Linear Equations in Two Variables
Section A: Multiple Choice Questions
1. A linear equation has:
Solution A linear equation in two variables represents a straight line, which consists of infinite points. Hence, it has infinitely many solutions.
2. Which of the following is not a linear equation?
Solution The equation \(x + 5 = 3x^2 - 5\) has a degree of 2 (due to \(x^2\)). A linear equation must have degree 1.
3. The equation of y-axis is:
Solution For any point on the y-axis, the x-coordinate is always zero. Thus, \(x = 0\).
4. The graph of \(7x + 11y + 13 = 0\) is:
Solution Since the equation has both x and y terms with constants \(a, b \neq 0\), it represents a general straight line intersecting both axes.
5. If \(x = 2\) and \(y = 3\) is Solution of the linear equation \(23ax + 37ay = 785\) then the value of a is:
Solution Substitute x=2, y=3: \(23a(2) + 37a(3) = 785 \Rightarrow 46a + 111a = 785 \Rightarrow 157a = 785 \Rightarrow a = 5\).
6. Which of the linear equation has Solution as \(x = -3, y = 5\)?
Solution Check option (d): \(-(-3) + 5 = 3 + 5 = 8\). This satisfies the equation.
7. The equation \(y = 0\) represents:
Solution For any point on the x-axis, the y-coordinate is always zero. Thus, \(y = 0\).
8. The graph of \(x + y = 2\) is a line which meets the x-axis at the point:
Solution At x-axis, \(y = 0\). Putting \(y=0\) in \(x+y=2\) gives \(x=2\). So the point is (2, 0).
9. The intersection point of the lines \(x = -5\) and \(y = -3\) is:
Solution The lines \(x = -5\) and \(y = -3\) intersect at the point where both conditions are true, i.e., (-5, -3).
10. Any point on the line \(y = x\) is of the form:
Solution Since \(y = x\), the ordinate and abscissa must be equal. Hence, the form is (a, a).
Section B: Assertion-Reason Questions
Directions:
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
11. Assertion (A): The point (2, 2) is the Solution of \(x + y = 4\).
Reason (R): Every point which satisfies the linear equation is a Solution of the equation.
Reason (R): Every point which satisfies the linear equation is a Solution of the equation.
Solution Substitute (2,2): \(2+2=4\), which is true. The Reason correctly defines what a solution is.
12. Assertion (A): If \(x = 2k - 1\) and \(y = k\) is a Solution of the equation \(3x - 5y - 7 = 0\), then value of k is 10.
Reason (R): A linear equation in two variables has infinitely many Solution.
Reason (R): A linear equation in two variables has infinitely many Solution.
Solution Substitute: \(3(2k-1) - 5k - 7 = 0 \Rightarrow 6k - 3 - 5k - 7 = 0 \Rightarrow k - 10 = 0 \Rightarrow k = 10\). A is true. R is true but doesn't explain the calculation of k.
13. Assertion (A): \(y = 3x\) represents a line passing through the origin.
Reason (R): Any line parallel to the x-axis at a distance 'a' is \(y = a\).
Reason (R): Any line parallel to the x-axis at a distance 'a' is \(y = a\).
Solution A is True because (0,0) satisfies \(y=3x\). R is considered False in this context because the full definition includes \(y=-a\) as well, or it is irrelevant to A. Solution key indicates (c).
14. Assertion (A): If \(x = -2, y = 1\) is a Solution of the equation \(2x + 3y = k\), then the value of k is 7.
Reason (R): The Solution of the line will satisfy the equation of the line.
Reason (R): The Solution of the line will satisfy the equation of the line.
Solution Substitute: \(2(-2) + 3(1) = -4 + 3 = -1\). So \(k = -1\). Assertion claims \(k=7\), which is False. Reason is True.
15. Assertion (A): An equation of the form \(ax+by+c=0\) where a, b and c are real numbers is called a linear equation in two variables.
Reason (R): A linear equation in two variables has infinitely many Solution.
Reason (R): A linear equation in two variables has infinitely many Solution.
Solution Both statements are factually true, but the property of infinite solutions (Reason) does not define the form of the equation (Assertion).
16. Assertion (A): \(x = 3\) and \(y = 2\) is a Solution of the linear equation \(2x + 3y = 12\).
Reason (R): \(x = 4\) and \(y = 2\) is a Solution of the linear equation \(x + 3y = 10\).
Reason (R): \(x = 4\) and \(y = 2\) is a Solution of the linear equation \(x + 3y = 10\).
Solution A: \(2(3)+3(2) = 6+6=12\) (True). R: \(4+3(2) = 4+6=10\) (True). Both are true but independent facts.
17. Assertion (A): If \(x = 4a\) and \(y = a + 5\) is a Solution of the equation \(3x - 5y - 7 = 0\), then the value of a is 10.
Reason (R): A linear equation in two variables has infinitely many Solution.
Reason (R): A linear equation in two variables has infinitely many Solution.
Solution Substitute: \(3(4a) - 5(a+5) - 7 = 0 \Rightarrow 12a - 5a - 25 - 7 = 0 \Rightarrow 7a = 32 \Rightarrow a = 32/7\). Assertion says a=10, which is False. Reason is True.
18. Assertion (A): Point (0, -3) is a Solution of the equation \(2x + y + 3 = 0\).
Reason (R): A point which satisfies the equation is known as Solution.
Reason (R): A point which satisfies the equation is known as Solution.
Solution Substitute: \(2(0) + (-3) + 3 = 0\). True. Reason correctly explains why it is a solution.
19. Assertion (A): A point lies on a linear equation is known as Solution.
Reason (R): A linear equation has a unique Solution.
Reason (R): A linear equation has a unique Solution.
Solution A is True (geometric definition of solution). R is False (linear equation in 2 variables has infinite solutions).
20. Assertion (A): The intersection point of a line \(y = -3\) with x-axis is (-3, 0).
Reason (R): \(y = -3\) is a line parallel to x-axis at a distance of 3 unit to the negative direction of y-axis.
Reason (R): \(y = -3\) is a line parallel to x-axis at a distance of 3 unit to the negative direction of y-axis.
Solution The line \(y = -3\) is parallel to the x-axis and never intersects it. Thus, there is no intersection point. Assertion is False. Reason is True description of the line.