(C) Step $1$: Analyze $S_1$. For $n=1$, $1^2+1+2 = 4$, which is divisible by $4$. Thus, $S_1$ is True.Step $2$: Analyze $S_2$. For $x=1$, $1-17 = -16

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(C) Step $1$: Analyze $S_1$. For $n=1$, $1^2+1+2 = 4$, which is divisible by $4$. Thus, $S_1$ is True.Step $2$: Analyze $S_2$. For $x=1$, $1-17 = -16 Step $3$: Analyze $S_3$. The equation $x^2+3x-10=0$ factors to $(x+5)(x-2)=0$, giving $x=-5$ or $x=2$. This is not true for all $n \in N$. Thus, $S_3$ is False.Step $4$: Analyze $S_4$. For all $n \in N$, $n \ge 1$, so $n^2 \ge 1$. Thus, $S_4$ is True.Step $5$: Only $S_3$ is False. Therefore, the correct option is $C$.

Class 11 Mathematics Mathematical Reasoning Mathematical logic

Medium

Which of the following statements is/are False?
$S_1: \exists n \in N$, such that $n^2 + n + 2$ is divisible by $4$.
$S_2: \exists x \in N$, such that $x - 17 < 20$.
$S_3: \forall n \in N, x^2 + 3x - 10 = 0$.
$S_4: \forall n \in N, n^2 \ge 1$.

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A
$S_1$ and $S_2$.
B
$S_1$ and $S_3$.
C
Only $S_3$.
D
$S_2$ and $S_4$.

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Which of the following statements is/are False?$S_1: \exists n \in N$, such that $n^2 + n + 2$ is divisible by $4$.$S_2: \exists x \in N$, such that $x - 17 < 20$.$S_3: \forall n \in N, x^2 + 3x - 10 = 0$.$S_4: \forall n \in N, n^2 \ge 1$.

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Write the negation of the following statement:$q:$ All cats scratch.

In a switching circuit, if the combination $(S_1 \land S_2)$ is connected in parallel to the combination $(S'_1 \land S'_2)$, then the room is lit only when ...

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Among the statements:$(S1) \quad (( p \vee q )$ $\Rightarrow r ) \Leftrightarrow ( p$ $\Rightarrow r )$$(S2) \quad (( p \vee q )$ $\Rightarrow r ) \Leftrightarrow (( p$ $\Rightarrow r ) \vee ( q$ $\Rightarrow r ))$

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Which of the following statements is/are False?
$S_1: \exists n \in N$, such that $n^2 + n + 2$ is divisible by $4$.
$S_2: \exists x \in N$, such that $x - 17 < 20$.
$S_3: \forall n \in N, x^2 + 3x - 10 = 0$.
$S_4: \forall n \in N, n^2 \ge 1$.

Write the negation of the following statement:
$q:$ All cats scratch.

By giving a counterexample,show that the following statement is not true.
$p:$ If all the angles of a triangle are equal,then the triangle is an obtuse-angled triangle.

Among the statements:
$(S1) \quad (( p \vee q )$ $\Rightarrow r ) \Leftrightarrow ( p$ $\Rightarrow r )$
$(S2) \quad (( p \vee q )$ $\Rightarrow r ) \Leftrightarrow (( p$ $\Rightarrow r ) \vee ( q$ $\Rightarrow r ))$