Assertion Reasoning Quiz

Assertion: The slope of the line joining \((2,-1)\) and \((6,7)\) is \(2\).
Reason: For points \((x_1,y_1)\) and \((x_2,y_2)\), the slope is \(\dfrac{y_2-y_1}{x_2-x_1}\), and here \(\dfrac{7-(-1)}{6-2}=2\).

Assertion (A): The slope of a line making an angle of \( 90^\circ \) with the positive direction of the x-axis is not defined.

Reason (R): The slope of a line with inclination \( \theta \) is given by \( m = \tan \theta \), and \( \tan 90^\circ \) is not defined.

Assertion (A): The distance between the parallel lines \( x + y - 2 = 0 \) and \( x + y - 4 = 0 \) is \( 2 \) units.

Reason (R): The distance between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by \( d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \).

Assertion (A): The lines \( x + y = 1 \), \( x - y = 1 \), and \( 2x = 2 \) are concurrent.

Reason (R): Three lines are said to be concurrent if they pass through a common point.

Assertion (A): The line passing through \( (2, 3) \) and \( (4, 5) \) is perpendicular to the line passing through \( (3, 4) \) and \( (5, 2) \).

Reason (R): Two non-vertical lines are perpendicular to each other if and only if the product of their slopes is \( 1 \) (i.e., \( m_1 m_2 = 1 \)).

Assertion: A line having inclination \(120^\circ\) has a positive slope.
Reason: The slope is \(m=\tan120^\circ=-\sqrt{3}\).