Assertion (A) : The sum of the last ten coeffcients in the expansion of $(1 + x)^{19}$ , when expanded in ascending powers of $x$ is $2^{18}$
Reason (R): $n C_{r} =^{n} C_{n - r} (\begin{matrix} n r > - 2 \end{matrix}) \forall n \in N$

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 A Both A and R are individually true and R is the correct explanation of A.
$(1 + x)^{(} 19) = 1 +^{19} C_{1} x^{1} +^{19} C_{2} x^{2} . . . +^{19} C_{19} x^{19}$
Substituting $x = 1$ we get
$2^{19} = (1 +^{19} C_{19}) + (^{19} C_{18} +^{19} C_{1}) + . . . + (^{19} C_{10} +^{19} C_{9})$
$2^{19} = 2 (^{19} C_{19} +^{19} C_{18} + . . .^{19} C_{10})$
$2^{18} =^{19} C_{19} +^{19} C_{18} + . . .^{19} C_{10}$
Hence sum of the coefficients of the last $10$ terms is $2^{18}$

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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.

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] .

(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 (A) : The sum of the last ten coeffcients in the expansion of (1+x)19, when expanded in ascending powers of x is 218Reason (R): nCr=nCn−r(nr>−2)∀n∈N

Assertion (A) : The sum of the last ten coeffcients in the expansion of $(1 + x)^{19}$ , when expanded in ascending powers of $x$ is $2^{18}$
Reason (R): $n C_{r} =^{n} C_{n - r} (\begin{matrix} n r > - 2 \end{matrix}) \forall n \in N$