Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables

Assertion and Reason Questions

1. Assertion (A): The pair of equations $2x + 3y = 5$ and $4x + 6y = 10$ represents the same line.
   Reason (R): Two equations are equivalent if one can be obtained by multiplying the other by a non-zero constant.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: A) Both A and R are true, and R is the correct explanation of A.
   • Explanation: The second equation is obtained by multiplying the first equation by 2, thus they represent the same line.

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2. Assertion (A): The equations $x + y = 3$ and $2x + 2y = 6$ have infinitely many solutions.
   Reason (R): The two equations represent parallel lines.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: C) A is true, but R is false.
   • Explanation: The second equation is a multiple of the first, meaning they represent the same line, not parallel lines.

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3. Assertion (A): The graph of the equations $y = 2x + 1$ and $y = -x + 4$ intersects at one point.
   Reason (R): Two linear equations can have at most one solution.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: A) Both A and R are true, and R is the correct explanation of A.
   • Explanation: The two lines will intersect at one point, indicating they have one unique solution.

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4. Assertion (A): The equations $3x + 4y = 12$ and $6x + 8y = 24$ are inconsistent.
   Reason (R): Inconsistent equations have no common solution.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: D) A is false, but R is true.
   • Explanation: The equations are consistent as the second is a multiple of the first; they represent the same line.

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5. Assertion (A): If a pair of linear equations has infinitely many solutions, then the equations are dependent.
   Reason (R): Dependent equations are those that represent the same line.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: A) Both A and R are true, and R is the correct explanation of A.
   • Explanation: Dependent equations yield infinitely many solutions as they represent the same line.

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6. Assertion (A): The pair of equations $x + y = 2$ and $2x + 3y = 5$ has a unique solution.
   Reason (R): The slopes of the lines represented by these equations are different.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: A) Both A and R are true, and R is the correct explanation of A.
   • Explanation: Different slopes indicate the lines intersect at exactly one point.

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7. Assertion (A): The equation $2x - 3y = 6$ can be expressed in the form $y = mx + c$ .
   Reason (R): Any linear equation can be rewritten in slope-intercept form.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: A) Both A and R are true, and R is the correct explanation of A.
   • Explanation: The equation can be rearranged to $y = \frac{2}{3}x - 2$ .

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8. Assertion (A): The equations $x - 2y = 4$ and $2x - 4y = 8$ are inconsistent.
   Reason (R): Inconsistent equations are those that have parallel lines.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: D) A is false, but R is true.
   • Explanation: The equations are dependent since the second equation is a multiple of the first, indicating the same line.

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9. Assertion (A): If the coefficients of a linear equation in two variables are all zero, then it represents a unique solution.
   Reason (R): A unique solution exists only when the equation is non-degenerate.
   • A) Both A and R are true, and R is the correct explanation of A.
   • B) Both A and R are true, but R is not the correct explanation of A.
   • C) A is true, but R is false.
   • D) A is false, but R is true.
   • Answer: D) A is false, but R is true.
   • Explanation: The equation represents all points (infinitely many solutions), not a unique solution.

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10. Assertion (A): The pair of equations $x + y = 1$ and $x + y = 3$ has no solution.
    Reason (R): The lines represented by these equations are parallel.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: The lines are parallel since they have the same slope but different intercepts.

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11. Assertion (A): The solution to the equations $2x + 3y = 6$ and $4x + 6y = 12$ is $(0, 2)$ .
    Reason (R): The equations are equivalent.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: D) A is false, but R is true.
• Explanation: The equations are equivalent, representing the same line, thus having infinitely many solutions, not a unique point.

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12. Assertion (A): A consistent pair of linear equations can have either one solution or infinitely many solutions.
    Reason (R): Consistent equations are those that intersect at one point or coincide completely.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: Consistent equations either intersect at a unique point or represent the same line.

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13. Assertion (A): The equations $5x + 2y = 10$ and $5x - 2y = 2$ can be solved using substitution.
    Reason (R): The substitution method can be applied to any pair of linear equations.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: Any pair of linear equations can be solved using substitution, and these equations are no exception.

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14. Assertion (A): If two linear equations have the same slope but different intercepts, they will have no solution.
    Reason (R): Such equations represent parallel lines.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: Parallel lines never intersect, leading to no solutions.

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15. Assertion (A): The point of intersection of two linear equations is the solution to the system of equations.
    Reason (R): A point of intersection signifies that the equations share the same $x$ and $y$ values.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: The intersection point provides the values of $x$ and $y$ that satisfy both equations.

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16. Assertion (A): The graphical representation of the equations $x + y = 4$ and $y = 2x - 4$ will intersect at one point.
    Reason (R): The equations have different slopes.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: Different slopes imply the lines will intersect at one point.

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17. Assertion (A): The solution to the equations $4x + y = 8$ and $x - y = 2$ can be found using elimination.
    Reason (R): The elimination method is useful when the coefficients of one variable can be made equal.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: The elimination method can be applied as the equations can be manipulated to eliminate one variable.

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18. Assertion (A): The equations $x + 2y = 6$ and $2x + 4y = 12$ are independent.
    Reason (R): Independent equations represent lines that intersect at one point.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: D) A is false, but R is true.
• Explanation: The equations are dependent, representing the same line.

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19. Assertion (A): The system of equations $y = 3x + 1$ and $y = 3x - 2$ has no solution.
    Reason (R): The equations represent parallel lines.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: The equations have the same slope and different intercepts, indicating they are parallel.

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20. Assertion (A): A pair of linear equations can have exactly two solutions.
    Reason (R): Linear equations represent straight lines.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: D) A is false, but R is true.
• Explanation: Linear equations can have at most one solution or infinitely many solutions; they cannot have exactly two solutions.

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21. Assertion (A): The equations $7x + 3y = 21$ and $3x + 7y = 21$ can be solved simultaneously.
    Reason (R): There is no specific condition that prevents solving a pair of equations.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: A) Both A and R are true, and R is the correct explanation of A.
• Explanation: Both equations can be solved simultaneously using methods like substitution or elimination.

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22. Assertion (A): The pair of equations $x + y = 0$ and $x - y = 0$ have a unique solution.
    Reason (R): The slopes of the lines represented by these equations are the same.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: C) A is true, but R is false.
• Explanation: The lines intersect at one point (the origin), but their slopes are different.

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23. Assertion (A): The solution to the pair of equations $4x + 5y = 20$ and $8x + 10y = 40$ is $(2, 0)$ .
    Reason (R): The two equations are equivalent.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: D) A is false, but R is true.
• Explanation: The equations are equivalent and represent the same line, hence have infinitely many solutions, not just one.

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24. Assertion (A): The equations $x + y = 5$ and $x + y = 3$ intersect at a unique point.
    Reason (R): The two equations have different slopes.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: D) A is false, but R is true.
• Explanation: The equations represent parallel lines and will not intersect at any point.

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25. Assertion (A): The point $(1, 2)$ is a solution to the equations $2x + 3y = 8$ and $x - y = -1$ .
    Reason (R): A point is a solution if it satisfies both equations.

• A) Both A and R are true, and R is the correct explanation of A.
• B) Both A and R are true, but R is not the correct explanation of A.
• C) A is true, but R is false.
• D) A is false, but R is true.
• Answer: B) Both A and R are true, but R is not the correct explanation of A.
• Explanation: The point satisfies one equation but not the other.