CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables MCQs Set 32

Mathematics Objective Questions and Answers: Chapter 03 Pair of Linear Equations in Two Variables

Explore reliable objective questions for Chapter 03 Pair of Linear Equations in Two Variables tailored for Class 10 learners. Utilizing these Mathematics multiple-choice formats ensures thorough preparation and strengthens problem-solving speed for upcoming school assessments.

Download Chapter 03 Pair of Linear Equations in Two Variables MCQs with Answers

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Multiple Choice Questions

Question 1: The line 2y + 4 = 0 is:
(a) parallel to the y-axis, 2 units to its left
(b) parallel to the x-axis, 4 units below it
(c) parallel to the x-axis, 2 units below it
(d) passing through the origin
Show Answer & Explanation

Answer: (c) parallel to the x-axis, 2 units below it

Explanation:
1. 2y + 4 = 0 gives y = -2: a horizontal line 2 units below the x-axis.

Teacher's Note:
1. Each linear equation is a straight line. Two points fix a line; the solution of the pair is the point where the two lines meet.

Question 2: The graphs of x - y = 1 and 2x + y = 8 meet at the point:
(a) (2, 3)
(b) (1, 6)
(c) (3, 2)
(d) (4, 0)
Show Answer & Explanation

Answer: (c) (3, 2)

Explanation:
1. Adding the equations: 3x = 9, so x = 3 and y = 2. Check: 3 - 2 = 1 and 6 + 2 = 8.

Teacher's Note:
1. Each linear equation is a straight line. Two points fix a line; the solution of the pair is the point where the two lines meet.

Question 3: The lines x - y = 1 and 2x + y = 8 and the y-axis bound a triangle of area:
(a) 9 square units
(b) 13.5 square units
(c) 27 square units
(d) 6.75 square units
Show Answer & Explanation

Answer: (b) 13.5 square units

Explanation:
1. The lines meet the y-axis at (0, -1) and (0, 8), so the base on the y-axis is 9. Their intersection (3, 2) is at distance 3 from the y-axis. Area = (1/2)(9)(3) = 13.5.

Teacher's Note:
1. Each linear equation is a straight line. Two points fix a line; the solution of the pair is the point where the two lines meet.

Question 4: The graphs of 2x + 3y - 6 = 0, 4x - 3y - 6 = 0, x = 2 and y = 2/3 meet in:
(a) exactly one point
(b) four points
(c) two points
(d) infinitely many points
Show Answer & Explanation

Answer: (a) exactly one point

Explanation:
1. Adding the first two: 6x = 12, so x = 2 and then y = 2/3. This point also lies on x = 2 and y = 2/3, so all four lines pass through the single point (2, 2/3).

Teacher's Note:
1. Each linear equation is a straight line. Two points fix a line; the solution of the pair is the point where the two lines meet.

Question 5: Two numbers are such that twice the first plus thrice the second is 89 and four times the first exceeds five times the second by 13. The numbers are:
(a) 15 and 22
(b) 20 and 17
(c) 23 and 14
(d) 22 and 15
Show Answer & Explanation

Answer: (d) 22 and 15

Explanation:
1. 2x + 3y = 89 and 4x - 5y = 13. Doubling the first: 4x + 6y = 178; subtracting: 11y = 165, so y = 15 and x = 22.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 6: The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator increased by 1, the denominator becomes 8 times the numerator. The fraction is:
(a) \( \frac49 \)
(b) \( \frac58 \)
(c) \( \frac26 \)
(d) \( \frac37 \)
Show Answer & Explanation

Answer: (d) \( \frac37 \)

Explanation:
1. Let the fraction be x/y with y - x = 4 and y + 1 = 8(x - 2). Then x + 5 = 8x - 16, so x = 3 and y = 7.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 7: A two-digit number has digit sum 12, and if 18 is subtracted from it the digits are reversed. The number is:
(a) 57
(b) 66
(c) 75
(d) 84
Show Answer & Explanation

Answer: (c) 75

Explanation:
1. Tens digit y, units digit x: x + y = 12 and 10y + x - 18 = 10x + y, so y - x = 2. Hence y = 7 and x = 5, and the number is 75.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 8: The sum of a two-digit number and the number formed by reversing its digits is 165, and the digits differ by 3. The number could be:
(a) 69 only
(b) 69 or 96
(c) 78 or 87
(d) 59 or 95
Show Answer & Explanation

Answer: (b) 69 or 96

Explanation:
1. The digit sum is 165/11 = 15. With difference 3 the digits are 9 and 6, giving 69 or 96.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 9: Six years hence a man's age will be three times his son's age, and three years ago he was nine times as old as his son. The man's present age is:
(a) 30 years
(b) 36 years
(c) 33 years
(d) 45 years
Show Answer & Explanation

Answer: (a) 30 years

Explanation:
1. x + 6 = 3(y + 6) and x - 3 = 9(y - 3). The equations give x = 3y + 12 and x = 9y - 24. Equating, 6y = 36, so y = 6 and x = 30.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 10: A boat goes 12 km upstream and 40 km downstream in 8 hours, and 16 km upstream and 32 km downstream in the same time. The speeds of the boat in still water and of the stream are:
(a) 6 km/h and 2 km/h
(b) 8 km/h and 2 km/h
(c) 6 km/h and 4 km/h
(d) 10 km/h and 2 km/h
Show Answer & Explanation

Answer: (a) 6 km/h and 2 km/h

Explanation:
1. Put u = 1/(x - y) and v = 1/(x + y): 12u + 40v = 8 and 16u + 32v = 8. Solving, u = 1/4 and v = 1/8, so x - y = 4 and x + y = 8. Hence x = 6 and y = 2.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 11: Ramesh travels 760 km, taking 8 hours if he goes 160 km by train and the rest by car, and 12 minutes more if he goes 240 km by train and the rest by car. The speed of the train is:
(a) 100 km/h
(b) 60 km/h
(c) 90 km/h
(d) 80 km/h
Show Answer & Explanation

Answer: (d) 80 km/h

Explanation:
1. 160/x + 600/y = 8 and 240/x + 520/y = 41/5. Solving, 1/x = 1/80 and 1/y = 1/100: the train goes 80 km/h and the car 100 km/h.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 12: Two cars start from points 90 km apart. Going in the same direction they meet in 9 hours, and going towards each other they meet in 9/7 hours. The speeds of the cars are:
(a) 50 km/h and 20 km/h
(b) 45 km/h and 25 km/h
(c) 40 km/h and 30 km/h
(d) 60 km/h and 10 km/h
Show Answer & Explanation

Answer: (c) 40 km/h and 30 km/h

Explanation:
1. Same direction: 9(x - y) = 90, so x - y = 10. Opposite directions: (9/7)(x + y) = 90, so x + y = 70. Hence x = 40 and y = 30.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 13: In a cyclic quadrilateral ABCD, angle A = (2x + 11) degrees, B = (y + 12), C = (3y + 6) and D = (5x - 25). The angle D measures:
(a) 105 degrees
(b) 135 degrees
(c) 75 degrees
(d) 45 degrees
Show Answer & Explanation

Answer: (b) 135 degrees

Explanation:
1. Opposite angles add to 180: 2x + 3y = 163 and 5x + y = 193. Solving, x = 32 and y = 33. So D = 5(32) - 25 = 135 degrees (A = 75, B = 45, C = 105).

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 14: A vessel holds 24 litres of milk and 6 litres of water, and a second holds 15 litres of milk and 10 litres of water. Quantities taken from both are mixed so that the new vessel has 25 litres of milk and 10 litres of water. The quantity taken from the first vessel is:
(a) 15 litres
(b) 20 litres
(c) 25 litres
(d) 10 litres
Show Answer & Explanation

Answer: (b) 20 litres

Explanation:
1. Take x and y litres. Milk: (4/5)x + (3/5)y = 25 and water: (1/5)x + (2/5)y = 10. Solving, x = 20 and y = 15.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 15: A lady has 25 p and 50 p coins, 40 coins in all, worth Rs. 12.50. The number of 25 p coins is:
(a) 30
(b) 10
(c) 20
(d) 25
Show Answer & Explanation

Answer: (a) 30

Explanation:
1. x + y = 40 and 25x + 50y = 1250. Then 25x + 50(40 - x) = 1250 gives 2000 - 25x = 1250, so x = 30 (and y = 10).

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 16: In a class, if one student more stands in each row there are 2 rows fewer, and if one student fewer stands in each row there are 3 rows more. The number of students is:
(a) 40
(b) 72
(c) 50
(d) 60
Show Answer & Explanation

Answer: (d) 60

Explanation:
1. Rows x and students per row y: (y + 1)(x - 2) = xy and (y - 1)(x + 3) = xy. These simplify to x - 2y = 2 and x - 3y = -3. Subtracting, y = 5 and x = 12, so the class has 12 x 5 = 60 students.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 17: A man's monthly salary was Rs. 4500 after 5 years of service and Rs. 5550 after 12 years, with a fixed yearly increment. His starting salary was:
(a) Rs. 3500
(b) Rs. 4000
(c) Rs. 3750
(d) Rs. 3600
Show Answer & Explanation

Answer: (c) Rs. 3750

Explanation:
1. x + 5y = 4500 and x + 12y = 5550. Subtracting: 7y = 1050, y = 150 and x = 3750.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 18: A dealer sold a VCR and a TV for Rs. 38560, with profits of 12 percent on the VCR and 15 percent on the TV. Had the profits been 15 percent on the VCR and 12 percent on the TV, he would have got Rs. 38620. The cost price of the TV is:
(a) Rs. 18000
(b) Rs. 15000
(c) Rs. 16000
(d) Rs. 17000
Show Answer & Explanation

Answer: (c) Rs. 16000

Explanation:
1. 1.12x + 1.15y = 38560 and 1.15x + 1.12y = 38620. Adding: 2.27(x + y) = 77180, so x + y = 34000. Subtracting: 0.03(x - y) = 60, so x - y = 2000. Hence x = 18000 (VCR) and y = 16000 (TV).

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 19: The denominator of a fraction exceeds its numerator by 7. If 4 is added to both, the fraction becomes 1/2. The fraction is:
(a) \( \frac{4}{11} \)
(b) \( \frac{3}{10} \)
(c) \( \frac{2}{9} \)
(d) \( \frac{5}{12} \)
Show Answer & Explanation

Answer: (b) \( \frac{3}{10} \)

Explanation:
1. With x/(x + 7): (x + 4)/(x + 11) = 1/2, so 2x + 8 = x + 11 and x = 3. The fraction is 3/10.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 20: A number divided by the sum of its digits gives quotient 6 and remainder 3. When the digits are interchanged and the number is divided by the digit sum, the quotient is 4 and remainder 9. The number is:
(a) 75
(b) 57
(c) 63
(d) 84
Show Answer & Explanation

Answer: (a) 75

Explanation:
1. Digits a, b: 10a + b = 6(a + b) + 3 and 10b + a = 4(a + b) + 9. These give 4a - 5b = 3 and 2b - a = 3, so b = 5 and a = 7. The number is 75.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 21: 2 men and 3 boys finish a piece of work in 8 days, while 3 men and 2 boys finish it in 6 days. One boy alone would take:
(a) 20 days
(b) 100 days
(c) 60 days
(d) 120 days
Show Answer & Explanation

Answer: (d) 120 days

Explanation:
1. Let a man do 1/m and a boy 1/b of the work per day: 2/m + 3/b = 1/8 and 3/m + 2/b = 1/6. Solving, m = 20 and b = 120.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

Question 22: The sum of two numbers is 18 and the sum of their reciprocals is 1/4. The numbers are:
(a) 14 and 4
(b) 9 and 9
(c) 15 and 3
(d) 12 and 6
Show Answer & Explanation

Answer: (d) 12 and 6

Explanation:
1. x + y = 18 and 1/x + 1/y = 18/(xy) = 1/4, so xy = 72. The numbers satisfy t^2 - 18t + 72 = 0, giving 12 and 6.

Teacher's Note:
1. Name the unknown quantities, translate each condition of the problem into an equation, solve the pair and state the answer with units.

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Practice MCQs for Class 10 Mathematics Chapter 03 Pair of Linear Equations in Two Variables

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