Nature of 2 Straight Lines in a Plane

Q.

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

$2x+4y=3$

$12y+6x=6$

Q. Question 6
For what value of k, do the equations 3x – y + 8 = 0 and 6x – ky = - 16 represent coincident lines?
(A) $\frac{1}{2}$
(B) $-\frac{1}{2}$
(C) $2$
(D) $-2$​​​​​​​

Q.

Question 7
If the lines given by 3x + 2ky = 2 and 3x + 5y = 1 are parallel, then the value of k is
(a) $-\frac{5}{4}$
(b) $\frac{1}{2}$
(c) $\frac{15}{4}$
(d) $\frac{3}{2}$

Q. Which of the following pair of lines are parallel to each other?

1. $5x+7y=12$; $2x+14y=12$
2. $x+7y=6$; $2x+14y=12$
3. $x+2y=7$; $3x+6y=28$
4. $11x+2y=4$; $3x+6y=10$

Q.

Which of the following pairs of lines are parallel?

1. 7x + y = 132, x + 7y = 3

2. 4x + 3y = 4, 8x + 6y = 2235

3. None of these

4. 3x + y = 225, 2x + y = 1.1

Q.

Question 1 (iv)
Which of the following pairs of linear equations has a unique solution, no solution or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
x - 3y - 7 = 0 ; 3x - 3y - 15= 0

Q.

The straight line passes through the points $(0,0)$, $(-1,1)$ and $(1,-1)$ has an equation:

1. $2-x=3y$

2. $y=x$

3. $2x-y=0$

4. $x+y=0$

Q. Question 3
For what values of a and b will the following pair of linear equations have infinitely many solutions?
x + 2y = 1
(a - b)x + (a + b)y = a + b - 2

Q.

Question 2 (i)
For which values of ‘a’ and ‘b’ does the following pair of linear equations have an infinite number of solutions?
2x + 3y =7
(a - b)x + (a + b)y = 3a +b - 2

Q.

Question 7
If the lines given by 3x + 2ky = 2 and 3x + 5y = 1 are parallel, then the value of k is
(a) $-\frac{5}{4}$
(b) $\frac{1}{2}$
(c) $\frac{15}{4}$
(d) $\frac{3}{2}$

Q.

Unique point is obtained for the pair of equations $a_1$x + $b_1$y + $c_1$ = 0 and $a_2$x +$b_2$y + $c_2$ = 0 if

1. a₁/a₂ ≠ b₁/b₂ = c₁/c₂

2. a₁/a₂ = b₁/b₂ ≠ c₁/c₂

3. a₁/a₂ = b₁/b₂ = c₁/c2

4. a₁/a₂ ≠ b₁/b₂

Q. On comparing ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}and\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
$5x-4y+8=0;7x+6y-9=0$

1. Intersect at a point
2. Coincident
3. Parallel
4. None of these

Q. Represent the following equations graphically:
$x+y=10$
$10x+y+10y+x=110$

1. 
2. 
3. 
4. None of these

Q. Which of the following pairs of equations have the same solutions?

1. 6x = 3 and 3x = 1.5
2. 8x = 3 and 4x = 1
3. 10x = 9 and 5x = 18
4. 12x = 6 and 6x = 6

Q. One equation of a pair of dependent linear equations is $5x+7y=2$. The second equation can be

1. $10x-14y-4=0$
2. $10x+14y-4=0$
3. $10x+14y+1=0$
4. $10x-14y+4=0$

Q. A pair of linear equations in 2 variables has no solution, it represents a pair of ___ lines.

1. intersecting
2. equal
3. parallel
4. coincident

Q. The line $ax+by+c=0,$ is parallel to $x$ axis when $a$ = .

1. $2$
2. $0$
3. $1$

Q.

Which of the following pair of linear equations has infinite solutions?

1. $x+2y=7$; $3x+6y=21$

2. $4x+3y=7$; $3x+6y=25$

3. $y+2x=10$; $11x+6y=21$

4. $\frac{1}{2}x+2y=\frac{7}{11}$; $x+6y=21$

Q. Represent the following equations graphically:
$x+y=10$
$10x+y+10y+x=110$

1. 
2. 
3. 
4. None of these

Q. Coincident lines have infinite solutions.

1. False
2. True

Q. Question 1(iii)
Do the following pair of linear equations have no solution? Justify your answer.
3x + y – 3 = 0 and 2x + $\frac{2}{3}$ y = 2

Q. $\frac{2}{x}-\frac{3}{y}=15:\frac{8}{x}+\frac{5}{y}=77$ Find the value of $x$ and $y$.

Q. The number of solutions of the given pair of linear equations $3x-9y=10$ and $9x-27y=30$ is:

[1 mark]

1. One
2. Infinite
3. Two
4. Zero

Q. Solve the equations graphically
$2x-y=2$
$4x-y=4$

Q.

Question 2 (ii)
On comparing the ratios $\frac{a_1}{a_2}$ , $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
9x + 3y + 12 = 0
18x + 6y + 24 = 0

Q. Consider two equations in the variables x and y written in the standard form:
$5x+6y+4=0and$
$10x+12y+7=0$
What is the nature of these two lines?

1. Coincident
2. Intersecting
3. Parallel
4. Coincident or parallel

Q. Obtain real solutions of simultaneous equations.
$xy+3y^2-x+4y-7=0$
$2xy+y^2-2x-2y+1=0$.

Q. Draw the graph of equation 3x+2y = 6. Find the sum of the co-ordinates of the point where the graph intersects the Y-axis.

Q. Graphically, the pair of equations $6x+3y+10=0$ and $2x+y+9=0$ represent two lines which are:

1. intersecting at exactly one point.
2. intersecting at exactly two points.
3. coincident.
4. parallel.

Q. Solve the equation: $x+y=1$ and $2x+3y=3$ and then identify the system of equation.

1. Dependent
2. All the above
3. Consistent
4. Inconsistent