Pair of Linear Equations in Two Variables
Section A: Multiple Choice Questions
1. The solution of the equations \(x - y = 2\) and \(x + y = 4\) is:
Solution Adding the equations gives \(2x = 6 \Rightarrow x = 3\). Substituting into the second equation gives \(3 + y = 4 \Rightarrow y = 1\).
2. If the lines \(3x + 2ky - 2 = 0\) and \(2x + 5y + 1 = 0\) are parallel, then what is the value of k?
Solution For parallel lines, \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\).
\(\frac{3}{2} = \frac{2k}{5} \Rightarrow 4k = 15 \Rightarrow k = \frac{15}{4}\).
\(\frac{3}{2} = \frac{2k}{5} \Rightarrow 4k = 15 \Rightarrow k = \frac{15}{4}\).
3. The pair of equations \(3x - 5y = 7\) and \(-6x + 10y = 7\) have
Solution \(\frac{a_1}{a_2} = \frac{3}{-6} = -\frac{1}{2}\); \(\frac{b_1}{b_2} = \frac{-5}{10} = -\frac{1}{2}\); \(\frac{c_1}{c_2} = \frac{7}{7} = 1\).
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), there is no solution.
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), there is no solution.
4. If a pair of linear equations is consistent, then the lines will be
Solution Consistent means having at least one solution. Intersecting lines have 1 solution, coincident lines have infinite. Both are consistent.
5. The pair of equations \(5x + 7y = 5\) and \(2x - 3y = 7\) are
Solution \(\frac{a_1}{a_2} = \frac{5}{2}\), \(\frac{b_1}{b_2} = \frac{7}{-3}\). Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), they are consistent (unique solution).
6. Which of the following pair of equations are dependent?
Solution Dependent equations represent the same line (coincident).
For (a): \(\frac{2}{4} = \frac{3}{6} = \frac{9}{18} = \frac{1}{2}\). All ratios equal.
For (a): \(\frac{2}{4} = \frac{3}{6} = \frac{9}{18} = \frac{1}{2}\). All ratios equal.
7. The value of c for which the pair \(cx - y = 2\) and \(6x - 2y = 3\) will have infinitely many solutions is:
Solution For infinite solutions: \(\frac{c}{6} = \frac{-1}{-2} = \frac{2}{3}\).
\(\frac{-1}{-2} = \frac{1}{2}\) but \(\frac{2}{3} \neq \frac{1}{2}\). The condition \(\frac{b_1}{b_2} = \frac{c_1}{c_2}\) is not met. So, no value of c works.
\(\frac{-1}{-2} = \frac{1}{2}\) but \(\frac{2}{3} \neq \frac{1}{2}\). The condition \(\frac{b_1}{b_2} = \frac{c_1}{c_2}\) is not met. So, no value of c works.
8. A fraction becomes \(1/3\) when 1 is subtracted from the numerator and it becomes \(1/4\) when 8 is added to the denominator. The fraction is:
Solution Let fraction be \(x/y\).
1) \(\frac{x-1}{y} = \frac{1}{3} \Rightarrow 3x - y = 3\).
2) \(\frac{x}{y+8} = \frac{1}{4} \Rightarrow 4x - y = 8\).
Solving gives \(x=5, y=12\).
1) \(\frac{x-1}{y} = \frac{1}{3} \Rightarrow 3x - y = 3\).
2) \(\frac{x}{y+8} = \frac{1}{4} \Rightarrow 4x - y = 8\).
Solving gives \(x=5, y=12\).
9. The graph of equations \(x + 2y - 4 = 0\) and \(2x + 4y - 12 = 0\) represents:
Solution \(\frac{a_1}{a_2} = \frac{1}{2}\), \(\frac{b_1}{b_2} = \frac{2}{4} = \frac{1}{2}\), \(\frac{c_1}{c_2} = \frac{-4}{-12} = \frac{1}{3}\).
Since \(a_1/a_2 = b_1/b_2 \neq c_1/c_2\), lines are parallel.
Since \(a_1/a_2 = b_1/b_2 \neq c_1/c_2\), lines are parallel.
10. Which of the following is a condition for unique solution of a pair of linear equations?
Solution The condition for a unique solution (intersecting lines) is \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).
Section B: Assertion-Reason Questions
Directions:
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
11. Assertion (A): A pair of linear equations has no solution if it is represented by intersecting lines.
Reason (R): Intersecting lines represent a pair of equations having a unique solution.
Reason (R): Intersecting lines represent a pair of equations having a unique solution.
Solution Intersecting lines always have a unique solution. Assertion says "no solution", which is false. Reason is true.
12. Assertion (A): If the pair of equations has the same solution, then it is called consistent.
Reason (R): Consistent system has at least one solution.
Reason (R): Consistent system has at least one solution.
Solution A pair with the same solution (coincident) is consistent (infinitely many solutions). Reason defines consistent correctly. Both true and R explains A.
13. Assertion (A): The pair of equations \(2x + 3y = 6\) and \(4x + 6y = 10\) are consistent.
Reason (R): They represent parallel lines.
Reason (R): They represent parallel lines.
Solution \(\frac{2}{4} = \frac{3}{6} \neq \frac{6}{10}\). They are parallel and inconsistent. Assertion is False. Reason is True.
14. Assertion (A): The graph of a pair of linear equations is a pair of parallel lines if there is no solution.
Reason (R): Parallel lines never intersect and hence are inconsistent.
Reason (R): Parallel lines never intersect and hence are inconsistent.
Solution Both A and R are true and R explains why "no solution" corresponds to "parallel lines".
15. Assertion (A): If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), the system is inconsistent.
Reason (R): Such a system represents parallel lines.
Reason (R): Such a system represents parallel lines.
Solution The condition describes parallel lines, which never meet (inconsistent). Both true and R explains A.
16. Assertion (A): The equations \(x + 2y = 3\) and \(2x + 4y = 6\) represent intersecting lines.
Reason (R): Their slopes are same.
Reason (R): Their slopes are same.
Solution \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6}\). These are coincident lines, not intersecting. A is False. Reason is True (slopes are same for coincident/parallel lines).
17. Assertion (A): The pair \(2x + 3y = 8\) and \(4x + 6y = 10\) has a unique solution.
Reason (R): Lines with different slopes intersect at one point.
Reason (R): Lines with different slopes intersect at one point.
Solution \(\frac{2}{4} = \frac{3}{6} \neq \frac{8}{10}\). Parallel (no solution). Assertion is False. Reason is True.
18. Assertion (A): The system \(x + y = 5\) and \(2x + 2y = 10\) has infinitely many solutions.
Reason (R): Both equations represent the same line.
Reason (R): Both equations represent the same line.
Solution \(\frac{1}{2} = \frac{1}{2} = \frac{5}{10}\). Coincident lines -> infinite solutions. Both true and R explains A.
19. Assertion (A): If graphical representation results in two parallel lines, system is inconsistent.
Reason (R): In an inconsistent system, there is no point common to both lines.
Reason (R): In an inconsistent system, there is no point common to both lines.
Solution Both are true definitions. No common point = inconsistent.
20. Assertion (A): A system of equations with infinitely many solutions must be inconsistent.
Reason (R): Coincident lines have no unique solution.
Reason (R): Coincident lines have no unique solution.
Solution A system with infinite solutions is **consistent** (dependent). Assertion is False. Reason is true (they have infinite, not unique).