Assertion -If -z- - -2 - 1 -then -z2 - 2z cos B - - 1- Reason- -z1 - z2- - -z1- - -z2- -cos B- - 1-

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Byju's Answer

Standard XII

Physics

Antiderivative

Assertion :If...

Question

Assertion :If $| z | < \sqrt{2} - 1$ ,then $| z^{2} + 2 z c o s B | < 1$ . Reason: $| z_{1} + z_{2} | < | z_{1} | + | z_{2} |$ & $| c o s B | < 1$ .

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 (R) is false.

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(A) is false (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).
$| z^{2} + 2 z c o s B | < | z |^{2} + | 2 z c o s B |$
(By reason)
$= | z |^{2} + 2 | z | | c o s B | < (\sqrt{2} - 1)^{2} + 2 (\sqrt{2} - 1) . 1$
$= 3 - 2 \sqrt{2} + 2 \sqrt{2} - 2 = 1$
$∴ | z^{2} + 2 z c o s B | < 1$ (Assertion (A) is true)

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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]

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)

\sqrt{8}

is an irrational number.
Reason (R)
If m is a natural number which is not a perfect square, then

\sqrt{m}

is irrational.

(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 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 probability of winning a game is

\frac{8}{15}

, then the probability of losing the game is

\frac{7}{15}

.

Reason (R)
For any event E, we have P(E) + P(not E) = 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): Some algae are unicellular.
Reason(R): They belong to the kingdom Monera.

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Assertion :If |z|<√2−1 ,then |z2+2zcosB|<1. Reason: |z1+z2|<|z1|+|z2| & |cosB|<1.

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).|z2+2zcosB|<|z|2+|2zcosB|(By reason)=|z|2+2|z||cosB|<(√2−1)2+2(√2−1).1=3−2√2+2√2−2=1∴|z2+2zcosB|<1 (Assertion (A) is true)

Assertion :If $| z | < \sqrt{2} - 1$ ,then $| z^{2} + 2 z c o s B | < 1$ . Reason: $| z_{1} + z_{2} | < | z_{1} | + | z_{2} |$ & $| c o s B | < 1$ .