Question

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 \(x-17 < 20\).
\(S_3:\forall \,n\in N,\,x^2+3x-10 = 0\).
\(S_4:\forall \,n\in N,\,n^2\geq 1\).

• A. \(S_1\) and \(S_2\).
• B. \(S_1\) and \(S_3\).
• C. Only \(S_3\).
• D. \(S_2\) and \(S_4\).

Hint:

Find the intersection point, then use slope-intercept form with the given intercept.

Answer 1

The Correct Option is C

Step 1: Find the intersection:
From \(2x - y = 2\), \(y = 2x - 2\). Put it in \(x + 2y + 6 = 0\): \(x + 4x - 4 + 6 = 0\), so \(x = -\frac{2}{5}\) and \(y = -\frac{14}{5}\).

Step 2: Line with y intercept 5:
Use \(y = mx + 5\) and require it to pass through \(\left(-\frac{2}{5}, -\frac{14}{5}\right)\):
\[ -\frac{14}{5} = -\frac{2m}{5} + 5 \Rightarrow -14 = -2m + 25 \Rightarrow m = \frac{39}{2} \]

Step 3: Equation:
\[ y = \frac{39}{2}x + 5 \Rightarrow 2y = 39x + 10 \Rightarrow 39x - 2y + 10 = 0 \]
Check with option (A): it has a y intercept of 5 but passes the wrong sign of the slope (it gives \(y = -\frac{39}{2}x + 5\)), so it does not pass through the point.

Final Answer:
The required line is \(39x - 2y + 10 = 0\), option (B). \[ \boxed{39x - 2y + 10 = 0} \]