Class 9 Two Variables, One Line Mock Test (CBSE)

Topics in Two Variables, One Line

• Linear equations in two variables
• Standard form of a linear equation in two variables
• Solutions of a linear equation in two variables
• Graph of a linear equation in two variables
• Slope
• Slope-intercept form
• Pair of linear equations in two variables
• Substitution method
• Elimination method
• Graphical method for solving a pair of linear equations
• Unique, no and infinitely many solutions
• Intersecting, parallel and coincident lines

Two Variables, One Line practice questions with answers

1. Q1Multiple choice·1 mark
   The pair of linear equations $3x - 5y = 7$ and $6x - 10y = 14$ has:

   1. (a)Two solutions
   2. (b)A unique solution
   3. (c)No solution
   4. (d)Infinitely many solutions
   Show answer and step marks

   1. Correct option: (d) Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2}$, the equations represent coincident lines and have infinitely many solutions.+1

   Answer: (d)
2. Q2Multiple choice·1 mark
   For what value of $k$ does the pair of linear equations $kx + 3y = 3$ and $12x + ky = 6$ have infinitely many solutions?

   1. (a)$6$
   2. (b)$3$
   3. (c)$12$
   4. (d)$-6$
   Show answer and step marks

   1. Correct option: (a) $6$+1

   Answer: (a)
3. Q3Assertion and reason·1 mark
   Assertion (A): The pair of lines given by $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ are parallel. Reason (R): A pair of linear equations satisfying $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ represents parallel lines.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer and step marks

   1. Correct option: (a)+1

   Answer: (a)
4. Q4Short answer·3 marks
   The total cost of a notebook and two pens is ₹ 50. Write a linear equation in two variables to represent this statement. Find any two different solutions for this equation.
   Show answer and step marks

   1. Let the cost of a notebook be ₹ $x$ and the cost of a pen be ₹ $y$.+1
   2. According to the given condition, the linear equation in two variables is $x + 2y = 50$.+1
   3. First solution: any pair satisfying $x + 2y = 50$, e.g. $y = 10 \implies x = 30$, giving $(30, 10)$; or any other valid pair.+0.5
   4. Second solution: a different pair satisfying the equation, e.g. $y = 15 \implies x = 20$, giving $(20, 15)$.+0.5

   Answer: $x + 2y = 50$; any two different pairs satisfying it, e.g. $(30, 10)$ and $(20, 15)$ (also $(10, 20)$, $(40, 5)$, etc.)
5. Q5Short answer·3 marks
   Write each of the following equations in the form $ax + by + c = 0$ and indicate the values of $a$, $b$, and $c$ in each case: (i) $2x + 3y = 4.3$ (ii) $x - 4 = \frac{y}{3}$ (iii) $3x = -2y$
   Show answer and step marks

   1. (i) Rewrite $2x + 3y = 4.3$ as $2x + 3y - 4.3 = 0$.+0.5
   2. (i) $a = 2$, $b = 3$, $c = -4.3$ (any non-zero multiple, e.g. $a = 20$, $b = 30$, $c = -43$, is also correct).+0.5
   3. (ii) Rewrite $x - 4 = \frac{y}{3}$ as $3x - y - 12 = 0$ (or equivalently $x - \frac{y}{3} - 4 = 0$).+0.5
   4. (ii) $a = 3$, $b = -1$, $c = -12$ (or $a = 1$, $b = -\frac{1}{3}$, $c = -4$, or any non-zero multiple).+0.5
   5. (iii) Rewrite $3x = -2y$ as $3x + 2y + 0 = 0$.+0.5
   6. (iii) $a = 3$, $b = 2$, $c = 0$ (or any non-zero multiple).+0.5

   Answer: (i) $2x + 3y - 4.3 = 0$: $a=2, b=3, c=-4.3$; (ii) $3x - y - 12 = 0$: $a=3, b=-1, c=-12$; (iii) $3x + 2y + 0 = 0$: $a=3, b=2, c=0$ (any non-zero multiple of an equation is also correct)
6. Q6Long answer·5 marks
   Consider the linear equation $2x + 3y = 12$. (i) Find any four distinct solutions for this equation. (ii) Determine the coordinates of the points where the line represented by this equation intersects the $x$-axis and the $y$-axis. (iii) Check whether the points $(9, -2)$ and $(1, 3)$ are solutions of the equation, and show your algebraic verification.
   Show answer and step marks

   1. (i) First solution: any pair $(x, y)$ with $2x + 3y = 12$, e.g. $(0, 4)$, $(3, 2)$, $(6, 0)$, $(-3, 6)$.+0.5
   2. (i) Second solution, a different valid pair.+0.5
   3. (i) Third solution, a different valid pair.+0.5
   4. (i) Fourth solution, a different valid pair.+0.5
   5. (ii) $x$-axis: put $y = 0 \implies 2x = 12 \implies x = 6$, so the point is $(6, 0)$.+0.5
   6. (ii) $y$-axis: put $x = 0 \implies 3y = 12 \implies y = 4$, so the point is $(0, 4)$.+0.5
   7. (iii) Check $(9, -2)$: $2(9) + 3(-2) = 18 - 6 = 12$, so $(9, -2)$ is a solution.+1
   8. (iii) Check $(1, 3)$: $2(1) + 3(3) = 2 + 9 = 11 \neq 12$, so $(1, 3)$ is not a solution.+1

   Answer: (i) Any four pairs satisfying $2x + 3y = 12$, e.g. $(0, 4), (3, 2), (6, 0), (-3, 6)$; (ii) $x$-axis at $(6, 0)$, $y$-axis at $(0, 4)$; (iii) $(9, -2)$ is a solution, $(1, 3)$ is not.

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