Question Bank — Pair of Linear Equations in Two Variables

Section A — MCQs (1 mark each)

1. If the lines represented by the equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) satisfy \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), then the pair of equations is
   (a) consistent (b) inconsistent (c) dependent (d) none of these

2. The pair of equations \(x + 2y - 4 = 0\) and \(2x + 4y - 12 = 0\) represents
   (a) intersecting lines (b) parallel lines (c) coincident lines (d) none of these

3. For the equations \(2x + 3y - 9 = 0\) and \(4x + 6y - 18 = 0\), the ratios satisfy
   (a) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) (b) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) (c) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) (d) none of these

4. Assertion (A): The pair \(3x + 4y - 20 = 0\) and \(x - 2y = 0\) has a unique solution.
   Reason (R): If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the lines intersect at exactly one point.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false. (d) A is false but R is true.

5. Assertion (A): The equations \(2x + 3y - 9 = 0\) and \(4x + 6y - 18 = 0\) have infinitely many solutions.
   Reason (R): When the lines are coincident, the pair is dependent and consistent.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false. (d) A is false but R is true.

6. The solution of the pair \(x - 2y = 0\), \(3x + 4y - 20 = 0\) obtained graphically is
   (a) (4, 2) (b) (2, 1) (c) (0, 5) (d) (5, 0)

7. If a pair of linear equations is inconsistent, then the lines representing them are
   (a) intersecting (b) parallel (c) coincident (d) none of these

8. In the substitution method, if after substitution we obtain a statement like \(18 = 18\), the pair has
   (a) unique solution (b) no solution (c) infinitely many solutions (d) none of these

9. The pair \(5x - 8y + 1 = 0\) and \(\frac{3}{5}x - \frac{24}{5}y + \frac{3}{5} = 0\) has
   (a) no solution (b) unique solution (c) infinitely many solutions (d) none of these

10. Which of the following pairs is consistent?
    (a) \(2x - 3y = 8\), \(4x - 6y = 9\) (b) \(3x + 2y = 5\), \(2x - 3y = 7\)
    (c) \(x + y = 5\), \(2x + 2y = 10\) (d) none of these

Section B — Very Short Answer (2 marks each)

1. On comparing the ratios \(\frac{a_1}{a_2}\), \(\frac{b_1}{b_2}\) and \(\frac{c_1}{c_2}\), state whether the lines representing \(5x - 4y + 8 = 0\) and \(7x + 6y - 9 = 0\) intersect, are parallel or coincident.

2. Check whether the pair \(3x + 2y = 5\) and \(2x - 3y = 7\) is consistent or inconsistent.

3. Form the pair of linear equations for: “10 students took part in a quiz. Number of girls is 4 more than number of boys.” Find the number of boys and girls graphically.

4. Using the substitution method, solve \(x + y = 14\) and \(x - y = 4\).

5. State the condition under which the lines representing a pair of linear equations are coincident.

6. Write another linear equation so that \(2x + 3y - 8 = 0\) and the new equation represent parallel lines.

Section C — Short Answer (3 marks each)

1. Solve graphically: \(x + 3y = 6\) and \(2x - 3y = 12\).

2. Solve by substitution: \(7x - 15y = 2\) and \(x + 2y = 3\).

3. Half the perimeter of a rectangular garden is 36 m and length is 4 m more than width. Find the dimensions.

4. Solve by elimination: \(2x + 3y = 8\) and \(4x + 6y = 7\).

5. The difference between two numbers is 26 and one number is three times the other. Find the numbers by substitution.

Section D — Long Answer (5 marks each)

1. The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each saves ₹2000 per month, find their monthly incomes using the elimination method.

2. Form the pair of equations for: “Five years hence, Jacob’s age will be three times his son’s age. Five years ago, Jacob’s age was seven times his son’s age.” Solve by substitution and find their present ages.

3. Draw the graphs of \(x - y + 1 = 0\) and \(3x + 2y - 12 = 0\). Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis. Shade the triangular region and justify the type of pair.

Section E — Case/Source-Based (4 marks each)

Case 1

Champa went to a sale to purchase pants and skirts. She told her friends: “The number of skirts is two less than twice the number of pants. Also, the number of skirts is four less than four times the number of pants.”

(i) Form the pair of linear equations.
(ii) Solve the pair graphically.
(iii) State the number of pants and skirts purchased.
(iv) Verify whether the solution satisfies both statements.

Case 2

A lending library charges a fixed amount for the first three days and an additional charge per day thereafter. Saritha paid ₹27 for keeping a book for seven days while Susy paid ₹21 for keeping a book for five days.

(i) Form the pair of linear equations.
(ii) Solve by the elimination method.
(iii) Find the fixed charge and the charge for each extra day.
(iv) How much would a person pay for keeping a book for nine days?