CBSE-Class-10-Maths-Assertion-and-Reason-Chapter-3-Pair-of-Linear-Equations-in-Two-Variables

Class 10^th Maths Chapter Assertion and Reason Questions

Assertion and Reason Questions

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.

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.

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.

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.

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.

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.

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 $y = \frac{2}{3}x - 2$ .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Chapter 3 Pair of Linear Equations in Two Variables

Class 10th Maths Chapter Assertion and Reason Questions

Assertion and Reason Questions

Class 10^th Maths Chapter Assertion and Reason Questions