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Chapter 3: Pair of Linear Equations

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This page provides comprehensive Chapter 3: Pair of Linear Equations - Assertion Reason Worksheet - SJMaths. Assertion and Reason type questions for Class 10 Pair of Linear Equations in Two Variables. Practice for CBSE Board Exams.

Assertion-Reason Worksheet

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Directions:

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:

  • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • (B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
  • (C) Assertion (A) is true but Reason (R) is false.
  • (D) Assertion (A) is false but Reason (R) is true.
  1. Question 1:
    Assertion (A): The pair of equations $x + 2y - 5 = 0$ and $-3x - 6y + 15 = 0$ have infinitely many solutions.
    Reason (R): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, then the pair of equations has infinitely many solutions.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Check ratios: $\frac{1}{-3} = -\frac{1}{3}$, $\frac{2}{-6} = -\frac{1}{3}$, $\frac{-5}{15} = -\frac{1}{3}$.
    Step 2: Since all ratios are equal, the system has infinitely many solutions. A is true.
    Step 3: R is the correct condition for infinite solutions.
  2. Question 2:
    Assertion (A): The graph of $x = 2$ is a line parallel to the y-axis.
    Reason (R): The graph of $y = k$ is a line parallel to the x-axis.
    (A)
    (B)
    (C)
    (D)
    Solution: (B)
    Step 1: $x = 2$ is a vertical line passing through $x=2$, so it is parallel to y-axis. A is true.
    Step 2: $y = k$ is a horizontal line, parallel to x-axis. R is true.
    Step 3: R explains properties of $y=k$, not $x=k$. So R does not explain A.
  3. Question 3:
    Assertion (A): The value of $k$ for which the system $kx - y = 2$ and $6x - 2y = 3$ has a unique solution is $k = 3$.
    Reason (R): For a unique solution, $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: For unique solution: $\frac{k}{6} \neq \frac{-1}{-2} \Rightarrow \frac{k}{6} \neq \frac{1}{2} \Rightarrow k \neq 3$.
    Step 2: Assertion says $k=3$, which is the condition for *no* unique solution (parallel lines). So A is false.
    Step 3: Reason is the correct condition.
  4. Question 4:
    Assertion (A): If the lines $3x + 2ky - 2 = 0$ and $2x + 5y + 1 = 0$ are parallel, then $k = \frac{15}{4}$.
    Reason (R): For parallel lines, $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Apply condition: $\frac{3}{2} = \frac{2k}{5}$.
    Step 2: $4k = 15 \Rightarrow k = \frac{15}{4}$.
    Step 3: Check $c_1/c_2$: $\frac{-2}{1} = -2$. Since $\frac{3}{2} \neq -2$, lines are parallel. A is true, R explains A.
  5. Question 5:
    Assertion (A): The linear equations $x - y = 3$ and $3x - 3y = 9$ are consistent.
    Reason (R): A pair of linear equations is consistent if it has a unique solution only.
    (A)
    (B)
    (C)
    (D)
    Solution: (C)
    Step 1: Ratios: $\frac{1}{3} = \frac{-1}{-3} = \frac{3}{9}$. All equal $\Rightarrow$ Coincident lines $\Rightarrow$ Infinite solutions.
    Step 2: Infinite solutions means the system is consistent (dependent). A is true.
    Step 3: Reason is false because consistent systems can have unique OR infinite solutions.
  6. Question 6:
    Assertion (A): The point $(2, -3)$ lies on the graph of the linear equation $2x + 3y = -5$.
    Reason (R): Every point on the graph of a linear equation is a solution of the equation.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: Substitute $(2, -3)$ in LHS: $2(2) + 3(-3) = 4 - 9 = -5$.
    Step 2: LHS = RHS, so the point lies on the line. A is true.
    Step 3: R is the definition of the graph of an equation. R explains A.
  7. Question 7:
    Assertion (A): The lines $2x - 3y = 7$ and $4x - 6y = 15$ are parallel.
    Reason (R): Two lines are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$. $\frac{b_1}{b_2} = \frac{-3}{-6} = \frac{1}{2}$. $\frac{c_1}{c_2} = \frac{7}{15}$.
    Step 2: Since $\frac{1}{2} = \frac{1}{2} \neq \frac{7}{15}$, lines are parallel. A is true.
    Step 3: R correctly states the condition.
  8. Question 8:
    Assertion (A): The pair of equations $2x + 3y = 5$ and $4x + 6y = 10$ is inconsistent.
    Reason (R): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident.
    (A)
    (B)
    (C)
    (D)
    Solution: (D)
    Step 1: Ratios: $\frac{2}{4} = \frac{3}{6} = \frac{5}{10} = \frac{1}{2}$.
    Step 2: Lines are coincident, meaning they have infinite solutions. Thus, they are consistent (dependent). A is false.
    Step 3: R is true.
  9. Question 9:
    Assertion (A): The linear equation $4x + 3y = 18$ has infinitely many solutions.
    Reason (R): A linear equation in two variables has infinitely many solutions.
    (A)
    (B)
    (C)
    (D)
    Solution: (A)
    Step 1: A single linear equation represents a line, which contains infinite points. A is true.
    Step 2: R is a standard property of linear equations in two variables. R explains A.
  10. Question 10:
    Assertion (A): If the speed of a boat in still water is $x$ km/h and speed of stream is $y$ km/h, then speed upstream is $x - y$ km/h.
    Reason (R): Speed downstream is given by $x + y$ km/h.
    (A)
    (B)
    (C)
    (D)
    Solution: (B)
    Step 1: Upstream means going against the current, so speed reduces ($x-y$). A is true.
    Step 2: Downstream means going with the current, so speed increases ($x+y$). R is true.
    Step 3: R is a correct statement but does not explain why upstream speed is $x-y$.
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