1. Which of the following pair of equations has a unique solution?
Answer:
Explanation:
The given lines have different slopes and hence they intersect at a unique point.
2. The system of equations x + y = 5 and 2x + 2y = 10 has:
Answer:
Explanation:
The second equation is just a multiple of the first. Thus, they represent the same line, leading to infinite solutions.
3. If the lines represented by the equations ax + by = c and bx – ay = 1 are perpendicular, then:
Answer:
Explanation:
The product of the slopes of two perpendicular lines is -1.
4. The point of intersection of the lines 2x – y = 3 and x + y = 2 is:
Answer:
Explanation:
On solving the two equations, we get x=1 and y=1.
5. The lines represented by the equations x – 2y = 3 and 2x – 4y = 6 are:
Answer:
Explanation:
Both equations represent the same line.
6. The condition for the equations ax + by = c and bx + ay = d to represent coincident lines is:
Answer:
Explanation:
For coincident lines, the ratios of coefficients should be equal.
7. The solution to the system of equations x – y = 4 and x + y = 2 is:
Answer:
Explanation:
Solving the equations simultaneously, we get x = 3 and y = -1.
8. Which of the following pairs of equations are parallel?
Answer:
Explanation:
The lines have the same slope but different y-intercepts, hence they are parallel.
9. If a system of two equations has a unique solution, then the lines are:
Answer:
Explanation:
If there's a unique solution, the two lines intersect at just one point.
10. The system of equations 2x – y = 3 and 4x – 2y = 6 has:
Answer:
Explanation:
Both equations represent parallel lines as they have the same slope.
11. The graph of the equation 3x – 4y = 12 intersects the x-axis at:
Answer:
Explanation:
Setting y = 0 in the equation, we get x = 4.
12. The solution to the system of equations x + y = 5 and x – y = 1 is:
Answer:
Explanation:
Solving the equations simultaneously, we get x = 3 and y = 2.
13. If the lines represented by the equations 2x + 3y = 5 and ax + by = 7 are parallel, then:
Answer:
Explanation:
Parallel lines have the same slope. Thus, a/b should be equal to the ratio of the coefficients of x and y in the first equation.
14. If the pair of equations x – y = 2 and cx – dy = 2 has no solution, then:
Answer:
Explanation:
For no solution, the lines should be parallel. Thus, the ratio of coefficients should be the same.
15. The system of equations 3x – 2y = 8 and 6x – 4y = 16 has:
Answer:
Explanation:
Both equations represent the same line, leading to infinite solutions.
16. The condition for the lines 3x – 5y = 7 and kx – 5y = 14 to be parallel is:
Answer:
Explanation:
The lines are parallel if they have the same slope. Thus, k should be 6 to maintain the same ratio of x and y coefficients.
17. The graph of the equation y = 3 intersects the x-axis at:
Answer:
Explanation:
The equation represents a horizontal line passing through the point (0,3). It doesn't intersect the x-axis.
18. The solution of the pair of equations 2x + y = 7 and x – 2y = 3 is:
Answer:
Explanation:
Solving the equations simultaneously, we get x = 3 and y = 1.
19. If the pair of equations 2x + 3y = 6 and 4x + ky = 8 has a unique solution, then k is not equal to:
Answer:
Explanation:
For a unique solution, the lines should not be parallel. Therefore, k should not equal the ratio of coefficients from the first equation.
20. The pair of equations y = x and y = -x represents lines that are:
Answer:
Explanation:
The lines y = x and y = -x have slopes that are negative reciprocals of each other, hence they are perpendicular.