Graphical Method PYQs

Class 10 Previous Year Questions (2014 – 2026)

Topic Overview

Graphical representation is a powerful way to solve linear equations. Expect questions on finding intersection points, vertices of triangles formed with X and Y axes, and shading areas.

Q1 2024

00:00

Assertion (A): The graph of $x + 2y = 5$ and $2x + 4y = 14$ shows parallel lines.
Reason (R): Parallel lines never intersect, so the system has no solution.

For $x + 2y = 5$ and $2x + 4y = 14$:

$a_1/a_2 = 1/2, b_1/b_2 = 2/4 = 1/2, c_1/c_2 = 5/14$.

Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, the lines are parallel.

Reason (R) correctly explains the property of parallel lines.

Ans: (A) Both A and R are true; R is the correct explanation

Q2 2023

00:00

The pair $y = 0$ and $y = -7$ has:

(a)(A) one solution

(b)(B) two solutions

(c)(C) infinitely many solutions

(d)(D) no solution

$y = 0$ is the X-axis.

$y = -7$ is a line parallel to the X-axis.

Horizontal parallel lines never intersect, so no solution.

Ans: (D) no solution

Q3 2020

00:00

Lines: $2x + y - 5 = 0$ and $4x + 2y - 10 = 0$. Are they parallel, intersecting or coincident? Give reason.

$a_1/a_2 = 2/4 = 1/2$

$b_1/b_2 = 1/2$

$c_1/c_2 = -5/(-10) = 1/2$

Since $a_1/a_2 = b_1/b_2 = c_1/c_2$, they are coincident lines.

Ans: Coincident lines

Q4 2016

00:00

Check whether $6x - 3y + 10 = 0$ and $2x - y + 9 = 0$ are consistent or inconsistent.

$a_1/a_2 = 6/2 = 3$

$b_1/b_2 = -3/(-1) = 3$

$c_1/c_2 = 10/9$

Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, lines are parallel and system is inconsistent.

Ans: Inconsistent

Q5 2014

00:00

Is the system $x + 2y - 8 = 0$ and $2x + 4y = 16$ consistent or inconsistent? Justify.

$a_1/a_2 = 1/2, b_1/b_2 = 2/4 = 1/2, c_1/c_2 = -8/(-16) = 1/2$.

Since all ratios are equal, system has infinitely many solutions and is consistent.

Ans: Consistent

Q6 2024

00:00

Solve graphically: $x - y + 1 = 0$ and $x + y = 5$. Find vertices of triangle formed with the y-axis.

For $x-y+1=0$: points $(0,1), (1,2), (-1,0)$.

For $x+y=5$: points $(0,5), (5,0), (2,3)$.

Intersection is $(2,3)$. Vertices on Y-axis are $(0,1)$ and $(0,5)$.

Triangle vertices: $(0,1), (0,5), (2,3)$.

Ans: (0,1), (0,5), (2,3)

Q7 2024

00:00

Solve graphically: $x - y + 1 = 0$ and $x + y = 5$. Shade the region bounded by these lines and the y-axis. Find the area of the shaded region.

Intersection point is $(2,3)$.

Vertices on y-axis: $(0,1)$ and $(0,5)$.

Base of triangle (on y-axis) $= 5 - 1 = 4$ units.

Height of triangle (x-coordinate of intersection) $= 2$ units.

Area $= 1/2 \times 4 \times 2 = 4$ sq. units.

Ans: 4 sq. units

Q8 2022

00:00

Solve graphically: $x + 2y = 6$ and $2x - 5y = 12$. Find the vertices of triangle formed by these lines and the x-axis.

Intersection is $(6,0)$.

Points on x-axis: $x+2y=6 \Rightarrow (6,0)$. $2x-5y=12 \Rightarrow (6,0)$.

Wait, both lines meet the x-axis at $(6,0)$. The question likely meant y-axis or a different line.

Let's check y-axis intersections: $(0,3)$ and $(0,-2.4)$.

Vertices: $(6,0), (0,3), (0,-2.4)$.

Ans: (6,0), (0,3), (0,-2.4)

00:00

Assertion (A): The graph of $x + 2y = 5$ and $2x + 4y = 14$ shows parallel lines.
Reason (R): Parallel lines never intersect, so the system has no solution.

For $x + 2y = 5$ and $2x + 4y = 14$:

$a_1/a_2 = 1/2, b_1/b_2 = 2/4 = 1/2, c_1/c_2 = 5/14$.

Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, the lines are parallel.

Reason (R) correctly explains the property of parallel lines.

Ans: (A) Both A and R are true; R is the correct explanation

Q2 2023

00:00

The pair $y = 0$ and $y = -7$ has:

(a)(A) one solution

(b)(B) two solutions

(c)(C) infinitely many solutions

(d)(D) no solution

$y = 0$ is the X-axis.

$y = -7$ is a line parallel to the X-axis.

Horizontal parallel lines never intersect, so no solution.

Ans: (D) no solution

Q3 2020

00:00

Lines: $2x + y - 5 = 0$ and $4x + 2y - 10 = 0$. Are they parallel, intersecting or coincident? Give reason.

$a_1/a_2 = 2/4 = 1/2$

$b_1/b_2 = 1/2$

$c_1/c_2 = -5/(-10) = 1/2$

Since $a_1/a_2 = b_1/b_2 = c_1/c_2$, they are coincident lines.

Ans: Coincident lines

Q4 2016

00:00

Check whether $6x - 3y + 10 = 0$ and $2x - y + 9 = 0$ are consistent or inconsistent.

$a_1/a_2 = 6/2 = 3$

$b_1/b_2 = -3/(-1) = 3$

$c_1/c_2 = 10/9$

Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, lines are parallel and system is inconsistent.

Ans: Inconsistent

Q5 2014

00:00

Is the system $x + 2y - 8 = 0$ and $2x + 4y = 16$ consistent or inconsistent? Justify.

$a_1/a_2 = 1/2, b_1/b_2 = 2/4 = 1/2, c_1/c_2 = -8/(-16) = 1/2$.

Since all ratios are equal, system has infinitely many solutions and is consistent.

Ans: Consistent

Q6 2024

00:00

Solve graphically: $x - y + 1 = 0$ and $x + y = 5$. Find vertices of triangle formed with the y-axis.

For $x-y+1=0$: points $(0,1), (1,2), (-1,0)$.

For $x+y=5$: points $(0,5), (5,0), (2,3)$.

Intersection is $(2,3)$. Vertices on Y-axis are $(0,1)$ and $(0,5)$.

Triangle vertices: $(0,1), (0,5), (2,3)$.

Ans: (0,1), (0,5), (2,3)

Q7 2024

00:00

Solve graphically: $x - y + 1 = 0$ and $x + y = 5$. Shade the region bounded by these lines and the y-axis. Find the area of the shaded region.

Intersection point is $(2,3)$.

Vertices on y-axis: $(0,1)$ and $(0,5)$.

Base of triangle (on y-axis) $= 5 - 1 = 4$ units.

Height of triangle (x-coordinate of intersection) $= 2$ units.

Area $= 1/2 \times 4 \times 2 = 4$ sq. units.

Ans: 4 sq. units

Q8 2022

00:00

Solve graphically: $x + 2y = 6$ and $2x - 5y = 12$. Find the vertices of triangle formed by these lines and the x-axis.

Intersection is $(6,0)$.

Points on x-axis: $x+2y=6 \Rightarrow (6,0)$. $2x-5y=12 \Rightarrow (6,0)$.

Wait, both lines meet the x-axis at $(6,0)$. The question likely meant y-axis or a different line.

Let's check y-axis intersections: $(0,3)$ and $(0,-2.4)$.

Vertices: $(6,0), (0,3), (0,-2.4)$.

Ans: (6,0), (0,3), (0,-2.4)