Chapter 4: Quadratic Equations

Important questions — the ones the board asked most (20)

1. Q1Long answer5 marksAsked: 2026

   33. (A) A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains. OR (B) The sum of the areas of two squares is 640 m^2. If the difference in their perimeters is 64 m, find the sides of the two squares.

   

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   Model answer · Quadratic equations, speed problems

   1. 1.The speed of the faster train is 50 km/h and the speed of the slower train is 40 km/h. The working is shown below. Given: the journey is 200 km, the faster train takes 1 hour less than the slower train, and the slower train's speed is 10 km/h less than the faster train's speed. Let the speed of the faster train be x km/h. Then the speed of the slower train = (x − 10) km/h. Time = distance ÷ speed, so the slower train takes 200/(x − 10) h and the faster train takes 200/x h. According to the question: 200/(x − 10) − 200/x = 12 marks
   2. 2.Multiplying both sides by x(x − 10): 200x − 200(x − 10) = x(x − 10) 2000 = x² − 10x So x satisfies the quadratic equation x² − 10x − 2000 = 01 mark
   3. 3.Factorising by splitting the middle term: x² − 50x + 40x − 2000 = 0 x(x − 50) + 40(x − 50) = 0 (x − 50)(x + 40) = 01 mark
   4. 4.Equating each factor to zero: x − 50 = 0, so x = 500.5 marks
   5. 5.x + 40 = 0 gives x = −40, which is rejected because a speed cannot be negative. Hence, speed of faster train = 50 km/h and speed of slower train = 50 − 10 = 40 km/h.0.5 marks

   • Begin with "Let the speed of the faster train be x km/h" and write the slower train's speed as (x − 10) km/h before you form the equation.
   • Write the equation 200/(x − 10) − 200/x = 1 on its own line, with the slower train's time first.
   • State that x = −40 is rejected because a speed cannot be negative, and give both speeds in km/h.

   Marks split as in the board’s published marking scheme

2. Q2Long answer5 marksAsked: 2026

   (a) A person on a tour has Rs 4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by Rs 70. Find the original duration of the tour. OR (b) The area of a right-angled triangle is 600 cm^2. If the base of the triangle exceeds the altitude by 10 cm, find all the three dimensions of the triangle.

   

   Medha Ma’am

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   Model answer · Quadratic equations word problem

   1. 1.(a) Given: total money for expenses = ₹ 4200. If the tour is extended by 3 days, the daily expenses fall by ₹ 70. Let the original duration of the tour be x days. Original daily expenses = ₹ 4200/x New duration = (x + 3) days, so new daily expenses = ₹ 4200/(x + 3) The daily expenses are cut down by ₹ 70, so 4200/x − 4200/(x + 3) = 702 marks
   2. 2.Taking the LCM on the left-hand side: 4200[(x + 3) − x] / [x(x + 3)] = 70 4200 × 3 = 70x(x + 3) 12600 = 70(x² + 3x) Dividing both sides by 70: x² + 3x = 180 ⟹ x² + 3x − 180 = 0, which is a quadratic equation in x.1.5 marks
   3. 3.Solving by factorisation, splitting the middle term: x² + 15x − 12x − 180 = 0 x(x + 15) − 12(x + 15) = 0 ⟹ (x + 15)(x − 12) = 01 mark
   4. 4.So the roots of the equation are x = −15 or x = 12. The duration of a tour cannot be negative, so x = −15 is rejected. Therefore x = 12. ∴ Original duration of the tour = 12 days.0.5 marks

   • Write both daily expenses, ₹ 4200/x and ₹ 4200/(x + 3), before you write the equation 4200/x − 4200/(x + 3) = 70.
   • Show the factorisation step (x + 15)(x − 12) = 0 on its own line.
   • Write that x = −15 is rejected because a duration cannot be negative, then finish with "Original duration of the tour = 12 days".

   Marks split as in the board’s published marking scheme

3. Q3Long answer5 marksAsked: 2026

   In a flight of 600 km, an aircraft slowed down its speed due to bad weather. Its average speed for the trip reduced by 200 km/h from its usual speed and time of flight increased by 30 minutes. Find the scheduled duration of the flight. OR Two pipes are used to fill a swimming pool. If the pipe of the larger diameter is used for 4 hours and the pipe of the smaller diameter for 9 hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool, separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.

   

   Medha Ma’am

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   Model answer · Quadratic equations, speed-time problems

   1. 1.Given: distance of the flight = 600 km, the speed is reduced by 200 km/h and the time increases by 30 minutes = 1/2 hour. Let the usual speed of the aircraft be x km/h. Usual time of flight = 600/x hours. Reduced speed = (x − 200) km/h, so the time taken at reduced speed = 600/(x − 200) hours. The time increased by 1/2 hour, so 600/(x − 200) − 600/x = 1/22 marks
   2. 2.Multiplying both sides by 2x(x − 200): 1200x − 1200(x − 200) = x(x − 200) 240000 = x² − 200x So the speed satisfies the quadratic equation x² − 200x − 240000 = 01 mark
   3. 3.Solving by factorisation, splitting the middle term: x² − 600x + 400x − 240000 = 0 x(x − 600) + 400(x − 600) = 0 (x − 600)(x + 400) = 01 mark
   4. 4.Equating each factor to zero, the roots are x = 600 or x = −400. Since x is the speed of the aircraft, it cannot be negative, so x = −400 is ignored. Therefore x = 600, i.e. the usual speed is 600 km/h.0.5 marks
   5. 5.Scheduled duration of the flight = 600/600 = 1 hour.0.5 marks

   • Change 30 minutes into 1/2 hour before you write 600/(x − 200) − 600/x = 1/2.
   • Write why x = −400 is rejected: the speed cannot be negative.
   • End with the time, not the speed: "Scheduled duration of the flight = 600/600 = 1 hour".

   Marks split as in the board’s published marking scheme

4. Q4Long answer5 marksAsked: 2026

   (a) In a class test, the sum of Anamika's marks obtained in Maths and Science is 30. Had she got 2 marks more in Maths and 3 marks less in Science, the product of the marks would have been 210. Find the marks she got in the two subjects. OR (b) The length of hypotenuse (in cm) of a right-angled triangle is 6 cm more than twice the length of its shortest side. If the length of its third side is 6 cm less than thrice the length of its shortest side, find the dimensions of the triangle.

   How the model answer begins

   1. Answering part (a). Let the marks Anamika got in Maths be x. Since the sum of her marks in Maths and Science is 30, her marks in Science = 30 – x. With 2 marks more in Maths, her Maths marks = x + 2. With 3 marks less…

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5. Q5Long answer5 marksAsked: 2025

   (a) The sum of areas of two squares is 2650 cm^2. If the sum of their perimeters is 280 cm, find the sides of the two given squares. OR (b) Express the equation 1/x - 1/(x-2) = 3, (x != 0, 2) as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.

   How the model answer begins

   1. (b) Given: 1/x − 1/(x − 2) = 3, where x ≠ 0, 2. We take the LCM x(x − 2) on the left-hand side: (x − 2 − x) / (x(x − 2)) = 3 −2 / (x² − 2x) = 3

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6. Q6Long answer5 marksAsked: 2025

   (a) The sum of the areas of two squares is 2650 cm^2. If the sum of their perimeters is 280 cm, find the sides of the two squares. or (b) Express the equation 1/x - 1/(x-2) = 3, (x not equal to 0, 2) in the standard form of a quadratic equation. Hence, find the roots of the quadratic equation so obtained.

   How the model answer begins

   1. Given: 1/x − 1/(x − 2) = 3, where x ≠ 0, 2. Taking the LCM x(x − 2) on the LHS: (x − 2 − x) / (x(x − 2)) = 3 −2 / (x² − 2x) = 3

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7. Q7Long answer5 marksAsked: 2025

   (a) Find two consecutive odd integers, sum of whose squares is 290. OR (b) A charity trust decides to build a rectangular hall having an area of 300 m^2. The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.

   How the model answer begins

   1. (a) Given: the sum of the squares of two consecutive odd integers is 290. Two consecutive odd integers differ by 2. Let the two consecutive odd integers be x and x + 2.

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8. Q8Long answer5 marksAsked: 2025

   (A) It is given that p^2x^2 + (p^2 - q^2)x - q^2 = 0; (p != 0) (i) Show that the discriminant (D) of above equation is a perfect square. (ii) Find the roots of the equation. OR (B) Three consecutive positive integers are such that the sum of the square of smallest and product of other two is 67. Find the numbers, using quadratic equation.

   How the model answer begins

   1. (A) (i) The given quadratic equation is p²x² + (p² − q²)x − q² = 0, where p ≠ 0. Comparing it with ax² + bx + c = 0: a = p², b = p² − q², c = −q² The discriminant is D = b² − 4ac D = (p² − q²)² − 4 × p² × (−q²) D = (p²…

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9. Q9Long answer5 marksAsked: 2024

   (a) A 2-digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number. OR (b) The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 cm^2. Find the sides of the squares.

   How the model answer begins

   1. The required number is 27. Here is how it is found. Given: the product of the digits of a 2-digit number is 14. When 45 is added to the number, its digits are reversed. Let the tens digit be x and the units digit be y.…

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10. Q10Long answer5 marksAsked: 2024

    (a) Find the value of 'k' for which the quadratic equation (k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0, k ≠ - 1 has real and equal roots. OR (b) The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.

    How the model answer begins

    1. Answering (b): The present age of the son is 4 years and the present age of the man is 32 years. Working: Given: the age of the man is twice the square of the age of his son. Let the present age of the son = x years.…

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11. Q11Long answer5 marksAsked: 2024

    A cottage industry produces a certain number of pottery articles in a day. It was observed that on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90, find the number of articles produced and the cost of each article.

    How the model answer begins

    1. The cottage industry produced 6 pottery articles that day, and each article cost ₹ 15. This is worked out below. Let the number of articles produced on that day = x. The cost of each article is 3 more than twice the…

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12. Q12Long answer5 marksAsked: 2024

    The area of a rectangular plot is 528 m^2. The length of the plot (in metres) is one more than twice the breadth. Find the length and breadth of the plot. Also, find the cost of levelling the plot at the rate of Rs 80 per square metre.

    How the model answer begins

    1. Let the breadth of the rectangular plot be x metres. Then, its length should be (2x + 1) metres.

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13. Q13Long answer5 marksAsked: 2023

    35. (a) A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what is its first average speed ? OR (b) Two pipes together can fill a tank in 15/8 hours. The pipe with larger diameter takes 2 hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.

    How the model answer begins

    1. Answering option (b): the pipe with smaller diameter fills the tank alone in 5 hours, and the pipe with larger diameter fills it alone in 3 hours. This is shown below. Let the time taken by the smaller diameter pipe to…

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14. Q14Long answer5 marksAsked: 2023

    (a) The sum of reciprocals of Roohi's age (in years) 3 years ago and 5 years hence from now is 1/3. Find her present age. OR (b) A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

15. Q15Long answer5 marksAsked: 2024

    (a) Find the value of 'k' for which the quadratic equation (k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0, k != -1 has real and equal roots. OR (b) The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.

    How the model answer begins

    1. The value of k for which the quadratic equation has real and equal roots is k = 3. It is found as follows. Given: (k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0, k ≠ -1 Here a = k + 1, b = -6(k + 1), c = 3(k + 9) For real and…

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16. Q16Long answer5 marksAsked: 2026

    Venkat can row his boat at 12 km/h in still water. He takes the tourists 15 km upstream and 18 km downstream and takes 3 hours for the entire trip. Find the speed of the stream.

17. Q17Long answer5 marksAsked: 2026

    By selling an article for ₹ 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

    How the model answer begins

    1. The cost price of the article is ₹ 120 with a loss of ₹ 72, or ₹ 80 with a loss of ₹ 32. The working is shown below. Given: selling price (SP) = ₹ 48, and the loss percent equals half of the cost price. Let the cost…

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18. Q18Long answer5 marksAsked: 2026

    35. Two water taps together can fill a tank in 8 8/9 hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

    How the model answer begins

    1. The tap of smaller diameter fills the tank alone in 20 hours and the tap of larger diameter in 16 hours. The working is shown below. Given: the two taps together fill the tank in 8 8/9 = 80/9 hours. Let the tap of…

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19. Q19Long answer5 marksAsked: 2026

    (a) ABCD is a rectangle of dimensions 80 cm * 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x. OR (b) A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

20. Q20Long answer5 marksAsked: 2026

    (a) ABCD is a rectangle of dimensions 80 cm x 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x. OR (b) A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

    How the model answer begins

    1. The value of x is 10 cm. Here is how it is found. ABCD is a rectangle with AB = 80 cm and BC = 60 cm. The rectangle PQRS is drawn inside it, leaving a strip of equal width x cm along every edge of ABCD. The strip takes…

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49 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   The roots of the quadratic equation (x - 1)^2 = 16 are :

   • (A) 5, 3
   • (B) 4, - 4
   • (C) 5, - 3 ✓
   • (D) - 5, 3

   Answer: (C)

2. 2. CBSE 2026

   If the roots of the quadratic equation sqrt(3)x^2 - kx + 2sqrt(3) = 0 are real and equal, then the value(s) of k is/are :

   • (A) +- sqrt(24) ✓
   • (B) 0
   • (C) 4
   • (D) - 5

   Answer: (A)

3. 3. CBSE 2026

   The roots of the quadratic equation 4x^2 - (a - 1)^2 = 0 are :

   • (A) a - 1, a + 1
   • (B) (a - 1)/2, (-a + 1)/2 ✓
   • (C) (a - 1)/2, (-a - 1)/2
   • (D) ± (a - 1)

   Answer: (B)

4. 4. CBSE 2026

   If the roots of the equation 2kx^2 - 6x + 3 = 0 are real and equal, then the value of k is :

   • (A) 3/2 ✓
   • (B) 1/2
   • (C) -3/2
   • (D) 2

   Answer: (A)

5. 5. CBSE 2026

   The value of k for which the equation 5x^2 - 2x + k = 0 has equal real roots, is :

   • (A) 5
   • (B) -1/5
   • (C) 1/5 ✓
   • (D) 0

   Answer: (C)

6. 6. CBSE 2026

   The value of k for which the equation 2kx^2 - 6x + 3 = 0 has real and equal roots, is :

   • (A) 3/2 ✓
   • (B) 1/2
   • (C) -3/2
   • (D) 2

   Answer: (A)

7. 7. CBSE 2026

   The value of k for which the equation kx^2 - 6x - 4 = 0 has real and equal roots, is

   • (A) 9/4
   • (B) -4
   • (C) -9/4 ✓
   • (D) -2

   Answer: (C)

8. 8. CBSE 2026

   The value of p for which roots of the quadratic equation x^2 - px + 6 = 0 are rational, is

   • (A) 1
   • (B) -5 ✓
   • (C) 25
   • (D) sqrt(5)

   Answer: (B)

9. 9. CBSE 2026

   The value of m for which the quadratic equation 3x^2 - 7x + m = 0 has real and equal roots, is

   • (A) 7
   • (B) 49/12 ✓
   • (C) 49/3
   • (D) 4

   Answer: (B)

10. 10. CBSE 2026

    The roots of the quadratic equation x^2 + 9 = 0 are

    • (A) real and equal
    • (B) not real ✓
    • (C) real and negative of each other
    • (D) rational numbers

    Answer: (B)

Questions parents and students ask

How many marks does "Quadratic Equations" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 5 marks and about 2 questions a year; by marks it ranks 9 in Maths.

Which are the most important (VVI) questions of "Quadratic Equations"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 69 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.