1. The pairs of equations x+2y-5 = 0 and -4x-8y+20=0 have:
a) Unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution
Answer: (c) Infinitely many solutions
Explanation:
a1/a2 = 1/-4
b1/b2 = 2/-8 = 1/-4
c1/c2 = -5/20 = -¼
This shows:
a1/a2 = b1/b2 = c1/c2
Therefore, the pair of equations has infinitely many solutions.
2. If a pair of linear equations is consistent, then the lines are:
(a) Parallel
(b) Always coincident
(c) Always intersecting
(d) Intersecting or coincident
Answer: (d) Intersecting or coincident
Explanation: Because the two lines definitely have a solution.
3. The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have
(a) Unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution
Answer: (d) No solution
Explanation: Given, 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0
a1/a2 = 9/18 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 12/26 = 6/13
Since, a1/a2 = b1/b2 ≠ c1/c2
So, the pairs of equations are parallel and the lines never intersect each other at any point, therefore there is no possible solution.
4. If the lines 3x+2ky – 2 = 0 and 2x+5y+1 = 0 are parallel, then what is the value of k?
(a) 4/15
(b) 15/4
(c) ⅘
(d) 5/4
Answer: (b) 15/4
Explanation: The condition for parallel lines is:
a1/a2 = b1/b2 ≠ c1/c2
Hence, 3/2 = 2k/5
k=15/4
(a) -6x+10y-4=0
(b) 6x-10y-4=0
(c) 6x+10y-4=0
(d) -6x+10y+4=0
Answer: (a) -6x+10y-4=0
Explanation: The condition for dependent linear equations is:
a1/a2 = b1/b2 = c1/c2
For option a,
a1/a2 = b1/b2 = c1/c2= ½
(a) -6x+10y-4=0
(b) 6x-10y-4=0
(c) 6x+10y-4=0
(d) -6x+10y+4=0
Answer: (a) -6x+10y-4=0
Explanation: The condition for dependent linear equations is:
a1/a2 = b1/b2 = c1/c2
For option a,
a1/a2 = b1/b2 = c1/c2= ½