CBSE Class 10 Mathematics Pairs of Linear Equations in Two Variables MCQs Set 10

Question. Which of the following is a linear equation?
(a) \(x - 2y = 7\)
(b) \(x^3 - 1 = 0\)
(c) \(x + \frac{6}{x} = 12\)
(d) \(\frac{x}{2} + \frac{3}{x} = 14\)
Answer: A

Question. Which of the following is a solution of \(2p + 3q = 5\)?
(a) \(p = -1, q = 1\)
(b) \(p = 1, q = -1\)
(c) \(p = 1, q = 1\)
(d) \(p = -1, q = -1\)
Answer: C

Question. Which of the following is the other name for a pair of linear equations in two variables?
(a) Consistent equations
(b) Simultaneous equations
(c) Inconsistent equations
(d) Dependent equations
Answer: B

Question. What is the condition that a system of simultaneous equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) must satisfy to have exactly one solution?
(a) \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\)
(b) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
(c) \(\frac{a_1}{a_2} = \frac{c_1}{c_2}\)
(d) \(\frac{b_1}{b_2} = \frac{c_1}{c_2}\)
Answer: B

Question. How many solutions do the equations satisfying \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) have?
(a) One
(b) Two
(c) Three
(d) Infinitely many
Answer: D

Question. What is a system of simultaneous equations called if it has no solution?
(a) Consistent system
(b) independent system
(c) Inconsistent system
(d) Dependent system
Answer: C

Question. How many solutions does the system of equations \(p + 2q = 4\) and \(2p + 4q - 12 = 0\) have?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: A

Question. What is a system of simultaneous equations called if its graph has intersecting lines?
(a) Inconsistent system
(b) Consistent system
(c) Dependent system
(d) Independent system
Answer: B

Question. What is the nature of the graphs of a dependent system?
(a) Parallel lines
(b) Perpendicular lines
(c) Intersecting lines
(d) Coincident lines
Answer: D

Question. What is the nature of the graphs of a system of linear equations with exactly one solution?
(a) Parallel lines
(b) Perpendicular lines
(c) Coincident lines
(d) Intersecting lines
Answer: D

Question. What is the number of solutions of the pair of linear equations \(4p - 6q + 18 = 0\) and \(2p - 3q + 9 = 0\)?
(a) 0
(b) 1
(c) 2
(d) Infinitely many
Answer: D

Question. Which of the following is a consistent system of simultaneous equations?
(a) \(m + 3n = 6\); \(2m + 6n = 12\)
(b) \(a + 3b = 6\); \(2a - 3b = 12\)
(c) \(x - 4y = 6\); \(2x - 8y = 12\)
(d) \(l - 2m = 6\); \(3l - 6m = 12\)
Answer: B

Question. Find the unique solution of the system of simultaneous equations \(2x - y = 2\) and \(4x - y = 4\).
(a) \(x = 0, y = 1\)
(b) \(x = 0, y = 0\)
(c) \(x = 1, y = 0\)
(d) \(x = 1, y = 1\)
Answer: C

Question. The sum of a two-digit number and the number obtained by reversing its digits is 154. If the digits differ by 4, find the number.
(a) 95
(b) 73
(c) 84
(d) 62
Answer: A

Question. Choose the dependent system from the following.
(a) \(m + n = 7\); \(3m + 3n = 21\)
(b) \(3x - 2y = 5\); \(2x - 3y = 7\)
(c) \(3x - 3y = 18\); \(x - y = 10\)
(d) \(2x + y = 6\); \(4x - 2y = 4\)
Answer: A

Question. Five years ago, a father's age was seven times his son's age. Five years from now, the father's age will be thrice the son's age. What are the respective present ages of father and son?
(a) 40 years, 10 years
(b) 10 years, 40 years
(c) 25 years, 5 years
(d) 30 years, 8 years
Answer: A

Question. Rajesh buys 7 books and 6 pens for ₹ 3800 and Amar buys 3 books and 5 pens of the same kind for ₹ 1750. What are the respective costs of a book and a pen?
(a) ₹ 350, ₹ 50
(b) ₹ 500, ₹ 75
(c) ₹ 250, ₹ 100
(d) ₹ 500, ₹ 50
Answer: D

Question. The system of simultaneous equations \(3m + n = 1\) and \((2k - 1)m + (k - 1)n = 2k + 1\), is inconsistent. What is the value of 'k'?
(a) 3
(b) 1
(c) 2
(d) 0
Answer: C

Question. What type of a system of equations is the pair of linear equations \(2x - 3y = 8\) and \(4x - 6y = 9\)?
(a) Consistent system
(b) Inconsistent system
(c) Dependent system
(d) Independent system
Answer: B

Question. If the pair of linear equations \(2x + ky - 3 = 0\) and \(6x + \frac{2}{3}y + 7 = 0\) has a unique solution, which of the following is true?
(a) \(k = \frac{2}{3}\)
(b) \(k \neq \frac{2}{3}\)
(c) \(k = \frac{2}{9}\)
(d) \(k \neq \frac{2}{9}\)
Answer: D

Question. If the length of a rectangle is increased by 2 m and breadth is reduced by 2 m, its area decreases by 28 sq. m. If the length is reduced by 1 m and the breadth is increased by 2 m, the area increases by 33 sq m. Find the actual measurements of the rectangle.
(a) \(l = 13m, b = 11 m\)
(b) \(l = 23 m, b = 11 m\)
(c) \(l = 23m, b = 20m\)
(d) \(l = 12 m, b = 10 m\)
Answer: B

Question. The side of a square is 4m more than the side of another square. The sum of their areas is 208 sq. m. What is the side of the larger square?
(a) 12m
(b) 8m
(c) 9m
(d) 5m
Answer: A

Question. Two numbers are in the ratio 2 :7. If 6 is added to each of the numbers, the ratio becomes 1 :3. Find the numbers.
(a) 14, 49
(b) 16, 56
(c) 18, 63
(d) 24, 84
Answer: D

Question. If the pair of equations \(3x + 5y = k\) and \(9x + 12y = 6\) has infinitely many solutions, which of the following is true?
(a) \(k = 2\)
(b) \(k = 6\)
(c) \(k \neq 6\)
(d) \(k = 3\)
Answer: A

Question. If the pair of linear equations \(3x + 5y = 3\) and \(6x + ky = 6\) do not have any solution, which of the following is true?
(a) \(k = 5\)
(b) \(k = 10\)
(c) \(k \neq 10\)
(d) \(k \neq 5\)
Answer: B

Question. When is the pair of linear equations \(7x - 3y = 4\); \(3x - \frac{k}{7}y = 4\) consistent?
(a) \(k = 9\)
(b) \(k = -9\)
(c) \(k \neq -9\)
(d) \(k \neq 7\)
Answer: C

Question. When does the pair of linear equations \(7x + ky = k\); \(14x + 2y = k + 1\) have infinitely many solutions?
(a) \(k = 1\)
(b) \(k \neq 1\)
(c) \(k = 2\)
(d) \(k = 4\)
Answer: A

Question. The pair of linear equations \(x + y = 3\); \(2x + 5y = 12\) has a unique solution \(x = x_1, y = y_1\). Find the value of \(x_1\).
(a) 1
(b) 2
(c) -1
(d) -2
Answer: A

Question. Which of the following solutions does the pair of linear equations \(x + 2y = 5\); \(3x + 12y = 10\) have?
(a) A unique solution
(b) No solution
(c) More than two solutions
(d) Infinitely many solutions
Answer: A

Question. If the sum of the ages (in years) of a father and his son is 65 and twice the difference of their ages (in years) is 50, what is the age of the father?
(a) 45 years
(b) 40 years
(c) 50 years
(d) 55 years
Answer: A

Question. Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹ 2850. Find the total cost of one chair and one table.
(a) ₹ 800
(b) ₹ 850
(c) ₹ 900
(d) ₹ 950
Answer: B

Question. If \(a + b = 5\) and \(3a + 2b = 20\), find \(3a + b\).
(a) 25
(b) 20
(c) 15
(d) 10
Answer: A

Question. Which of the respective values of 'x' and 'y' satisfy the following equations I and II?
(I) \(3x + y = 19\)
(II) \(x - y = 9\)
(a) 7, 2
(b) 7, -2
(c) -7, 2
(d) -7, -2
Answer: B