Assertion :The number of terms in the expansion of $(x_{1} + x_{2} + . . . . . . x_{10})^{5}$ are $^{14} C_{5}$ Reason: The number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2,.....,n items is equal to $^{n + r - 1} C_{n}$

Both (A) & (R) are individually true & (R) is correct explanation of (A),

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Both (A) & (R) are individually true but (R) is not the correct (proper) 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 A Both (A) & (R) are individually true & (R) is correct explanation of (A),
We know the number of terms in the expansion of $(x_{1} + x_{2} + . . . . . x_{r})^{n}$ are $^{n + r - 1} C_{r - 1} =^{n - + r - 1} C_{n}$
Now putting $n = 5, r = 10$ $∴^{n + r - 1} C_{n} =^{5 + 10 - 1} C_{5} =^{14} C_{5}$
Hence Assertion is followed by Reason.

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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]

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

52. Assertion a): Shah Alam II spent the initial years as an emperor far away from his capital.

Reason

(R): There was always a lurking danger of foreign invasion from the north-west frontier. [2003]

Q. Assertion(A): Some algae are unicellular.
Reason(R): They belong to the kingdom Monera.

Directions:- The following item consist of two statements, one labelled as ‘Assertion A’ and the other labelled as ‘Reason R’. You are to examine these two statements carefully and decide if the Assertion A and the Reason R are individually true and if so, whether the Reason is a correct explanation of the Assertion. Select your answers to these items using the codes given below and mark your answer sheet accordingly.

Assertion (A): “DNA Finger printing” has become a powerful tool to establish paternity and identity of criminals in rape and assault cases.

Reason (R): Trace evidences such as hairs, saliva and semen are adequate for DNA analysis.

Q. Assertion (A)

\frac{1242}{49}

is a non-terminating, repeating decimal.
Reason (R)
The rational number

\frac{p}{q}

is a terminating decimal if q = (2m × 5n) for some whole numbers m and n.

(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.

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Assertion :The number of terms in the expansion of (x1+x2+......x10)5 are 14C5 Reason: The number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2,.....,n items is equal to n+r−1Cn

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),We know the number of terms in the expansion of (x1+x2+.....xr)n are n+r−1Cr−1=n−+r−1CnNow putting n=5,r=10 ∴n+r−1Cn=5+10−1C5=14C5Hence Assertion is followed by Reason.

Assertion :The number of terms in the expansion of $(x_{1} + x_{2} + . . . . . . x_{10})^{5}$ are $^{14} C_{5}$ Reason: The number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2,.....,n items is equal to $^{n + r - 1} C_{n}$