Q1 1 mark MCQ p. 33
Which of these quadratic equations has equal roots?
Reveal official answer
Correct option: 3 — x² + 2x + 1 = 0
Question Bank · Mathematics
Quadratic Equations
Source: CBSE · Competency Focused Practice Questions, Mathematics (Volume 2), Grade 10, pp. 33–45. Questions and answers are reproduced verbatim from the official CBSE document.
Try the question yourself before revealing the answer — that's how marks stick.
Q2 1 mark MCQ p. 33
The quadratic equation x² − 7x + 10 = 0 can also be written as __________.
Reveal official answer
Correct option: 1 — (x − 5)(x − 2) = 0
Q3 1 mark MCQ p. 33
The quadratic equation x² + 8x + h = 0 has equal roots.Which of these is the value of h?
Reveal official answer
Correct option: 4 — 16
Q4 1 mark MCQ p. 33
The sum of a number z and its reciprocal is 4.Which of these correctly represents the above statement?
Reveal official answer
Correct option: 3 — z² + 1 = 4z
Q5 1 mark MCQ p. 33
Which among the following equations can be categorized as a quadratic equation?
Reveal official answer
Correct option: 3 — 3x² + 5x = 9x(x − 3)
Q6 1 mark MCQ p. 33
Look at the quadratic equation below:−y² + 8y − 18 = 0Which of these can be said about the nature of roots of the above quadratic equation?
Reveal official answer
Correct option: 3 — No real roots
Q7 1 mark MCQ p. 33
Which of these equations will definitely have NO real roots, for any value of a and b other than zero?i) 2x² − bx − b² = 0ii) a²x − ax + 2 = 0iii) x² + ax + b = 0
Reveal official answer
Correct option: 2 — only ii
Q8 1 mark MCQ p. 34
In the quadratic equation 6x² − gx + 2 = 0, the sum of the roots is equal to three times their product.What is the value of g?
Reveal official answer
Correct option: 4 — 6
Q9 1 mark Written p. 34
Given below is an equation, where p is a real number.px² + 4x + 4 = 0For what value of p will this equation not be a quadratic equation? Justify your answer.
Reveal official answer
• Mentions that for p = 0, the equation will not be a quadratic equation. [0.5]
• Reasons that as p = 0 will make the term with degree 2 zero, the equation will turn into a linear equation. [0.5]
Q10 1 mark Written p. 34
Frame a quadratic equation with roots that add up to 15 and has a product of 28.
Reveal official answer
• Frames a quadratic equation using x² − (α + β)x + αβ = 0 where α and β are roots of the quadratic equation. For example, x² − 15x + 28 = 0. [1]
Q11 2 marks Written p. 34
One of the roots of the quadratic equation ax² + 4x + a = 0 is (−2).Find the value of a. Show your work.
Reveal official answer
• Substitutes x with (−2) in the given equation as: (−2)²a + 4(−2) + a = 0 => 4a − 8 + a = 0 [1]
• Solves the above equation to find the value of a as 8/5. [1]
Q12 2 marks Written p. 34
Raveena and Siya had Rs 800 together. Each of them spent Rs 100 from their respective amounts. Now the product of the amount left with them is Rs 9000.Frame a quadratic equation to determine the money each one of them had initially.
Reveal official answer
• Assumes that Raveena had Rs x initially, hence Siya will have Rs (800 − x). Mentions that as each spent Rs 100, money left with them will be Rs (x − 100) and Rs (700 − x) respectively, thus frames the quadratic equation as: (x − 100)(700 − x) = 9000 [1]
• Simplifies the above equation as: x² − 800x + 79000 = 0 [1]
Q13 2 marks Written p. 34
The sum of the squares of two consecutive even numbers is 340.Express the above statement as a quadratic equation in the standard form.
Reveal official answer
• Assumes one number to be 2x and other to be (2x + 2). Frames the following quadratic equation: (2x)² + (2x + 2)² = 340 [1]
• Simplifies the above equation into the standard form as: 8x² + 8x − 336 = 0 or x² + x − 42 = 0 (Award full marks if the other number is assumed to be (2x − 2) instead of (2x + 2), hence final equation being x² − x − 42 = 0.) [1]
Q14 2 marks Written p. 34
Find the value(s) of g for which the equation 3x² + 3x + gx + 3 = 0 have real and equal roots. Show your work.
Reveal official answer
• Mentions that for the equation to have real and equal roots, value of discriminant should be 0 and writes: (3 + g)² − (4 × 3 × 3) = 0 [1]
• Solves the above equation and finds the values of g as 3 and (−9). [1]
Q15 2 marks Written p. 34
A store sells two types of toys: standard toys and premium toys. A customer buys a total of 16 toys of both the types such that the product of the number of toys of each type is 28.Frame a quadratic equation representing the above scenario. Solve it to find the number of toys of each type purchased by the customer. Show your work.
Reveal official answer
• Assumes the number of standard toys to be x and premium toys be (16 − x). Frames the quadratic equation as: x(16 − x) = 28 => x² − 16x + 28 = 0 [1]
• Solves the above equation to find the value of x as 14 and 2. Concludes that the number of standard toys and premium toys are 14 and 2 respectively or vice versa. [1]
Q16 2 marks Written p. 35
A ball is thrown vertically upward from the ground. The maximum height, h metres, the ball reaches with respect to time, t seconds, is represented by the polynomial h(t) = −5t² + 30t.How long does it take for the ball to hit the ground? Show your work.
Reveal official answer
• Writes the equation for height when the ball hits the ground as: −5t² + 30t = 0 [1]
• Simplifies the above equation as: t(−5t + 30) = 0 => t = 0 or 6 seconds. Writes that the ball takes 6 seconds to hit the ground. [1]
Q17 2 marks Written p. 35
What is the maximum and minimum number of real roots a quadratic equation can have? Justify your answer.
Reveal official answer
• Writes that the maximum number of real roots a quadratic equation can have is 2 and justifies the answer. For example, writes that when the value of discriminant is non negative, it has 2 real roots. [1]
• Writes that the minimum number of real roots a quadratic equation can have is 0 and justifies the answer. For example, writes that when the value of discriminant is negative, it has no real roots. [1]
Q18 2 marks Written p. 35
Write a quadratic equation with roots as 3 and 5 and coefficient of x as (−8). Show your work.
Reveal official answer
• Assumes a quadratic equation of the form ax² + bx + c = 0 and writes that: b = −8, 3 + 5 = −b/a, 3 × 5 = c/a [1]
• Solves the two equations to find the values of a and c as 1 and 15 respectively. The working may look as follows: 3 + 5 = −b/a = 8/a => a = 8/8 = 1 => 3 × 5 = c/a => c = 15. Writes the final quadratic equation as x² − 8x + 15 = 0. [1]
Q19 3 marks Written p. 35
The length of a rectangular garden is 6 metres more than its width. The total area of the garden is 72 m².Find the width and length of the garden. Show your work.
Reveal official answer
• Takes the width of the rectangular garden as x m. Writes the length of the rectangular garden as (x + 6) m. Equates the area of the rectangular garden to 72 m² as: x(x + 6) = 72 => x² + 6x − 72 = 0 [1]
• Solves the above quadratic equation to find the value of x as 6 and (−12). [1]
• Rejects x = (−12) and finds the width of garden as 6 meters. Thus finds the length of the garden as 12 meters. [1]
Q20 3 marks Written p. 35
The sum of the squares of two consecutive odd integers is 290.Find the integers. Show your work.
Reveal official answer
• Takes the pair of consecutive odd integers as x and (x + 2). Uses given condition to frame the equation as: x² + (x + 2)² = 290 => x² + 2x − 143 = 0 [1]
• Solves the above quadratic equation to find the value of x as 11 and (−13). [1]
• Considering x as 11, finds the integers as 11 and 13. Considering x as (−13), finds the integers as (−13) and (−11). [1]
Q21 3 marks Written p. 35
Gaurav saved a certain amount of money in May. He saved Rs 3540 in the month of June. He noticed that his total savings in May and June is equal to the square of his savings in May.Express this situation in the form of a quadratic equation and find Gaurav's savings in May. Show your work.
Reveal official answer
• Assumes May's savings to be Rs x and frames the quadratic equation as: x + 3540 = x² [1]
• Solves the above quadratic equation to find the values of x as 60 and (−59). [1.5]
• Rejects (−59) as savings cannot be negative. Finds Gaurav's savings in May as Rs 60. [0.5]
Q22 3 marks Written p. 36
A circular garden has a concentric circular fountain area with centre O. The distance between the circumference of the fountain area and that of the garden is 5 meters as shown in the figure below.The height of the fountain OP is 5 times the radius of the fountain area.Find the radius of the fountain area if the shortest distance between the top of the fountain and the circumference of the garden is 17 meters. Show your work.
Reveal official answer
• Assumes the radius of fountain area to be r meters. Identifies that the figure forms a right angled triangle, which can be written as follows using pythagoras theorem: (5r)² + (r + 5)² = (17)² => 25r² + r² + 25 + 10r = 289 => 13r² + 5r − 132 = 0 [1.5]
• Solves the above quadratic equation to find the value of r as 3 meters and (−44/13) meters. Rejects (−44/13) as radius cannot be negative. Thus finds the radius of the fountain area as 3 meters. [1.5]
Q23 3 marks Written p. 36
A person needs to arrange desks in a classroom such that the number of rows is the same as the number of columns of desks. After having made such an arrangement, he found that 34 desks were still left with him. When he doubled the number of rows and columns, he found that he was short of 14 desks.Find the total number of desks available with him. Show your work.
Reveal official answer
• Takes the former number of rows and columns of desks as x and finds the number of desks available as x² + 34. [0.5]
• Takes the latter number of rows and columns as 2x and finds the number of desks available as (2x)² − 14. [0.5]
• Equates the above two quadratic expressions and solves for x as: x² + 34 = 4x² − 14 => 3x² = 48 => x = 4, −4. Rejects (−4) as number of rows or columns of desks cannot be negative. [1]
• Finds the number of desks available with the person as: 4² + 34 = 50 or (2 × 4)² − 14 = 50 [1]
Q24 3 marks Written p. 36
i) Find the nature of roots for the quadratic equation x² + 4x + 9 = 0.ii) What will be the nature of new roots if 2x is added to the quadratic equation given in i).Show your work.
Reveal official answer
• i) Finds the discriminant (b² − 4ac) of the given quadratic equation where a = 1, b = 4 and c = 9 as: b² − 4ac = 4² − 4 × 1 × 9 = 16 − 36 = −20 [1]
• States that since discriminant is negative, no real roots exists for the given quadratic equation. [1]
• ii) Writes new value of b as 6 and finds new discriminant as: 6² − 4 × 1 × 9 = 36 − 36 = 0. Concludes that since new discriminant is 0, newly formed quadratic equation will have two real and equal roots. [1]
Q25 3 marks Written p. 36
The length of a rectangular garden is 38 m more than its width.If the area of the garden is 600 m², find the dimensions of the garden. Show your work.
Reveal official answer
• Assumes the width of rectangular garden to be x m. States that length of the rectangular field would be (x + 38) m. [0.5]
• Uses formula of area of rectangle to make the quadratic equation as x(x + 38) = 600. [1]
• Factorises the above equation as (x − 12)(x + 50) = 0. Finds the value of x as 12 and (−50). [1]
• States that since value of length cannot be negative, x = −50 is not possible. Hence, x = 12. Concludes that width of rectangular garden is 12 m and length is 12 + 38 = 50 m. [0.5]
Q26 5 marks Written p. 37
The length of a rectangular park is 5 meters more than its breadth. The area of the park is 104 m².Find the total cost of sowing the plants along the boundary of the park at the rate of Rs 150 per meter. Show your work.
Reveal official answer
• Assumes breadth of the park as x m and its length as (x + 5) m. Frames the quadratic equation as: x(x + 5) = 104 => x² + 5x − 104 = 0 [1]
• Solves the above equation to find the value of x as 8, (−13). Finds the breadth of the park as 8 m rejecting (−13) as breadth cannot be negative. [1.5]
• Finds the length of the park as 8 + 5 = 13 m. Finds the length of the boundary of park as 2(8 + 13) = 42 m. [1.5]
• Finds the cost of sowing plants along the boundary as Rs 150 × 42 = Rs 6300. [1]
Q27 5 marks Written p. 37
Shreya said, "I know a real number whose square when added to two-third of itself, gives 152."Does such a number exist? If yes, find the number. If no, justify your answer.
Reveal official answer
• Assumes the number to be m. Frames the quadratic equation as: m² + (2/3)m = 152 => 3m² + 2m − 456 = 0 [1.5]
• To check if m is a real number, calculates the value of discriminant as: (2)² − (4 × 3 × −456) = 5476. Concludes that since this value is positive, m is a real number. [1.5]
• Solves the equation 3m² + 2m − 456 = 0 and finds the value of m as 12 and (−38/3). [2]
Q28 1 mark Written p. 37
Answer the questions based on the information given below.A construction company PQR pvt. ltd. signed a contract to construct a hostel building having a total of 135 rooms. The manager of company called up a meeting to decide on the design of building in an efficient way. Two employees put forward their ideas.Employee 1 said, "the number of rooms on a floor should be 2 less than thrice the number of floors in the building."Employee 2 said, "the number of rooms on a floor should be 3 less than double the number of floors in the building."Represent the design suggested by employee 1 in the form of a quadratic equation. Show your work.
Reveal official answer
• Assumes the number of floors in the building to be x. Hence, writes the number of rooms on a floor as (3x − 2). [0.5]
• Frames the quadratic equation as: x(3x − 2) = 135 => 3x² − 2x − 135 = 0 [0.5]
Q29 1 mark Written p. 37
Answer the questions based on the information given below.A construction company PQR pvt. ltd. signed a contract to construct a hostel building having a total of 135 rooms. The manager of company called up a meeting to decide on the design of building in an efficient way. Two employees put forward their ideas.Employee 1 said, "the number of rooms on a floor should be 2 less than thrice the number of floors in the building."Employee 2 said, "the number of rooms on a floor should be 3 less than double the number of floors in the building."Represent the design suggested by employee 2 in the form of a quadratic equation. Show your work.
Reveal official answer
• Assumes the number of floors in the building to be x. Hence, writes the number of rooms on a floor as (2x − 3). [0.5]
• Frames the quadratic equation as: x(2x − 3) = 135 => 2x − 3x − 135 = 0 [0.5]
Q30 3 marks Written p. 37
Answer the questions based on the information given below.A construction company PQR pvt. ltd. signed a contract to construct a hostel building having a total of 135 rooms. The manager of company called up a meeting to decide on the design of building in an efficient way. Two employees put forward their ideas.Employee 1 said, "the number of rooms on a floor should be 2 less than thrice the number of floors in the building."Employee 2 said, "the number of rooms on a floor should be 3 less than double the number of floors in the building."Which employee suggested the practical design to construct the building? How many floors should be constructed according to that employee? Show your work and give valid reason.
Reveal official answer
• Solves the quadratic equation 2x² − 3x − 135 = 0 to get the values of x as 9 and (−7.5). Rejects (−7.5) as number of floors cannot be negative. [1]
• Solves the quadratic equation 3x² − 2x − 135 = 0 to get the values of x as 7 1/25 and (−6 19/50). [1]
• Writes that since number of floors cannot be in fraction or negative, employee 2's design is practical to construct the building. Writes that the number of floors that should be constructed is 9. [1]