(Each question 3 marks)

Question 13 Marks

If $P(n)$ is the statement "$n^2 + n$ is even", and if $P(r)$ is true, then $P(r + 1)$ is true.

Answer

$P(n): n^2+n$ is even Given, $P(r)$ is true $\Rightarrow r^2+r$ is even $\Rightarrow r^2+r=2 \lambda \ldots(1)$ Now, $(r+1)^2+(r+1)=r^2+1+2 r+r$
$+1=(r+1)^2+2 r+2=2 \lambda+2 r+2\left[\right.$ Using equation (1)] $=2(\lambda+r+1)=2 \lambda \Rightarrow(r+1)^2+(r+1)$ is even $\Rightarrow P(r$ +1 ) is true

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Question 23 Marks

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

Answer

$P(n): n^2-n+41$ is prime $P(1): 1-1+41$ is prime
$\Rightarrow P(1): 41$ is prime
$\therefore P(1)$ is true. $P(2): 2^2-2+41$ is prime
$\Rightarrow P(2)$ : 43 is prime
$\therefore P(2)$ is true. $P(3): 3^2-3+41$ is prime
$\Rightarrow P(3): 47$ is prime
$\therefore P(3)$ is true. $P(41):(41)^2-41+41$ is prime $P(41):(41)^2$ is prime
$\Rightarrow P(41)$ is not true.

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Question 33 Marks

If $P(n)$ is the statement " $2^n \geq 3 n$ " and if $P(r)$ is true, prove that $P(r+1)$ is true.

Answer

$P(n): 2^n \geq 3 n$ Given that $P(r)$ is true $\Rightarrow 2^r \geq 3 r$ Multiplying both sides by $2,2.2^r \geq 2.3 r 2^{r+1} \geq 6 r 2^{r+1} \geq 3 r+3 r 2^{r+1} \geq 3$ $+3 r$,
[Since $3 r \geq 3 \Rightarrow 3 r+3 r \geq 3+3 r] 2^{r+1} \geq 3 r(r+1) \Rightarrow P(r+1)$ is true.

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MATHS (Each question 3 marks)

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