Welcome to Part 1 of our new 8-part series on Chapter 3: Pair of Linear Equations in Two Variables. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master graphical methods, algebraic methods, and conditions for consistency.
Recommended Books for Deep Practice
Top 25 MCQs - Linear Equations
Question 1: Unique Solution Condition
The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 has:
a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 5/10 = 1/2
Since a1/a2 = b1/b2 = c1/c2, the lines are coincident and have infinitely many solutions.
Question 2: Parallel Lines
If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is:
3/2 = 2k/5
4k = 15
k = 15/4.
Question 3: Intersecting Lines
The pair of equations x = a and y = b graphically represents lines which are:
Question 4: Value of k (Unique Solution)
For what value of k do the equations x - 2y = 3 and 3x + ky = 1 have a unique solution?
1/3 ≠ -2/k
k ≠ -6.
Question 5: Solution of Equations
If x + y = 14 and x - y = 4, then the values of x and y are:
Substitute x in first eq: 9 + y = 14 => y = 5.
Question 6: Nature of Graphs
The graph of y = 6 is a line:
Question 7: Sum of Digits
The sum of the digits of a two-digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is:
(d) 36: Sum = 3+6 = 9. Add 27: 36 + 27 = 63 (digits reversed). Correct.
(c) 63: 63 + 27 = 90 (not reversed).
Question 8: Consistent Equations
A pair of linear equations which has a unique solution x = 2, y = -3 is:
2 - 4(-3) - 14 = 2 + 12 - 14 = 0 (Satisfied)
5(2) - (-3) - 13 = 10 + 3 - 13 = 0 (Satisfied)
Question 9: Value of c
For what value of c does the pair of equations cx - y = 2 and 6x - 2y = 3 have infinitely many solutions?
c/6 = -1/-2 = 2/3
From first part: c/6 = 1/2 => c = 3.
Check last part: 1/2 ≠ 2/3. Since b1/b2 ≠ c1/c2, infinite solutions are not possible for any value of c.
Question 10: Inconsistent Pair
If a pair of linear equations is inconsistent, then the lines will be:
Question 11: Age Problem
A father is 3 times as old as his son. Five years later, he will be two and a half times as old as his son. The father's current age is:
After 5 years: (3x + 5) = 2.5(x + 5)
3x + 5 = 2.5x + 12.5
0.5x = 7.5 => x = 15.
Father = 3(15) = 45.
Question 12: Area of Triangle
The area of the triangle formed by the lines x = 3, y = 4 and x = y is:
x=3, y=4 => (3,4)
x=3, x=y => (3,3)
y=4, x=y => (4,4)
Triangle with vertices (3,4), (3,3), (4,4). Base (vertical) = 1, Height (horizontal) = 1. Area = 1/2 * 1 * 1 = 0.5.
Question 13: Substitution
The value of x and y in 2x + 3y = 11 and 2x - 4y = -24 is:
7y = 35 => y = 5.
Substitute y=5: 2x + 15 = 11 => 2x = -4 => x = -2.
Question 14: Dependent Consistency
If a pair of linear equations is consistent, then the lines will be:
Question 15: Line Intersection
The lines represented by 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0 are:
Question 16: X-axis Equation
The equation of the x-axis is:
Question 17: Fraction Problem
A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. The fraction is:
(c) 5/12: (5-1)/12 = 4/12 = 1/3. Correct.
5/(12+8) = 5/20 = 1/4. Correct.
Question 18: Sum of Reciprocals
The sum of a number and its reciprocal is 2. The number is:
Question 19: No Solution Condition
The value of k for which the system of equations x + 2y = 3 and 5x + ky + 7 = 0 has no solution is:
1/5 = 2/k
k = 10.
Question 20: Algebraic Methods
Which of the following is NOT an algebraic method to solve a pair of linear equations?
Question 21: Unit's Digit
In a two-digit number, the unit's digit is twice the ten's digit. If 27 is added to the number, the digits interchange their places. The number is:
(c) 36: Unit (6) is twice ten's (3). Correct.
36 + 27 = 63 (Reversed). Correct.
Question 22: Boat Upstream/Downstream
If x is speed of boat in still water and y is speed of stream, then speed of boat upstream is:
Question 23: Coincident Lines k Value
Find k if lines 2x + 3y = 7 and 8x + 12y = k are coincident.
2/8 = 3/12 = 7/k
1/4 = 7/k => k = 28.
Question 24: Complementary Angles
If two angles are complementary and the larger angle exceeds the smaller by 18 degrees, the angles are:
x + y = 90 (Complementary)
x - y = 18
Adding both: 2x = 108 => x = 54.
y = 90 - 54 = 36.
Question 25: Infinite Solutions k
The value of k for which equations 3x - y + 8 = 0 and 6x - ky = -16 have infinitely many solutions is:
Ratio: 3/6 = -1/-k = 8/16
1/2 = 1/k
k = 2.
