Quadratic Equations CBQs — CBSE Class 10 (2027) with step-wise solutions

Sample Quadratic Equations CBQs

1. Question 1MCQ1 mark
   A school is installing a rectangular array of solar panels on its roof. The array must cover 160 m², and its length must be 6 m more than its width. If the width of the array is x m, which equation models the situation?
   • x² + 6x − 160 = 0
   • x² − 6x − 160 = 0
   • x² + 6x + 160 = 0
   • 2x² + 6x − 160 = 0
   Show step-wise solution

   1. [1 mark] Length = (x + 6) m, so area x(x + 6) = 160 ⇒ x² + 6x − 160 = 0.

   Answer: x² + 6x − 160 = 0

2. Question 2Assertion-Reason1 mark
   A rectangular yoga mat has an area of 2 m², and its length is 1 m more than its width. Assertion (A): The width of the mat is 1 m. Reason (R): The equation x² + x − 2 = 0 has roots 1 and −2, and a width cannot be negative.
   • (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] (A): With width x m, x(x + 1) = 2 ⇒ x² + x − 2 = 0 ⇒ (x + 2)(x − 1) = 0 ⇒ x = 1 or −2; rejecting −2 gives width 1 m. A and R are true and R explains A.

   Answer: (A)

3. Question 3Short answer2 marks
   A housing society marks out a rectangular badminton court in its parking area. The length of the court is 6 m more than its breadth and the court covers an area of 112 m². Form a quadratic equation and find the length and breadth of the court.
   Show step-wise solution

   1. [1 mark] Let the breadth be x m; length = (x + 6) m. Then x(x + 6) = 112 ⇒ x² + 6x − 112 = 0.
   2. [1 mark] (x + 14)(x − 8) = 0 ⇒ x = 8 or x = −14; a breadth cannot be negative, so x = 8. Breadth = 8 m and length = 14 m.

   Answer: Breadth 8 m, length 14 m

4. Question 4Case study4 marks
   The notice-board team of a school has a photograph 6 cm long and 4 cm wide. To make it more visible, they reprint it after increasing both its length and its breadth by the same amount x cm. The area of the reprinted photograph is exactly twice the area of the original. Based on the above information, answer the following questions: (i) Find the area of the original photograph and the area of the reprint. (ii) Form the quadratic equation in x. (iii) Find x and the dimensions of the reprinted photograph.
   Show step-wise solution

   1. [1 mark] (i) Original area = 6 × 4 = 24 cm²; area of the reprint = 2 × 24 = 48 cm².
   2. [1 mark] (ii) (6 + x)(4 + x) = 48 ⇒ 24 + 10x + x² = 48 ⇒ x² + 10x − 24 = 0.
   3. [1 mark] (iii) (x + 12)(x − 2) = 0 ⇒ x = 2 (x = −12 is rejected as a length cannot be negative).
   4. [1 mark] (iii) x = 2 cm, so the reprint is 6 + 2 = 8 cm long and 4 + 2 = 6 cm wide.

   Answer: (i) 24 cm² and 48 cm² (ii) x² + 10x − 24 = 0 (iii) x = 2 cm; reprint 8 cm by 6 cm

5. Question 5Long answer5 marks
   A school bus takes students on a 120 km trip from the school to a nature camp. The driver notes that if the average speed of the bus were 20 km/h more, the trip would take 1 hour less. Let the usual average speed of the bus be x km/h. Based on the above information, answer the following questions: (a) Form a quadratic equation in x for this situation. (b) Find the usual average speed of the bus. (c) On the return trip the bus is slowed by traffic and its average speed is 10 km/h less than its usual speed. Find the total time taken for the round trip.
   Show step-wise solution

   1. [1 mark] (a) Usual time = 120/x h and time at the higher speed = 120/(x + 20) h, so 120/x - 120/(x + 20) = 1.
   2. [1 mark] Simplifying: 120(x + 20) - 120x = x(x + 20), i.e. 2400 = x² + 20x, so x² + 20x - 2400 = 0.
   3. [1 mark] (b) (x + 60)(x - 40) = 0, so x = 40 or x = -60; speed cannot be negative, so the usual speed = 40 km/h.
   4. [1 mark] (c) Time to the camp = 120 ÷ 40 = 3 h; return speed = 40 - 10 = 30 km/h, so return time = 120 ÷ 30 = 4 h.
   5. [1 mark] Total time for the round trip = 3 + 4 = 7 hours.

   Answer: Usual speed 40 km/h; round trip takes 7 hours

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