Consider the statement - -P n - n2-n-41 is prime- Then which of the following is true -A- Both P 3 and P 5 are false-B- P 3 is false but P 5 is true-C- P 5 is false but P 3 is true-D- Both P 3 and P 5 are true-

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Direct Method of Validation

Consider the ...

Question

Consider the statement : '' $P (n) : n^{2} - n + 41$ is prime.'' Then which of the following is true ?

Both

P (3)

and

P (5)

are false.

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P (3)

is false but

P (5)

is true.

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P (5)

is false but

P (3)

is true.

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Both

P (3)

and

P (5)

are true.

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Solution

The correct option is D Both $P (3)$ and $P (5)$ are true.
$P (x) = x^{2} - x + 41$ , where $x$ is a prime number.
$P (3) = 3^{2} - 3 + 41 = 47$
$\Rightarrow P (3)$ is a prime
$P (5) = 5^{2} - 5 + 41 = 61$
$\Rightarrow P (5)$ is also prime
Since, both $P (3)$ and $P (5)$ are prime therefore, both $P (3)$ and $P (5)$ are true.

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P (n) :

2 n^{2} + 4 n - 8

is negative. "
Then which of the following options are correct?

Q. Assertion : Two events A and B are such that

P (A) = \frac{1}{3}

and

P (B) = \frac{2}{3}

then

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Reason : If

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Q. If P (n) is the statement "n² − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

Q. Statement

1

:
If

p

is a prime number

(p \neq 2)

, then

[(2 + \sqrt{5})^{p}] - 2^{p + 1}

is always divisible by

p

(where

[.]

denotes the greatest integer function).

Statement

2

:
If

n

is a prime, then

^{n} C_{1},^{n} C_{2},^{n} C_{3}, \dots,^{n} C_{n - 1}

must be divisible by

n

If p and q are simple statements, $p \Leftrightarrow\sim q$ is true when

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Direct Method of Validation

Standard XII Mathematics

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Consider the statement : ''P(n):n2−n+41 is prime.'' Then which of the following is true ?

The correct option is D Both P(3) and P(5) are true.P(x)=x2−x+41, where x is a prime number. P(3)=32−3+41=47 ⇒P(3) is a prime P(5)=52−5+41=61 ⇒P(5) is also prime Since, both P(3) and P(5) are prime therefore, both P(3) and P(5) are true.

Consider the statement : '' $P (n) : n^{2} - n + 41$ is prime.'' Then which of the following is true ?