Chapter 3 · Pair of Linear Equations in Two Variables

NCERT Exemplar Class 10 Maths Exercise 3.2

All 6 questions of NCERT Exemplar Class 10 Maths Exercise 3.2 (Pair of Linear Equations in Two Variables), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Do the following ​pair ​of linear equations have no solution? Justify ​your answer.

   1. (i)$2x+4y=3$ ​and $12y+6x=6$
   2. (ii)$x=2y$ and $y=2x$
   3. (iii)$3x+y-3=0$ ​and $2x+\frac{2}{3}y=2$.

   1. i2x + 4y = 3 and 12y + 6x = 6
   2. iix = 2y and y = 2x
   3. iii$3x+y-3=0$ and $2x+(2/3)y=2$
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   Only pair (i) has no solution (parallel lines). Pair (ii) has a unique solution $(0,0)$; pair (iii) has infinitely many solutions (coincident lines).

   Step-by-step solution →
2. Q2
   Do ​the ​following equations represent a pair ​of ​coincident lines? Justify ​your ​answer.

   1. (i)$3x+\frac{1}{7}y=3$ ​and $7x+3y=7$
   2. (ii)$-2x-3y=1$ ​and $6y+4x=-2$
   3. (iii)$\frac{x}{2}+y+\frac{2}{5}=0$ and $4x+8y+\frac{5}{16}=0$
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   Only pair (ii) represents coincident lines. Pair (i) gives intersecting lines and pair (iii) gives parallel (non-coincident) lines.

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3. Q3
   Are the following pair ​of linear equations consistent? Justify your answer.

   1. (i)-3x - ​4y = 12 and 4y + 3x = 12.
   2. (ii)(3/5)x - y ​= 1/2 and ​(1/5)x - 3y = 1/6.
   3. (iii)​2ax + by = a ​and 4ax + ​2by - 2a = 0; a, b not equal to 0.
   4. (iv)​x + 3y = 11 and 2(2x + 6y) = 22.
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   (i) Inconsistent; (ii) Consistent (unique solution); (iii) Consistent (infinitely many solutions); (iv) Inconsistent.

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4. Q4
   For the pair of equations $\lambda x+3y=-7$ and $2x+6y=14$ to have infinitely ​many solutions, the value ​of $\lambda$ should be 1. Is ​the ​statement ​true? Give reasons.
   Show answer

   False

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5. Q5
   For all ​real values ​of $c$, the pair of equations $x-2y=8$ ​and $5x-10y=c$ have ​a unique solution. Justify ​whether it ​is ​true or false.
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   False. Since $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{1}{5}$, the pair never has a unique solution for any $c$.

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6. Q6
   The line represented ​by $x=7$ is parallel to the $x$-axis. ​Justify ​whether ​the statement ​is true or ​not.
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   False

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