10 Question MCQ (including Assertion)

A graph represents the distance ( \(y\)-axis, in km ) versus time ( \(x\)-axis, in hours) for two cyclists, \(A\) and \(B\). The path of Cyclist \(A\) is a line starting from the origin \((0,0)\). The path of Cyclist \(B\) is a line starting from \((0,10)\), indicating a 10 km head start. The line for Cyclist \(A\) is steeper than the line for Cyclist B, and they intersect at a point. What can be concluded at the point of intersection?

Assertion (A): If the pair of lines are coincident, then we say that pair of lines is consistent and it has a unique solution.
Reason (R): If the pair of lines are parallel, then the pairs has no solution and is called inconsistent pair of equations.

A boat's speed in still water is \(x \mathrm{~km} / \mathrm{h}\), and the speed of the stream is \(y \mathrm{~km} / \mathrm{h}\). Its speed upstream is given by \(x-y=5\) and downstream by \(x+y=15\). What does the value of \(x\) represent?

A father's age is three times the sum of the ages of his two children. Let the father's age be \(x\) and the sum of his children's ages be \(y\). The equation is \(x=3 y\). If one child is 5 and the other is 7 , what is the father's age?

8. The perimeter of a rectangular garden is 40 m . The length \((l)\) is 4 m more than the width ( \(w\) ). Which pair of equations models this situation?

Assertion (A): The value of \(k\) for which the system of linear equations \(3 x-4 y=7\) and \(6 x-8 y=k\) have infinite number of solution is 14.
Reason (R): The graph of linear equations \(a_1 x+b_1 y+c_1=0\) and \(a_2 x+b_2 y+c_2=0\) gives a pair of intersecting lines if \(\frac{a 1}{a 2} \neq \frac{b 1}{b 2}\).

Assertion (A): Graphically, two coincident lines represent a pair of inconsistent equations.
Reason (R): Coincident lines have infinite points of intersection.

If a pair of linear equations is consistent, then the lines will be:

A mobile company offers two plans. Plan A is represented by the equation \(y=0.5 x+\) 20 and Plan B by \(y=0.7 x+10\), where \(x\) is the number of minutes used and \(y\) is the total cost. The graph of these equations results in two intersecting lines. What does the point of intersection represent?