Pair of Linear Equations in Two Variables: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. For what values of $p$ and $q$ will $2x+3y=7$ and $(p+q)x+(2p-q)y=21$ have infinitely many solutions?

H2. The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number.

H3. A boat goes 30 km upstream and 44 km downstream in 10 h; it goes 40 km upstream and 55 km downstream in 13 h. Taking upstream speed $u$ and downstream speed $v$, form linear equations in $\frac{1}{u}$ and $\frac{1}{v}$ and solve.

20 MCQs (1 mark each)

1. The pair $x+y=4$, $2x+2y=8$ has
   (a) one solution (b) two (c) infinitely many (d) none
2. The pair $x-2y=3$, $3x-6y=5$ is
   (a) consistent (b) inconsistent (c) dependent (d) unique
3. The lines $2x+3y=5$ and $4x+ky=10$ coincide if $k$ =
   (a) 3 (b) 6 (c) 9 (d) 2
4. For no solution of $kx+2y=5$, $3x+y=1$, $k$ =
   (a) 2 (b) 3 (c) 6 (d) $\frac{3}{2}$
5. The solution of $x+y=7$, $x-y=1$ is
   (a) (3, 4) (b) (4, 3) (c) (5, 2) (d) (2, 5)
6. $x=2$ and $y=3$ represent lines which
   (a) are parallel (b) meet at (2, 3) (c) meet at (3, 2) (d) coincide
7. If $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, the lines are
   (a) parallel (b) coincident (c) intersecting (d) perpendicular always
8. The graph of $y=0$ is
   (a) the y-axis (b) the x-axis (c) a line through (0, 1) (d) none
9. If $(2,1)$ is a solution of $3x-ky=4$, $k$ =
   (a) 1 (b) 2 (c) −2 (d) 3
10. The pair $217x+131y=913$, $131x+217y=827$ gives $x+y$ =
    (a) 4 (b) 5 (c) 6 (d) 3
11. The sum of two numbers is 20 and difference 4. The larger is
    (a) 8 (b) 10 (c) 12 (d) 16
12. A pair with infinitely many solutions is
    (a) inconsistent (b) consistent and dependent (c) consistent with unique solution (d) parallel
13. The lines $x+2y=4$ and $2x+4y=12$ are
    (a) intersecting (b) parallel (c) coincident (d) perpendicular
14. The line $2x+3y=6$ meets the y-axis at
    (a) (3, 0) (b) (0, 2) (c) (2, 0) (d) (0, 3)
15. 5 years hence, A will be twice B. Taking present ages $x$, $y$, the equation is
    (a) $x+5=2y$ (b) $x+5=2(y+5)$ (c) $x=2y+5$ (d) $x-5=2(y-5)$
16. For $3x+2y=7$ and $6x+4y=c$ to be inconsistent, $c$ ≠
    (a) 7 (b) 14 (c) 21 (d) 3.5
17. A two-digit number with tens digit $x$ and units digit $y$ is
    (a) $xy$ (b) $10y+x$ (c) $10x+y$ (d) $x+y$
18. The area of the triangle formed by $x+y=4$ with the axes is
    (a) 4 (b) 8 (c) 16 (d) 2
19. If $ax+by=c$ and $lx+my=n$ have a unique solution, then
    (a) $am\ne bl$ (b) $am=bl$ (c) $ab=lm$ (d) $an=cl$
20. The pair $y=x$ and $y=-x$ meets at
    (a) (1, 1) (b) (0, 0) (c) (1, −1) (d) nowhere

Answer key

1 (c) · 2 (b) · 3 (b) · 4 (c) · 5 (b) · 6 (b) · 7 (c) · 8 (b) · 9 (b) · 10 (b) · 11 (c) · 12 (b) · 13 (b) · 14 (b) · 15 (b) · 16 (b) · 17 (c) · 18 (b) · 19 (a) · 20 (b)

Hints: Q3 $\frac{2}{4}=\frac{3}{k}$. Q4 $\frac{k}{3}=\frac{2}{1}$. Q9 $6-k=4$. Q16 coincident if $c=14$, so inconsistent needs $c\ne 14$. Q18 $\frac{1}{2}\times 4\times 4=8$.

Assertion–Reason (10)

(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.

1. A: $x+2y=3$ and $2x+4y=7$ have no solution. R: $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ means parallel lines.
2. A: $x+y=5$ and $2x+2y=10$ are inconsistent. R: Coincident lines have infinitely many solutions.
3. A: $3x-y=4$ and $x+y=4$ have a unique solution. R: The lines are intersecting, since $\frac{3}{1}\ne \frac{-1}{1}$.
4. A: The point (1, 1) lies on $x+y=2$ and $2x-y=1$. R: (1, 1) is the solution of the pair.
5. A: $x=5$ is parallel to the x-axis. R: $x=5$ is a line parallel to the y-axis.
6. A: A pair of linear equations can have exactly two solutions. R: Two distinct straight lines meet in at most one point.
7. A: If $kx+3y=3$ and $12x+ky=6$ are inconsistent, $k=-6$. R: For inconsistency, $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$.
8. A: Every pair of linear equations has a solution. R: Two parallel lines never meet.
9. A: Elimination is convenient for $217x+131y=913$, $131x+217y=827$. R: Adding and subtracting gives $x+y$ and $x-y$ directly.
10. A: The graph of $2x+3y=0$ passes through the origin. R: $(0,0)$ satisfies $2x+3y=0$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (a) · 5 (d) · 6 (d) · 7 (a) · 8 (d) · 9 (a) · 10 (a)

Q7: $\frac{k}{12}=\frac{3}{k}$ ⇒ $k=\pm 6$; $k=6$ gives $\frac{3}{6}=\frac{3}{6}$ for c too (coincident), so only $k=-6$ is inconsistent.

Very short answer (1 mark each)

1. Write the condition for $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ to be coincident. Ans. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
2. Is (2, −1) a solution of $3x+2y=4$? Ans. $6-2=4$ ✓ yes
3. Give an equation parallel to $x-2y=3$. Ans. $2x-4y=7$
4. Where does $x+2y=6$ meet the x-axis? Ans. (6, 0)
5. Find $k$ if $2x+ky=5$ and $4x+6y=10$ are coincident. Ans. $k=3$
6. What is the solution of $x-y=0$ and $x+y=0$? Ans. (0, 0)
7. If $x+y=10$ and $x-y=2$, find $x^2-y^2$. Ans. 20

Short answer practice (2 marks each)

1. Find $k$ for which $kx+y=k^2$ and $x+ky=1$ have infinitely many solutions.

2. Solve $\frac{x}{2}+\frac{y}{3}=4$ and $\frac{x}{3}+\frac{y}{2}=\frac{13}{3}$ … (multiply by 6).

3. The ratio of incomes of two persons is 9 : 7 and of their expenditures 4 : 3. If each saves ₹2000 a month, find their incomes.

Short answer practice (3 marks each)

4. A fraction becomes $\frac{1}{3}$ when 1 is subtracted from both terms, and $\frac{1}{2}$ when 8 is added to both. Find the fraction.

5. Ten years hence, a man will be twice as old as his son; ten years ago he was four times as old. Find their present ages.

6. Show that $3x+2y=8$ and $6x-4y=9$ are consistent and solve them.

Long answer (5 marks)

7. Draw the graphs of $2x-y=2$ and $4x-y=8$. Find the coordinates of the vertices of the triangle formed by these lines and the y-axis, and find its area.