

Reason

(R): He wanted to save the kingdom from further degeneration and disintegration. [2003]

Q. Assertion (A)
8th term from the end of the AP 18, 14, 10, 6, ..., −30 is −2.

Reason (R)
The nth term from the end of an AP with last term l and common difference d is l −(n − 1)d.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Q. Assertion (A)
If the first term of an AP is 6, last term is 62 and the sum of the given terms is 510. Then, there are 14 terms in the given AP.

Reason (R)
If a is the first term, l is the last term and n is the number of terms of an AP, then

S_{n} = \frac{n}{2} (a + 1) .

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Q. Assertion (A)
The polynomial p(x) = x³ + x has one real zero.

Reason (R)
A polynomial of nth degree has at most n zeros.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Q. Assertion (A)
Sum of first n positive integers is given by

S_{n} = \frac{1}{2} n (n + 1) .

Reason (R)
In an AP with first term a and common difference d, the sum of nth terms is given by

S_{n} = \frac{n}{2} [2 a + (n - 1) d] .

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Assertion (A) : In the expansion of (x+x−2)n the coefficient of eighth term and nineteenth term are equal then n=25.Reason (R) : Middle term in the expansion of (x+a)n has greatest coefficient where n is an even integer.

Assertion (A) : In the expansion of $(x + x^{- 2})^{n}$ the coefficient of eighth term and nineteenth term are equal then $n = 25$ .

Reason (R) : Middle term in the expansion of $(x + a)^{n}$ has greatest coefficient where n is an even integer.