Pair of Linear Equations CBQs — CBSE Class 10 (2027) with step-wise solutions

Sample Pair of Linear Equations CBQs

1. Question 1MCQ1 mark
   On a town planner's coordinate map, one straight road lies along 2x + 3y = 12 and a second straight road lies along 4x + 6y = 30. Which statement about the two roads is correct?
   • They meet at exactly one point.
   • They never meet, because the lines are parallel.
   • They overlap completely, because the lines coincide.
   • They meet at exactly two points.
   Show step-wise solution

   1. [1 mark] a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = 12/30 = 2/5; since a1/a2 = b1/b2 ≠ c1/c2 the lines are parallel and the roads never meet.

   Answer: They never meet, because the lines are parallel.

2. Question 2Assertion-Reason1 mark
   Two friends describe the same purchase of x kg of rice and y kg of sugar. One writes 2x + 5y = 230 and the other writes 6x + 15y = 690. Assertion (A): The price of 1 kg of rice and 1 kg of sugar can be found uniquely from these two equations. Reason (R): For this pair, a1/a2 = b1/b2 = c1/c2.
   • (A) Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
   • (B) Both Assertion and Reason are true but Reason is NOT the correct explanation of Assertion.
   • (C) Assertion is true but Reason is false.
   • (D) Assertion is false but Reason is true.
   Show step-wise solution

   1. [1 mark] (D): 2/6 = 5/15 = 230/690 = 1/3, so R is true; the lines coincide and there are infinitely many solutions, so A is false.

   Answer: (D)

3. Question 3Short answer3 marks
   Ram and Sham are two friends, each owning a rectangular plot. Ram is the owner of a rectangular plot whose perimeter is 50 m and Sham is the owner of a rectangular plot whose perimeter is 100 m. Sham's plot has a length twice that of Ram's plot and a breadth 5 m more than that of Ram's plot. Write the linear equations for both the plots.
   Show step-wise solution

   1. [1 mark] Let x m be the length and y m be the breadth of Ram's plot; then 2x m is the length and (y + 5) m is the breadth of Sham's plot.
   2. [1 mark] Apply the perimeter formula: 2(x + y) = 50 and 2(2x + y + 5) = 100.
   3. [1 mark] Simplify to get x + y = 25 and 2x + y = 45.

   Answer: x + y = 25 and 2x + y = 45

4. Question 4Case study4 marks
   At a blood-donation camp in a college, each donor gives either a 350 mL bag or a 450 mL bag of blood, depending on the donor's body weight. On the camp day 40 donors gave blood and 16.4 litres (16400 mL) was collected in all. Let x donors give a 350 mL bag and y donors give a 450 mL bag. Based on the above information, answer the following questions: (i) Write the pair of linear equations that represents this information. (ii) Find the number of donors who gave a 450 mL bag. (iii) Find the number of donors who gave a 350 mL bag. The next camp has a target of 20 litres. If the same number of donors give 350 mL bags as this time, how many donors giving 450 mL bags are needed to reach the target exactly?
   Show step-wise solution

   1. [1 mark] (i) x + y = 40 and 350x + 450y = 16400, i.e. 7x + 9y = 328.
   2. [1 mark] (ii) Multiplying x + y = 40 by 7: 7x + 7y = 280; subtracting from 7x + 9y = 328: 2y = 48, so y = 24 donors.
   3. [1 mark] (iii) x = 40 - 24 = 16 donors gave 350 mL bags.
   4. [1 mark] (iii) 16 × 350 = 5600 mL; still needed = 20000 - 5600 = 14400 mL; 14400 ÷ 450 = 32 donors giving 450 mL bags.

   Answer: (i) x + y = 40, 350x + 450y = 16400 (ii) 24 donors (iii) 16 donors; 32 donors of 450 mL needed

5. Question 5Long answer5 marks
   A tourist houseboat on a river in Kerala travels 36 km downstream from a jetty to a village in 2 hours, and makes the return trip of 36 km upstream in 3 hours. The speed of the boat in still water and the speed of the stream remain constant during these trips. Let the speed of the boat in still water be x km/h and the speed of the stream be y km/h. Based on the above information, answer the following questions: (a) Form a pair of linear equations in x and y for this situation. (b) Find the speed of the boat in still water and the speed of the stream. (c) During the monsoon the speed of the stream doubles, while the speed of the boat in still water stays the same. How long will the 36 km upstream trip take then?
   Show step-wise solution

   1. [1 mark] (a) Downstream speed = x + y = 36 ÷ 2 = 18, so x + y = 18 ...(1). Upstream speed = x - y = 36 ÷ 3 = 12, so x - y = 12 ...(2).
   2. [1 mark] (b) Adding (1) and (2): 2x = 30, so x = 15.
   3. [1 mark] Substituting x = 15 in (1): y = 18 - 15 = 3. Speed of the boat in still water = 15 km/h and speed of the stream = 3 km/h.
   4. [1 mark] (c) New speed of the stream = 2 × 3 = 6 km/h, so the new upstream speed = 15 - 6 = 9 km/h.
   5. [1 mark] Time for the upstream trip = 36 ÷ 9 = 4 hours.

   Answer: Boat in still water 15 km/h, stream 3 km/h; monsoon upstream trip takes 4 hours

Study Pair of Linear Equations

• Pair of Linear Equations notes
• Pair of Linear Equations Topic Hub
• All Class 10 CBQs