Mathematics · Textbook solutions

Quadratic Equations

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 39 questions

4.2 Quadratic Equations

14 q

Solved Examples

Worked · 2

4.1 Eg.1 $Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹ 750. We would like to find out the number of toys produced on that day.$
4.1 Eg.2 $(x - 2)^2 + 1 = 2x - 3$

Exercise 4.1

Practice · 12

Check whether the following are quadratic equations :
Ex 4.1 Q1 (i) $(x + 1)^2 = 2(x - 3)$
Ex 4.1 Q1 (ii) $x^2 - 2x = (-2)(3 - x)$
Ex 4.1 Q1 (iii) $(x - 2)(x + 1) = (x - 1)(x + 3)$
Ex 4.1 Q1 (iv) $(x - 3)(2x + 1) = x(x + 5)$
Ex 4.1 Q1 (v) $(2x - 1)(x - 3) = (x + 5)(x - 1)$
Ex 4.1 Q1 (vi) $x^2 + 3x + 1 = (x - 2)^2$
Ex 4.1 Q1 (vii) $(x + 2)^3 = 2x(x^2 - 1)$
Ex 4.1 Q1 (viii) $x^3 - 4x^2 - x + 1 = (x - 2)^3$
Represent the following situations in the form of quadratic equations :
Ex 4.1 Q2 (i) $^2$
Ex 4.1 Q2 (ii) $The product of two consecutive positive integers is 306. We need to find the integers.$
Ex 4.1 Q2 (iii) $Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.$
Ex 4.1 Q2 (iv) $A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.$

4.3 Solution of a Quadratic Equation by Factorisation

14 q

Solved Examples

Worked · 4

4.2 Eg.3 $2x^2 - 5x + 3 = 0$
4.2 Eg.4 $6x^2 - x - 2 = 0$
4.2 Eg.5 $3x^2 - 2\sqrt{6}\,x + 2 = 0$
4.2 Eg.6 $Find the dimensions of the prayer hall discussed in Section 4.1.$

Exercise 4.2

Practice · 10

Find the roots of the following quadratic equations by factorisation:
Ex 4.2 Q1 (i) $x^2 - 3x - 10 = 0$
Ex 4.2 Q1 (ii) $2x^2 + x - 6 = 0$
Ex 4.2 Q1 (iii) $\sqrt{2}\,x^2 + 7x + 5\sqrt{2} = 0$
Ex 4.2 Q1 (iv) $2x^2 - x + \dfrac{1}{8} = 0$
Ex 4.2 Q1 (v) $100x^2 - 20x + 1 = 0$
Ex 4.2 Q2 $Solve the problems given in Example 1.$
Ex 4.2 Q3 $Find two numbers whose sum is 27 and product is 182.$
Ex 4.2 Q4 $Find two consecutive positive integers, sum of whose squares is 365.$
Ex 4.2 Q5 $The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.$
Ex 4.2 Q6 $A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90, find the number of articles produced and the cost of each article.$

4.4 Nature of Roots

11 q

Solved Examples

Worked · 3

4.3 Eg.7 $2x^2 - 4x + 3 = 0$
4.3 Eg.8 $A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?$
4.3 Eg.9 $3x^2 - 2x + \dfrac{1}{3} = 0$

Exercise 4.3

Practice · 8

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
Ex 4.3 Q1 (i) $2x^2 - 3x + 5 = 0$
Ex 4.3 Q1 (ii) $3x^2 - 4\sqrt{3}\,x + 4 = 0$
Ex 4.3 Q1 (iii) $2x^2 - 6x + 3 = 0$
Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots.
Ex 4.3 Q2 (i) $2x^2 + kx + 3 = 0$
Ex 4.3 Q2 (ii) $kx(x - 2) + 6 = 0$
Ex 4.3 Q3 $^2$
Ex 4.3 Q4 $Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.$
Ex 4.3 Q5 $^2$

What the board has asked from this chapter

Questions per paper · 67 papers · 2022 to 2026

2026

23 q

1 $kx^2 - 6x - 4 = 0$
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Nature of Roots and the Discriminant
1 $(x - 1)^2 = 16$
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Solving a Quadratic Equation by Factorisation
1 $9x^2 + 8kx + 16 = 0$
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Nature of Roots and the Discriminant
7 $4x^2 - (a - 1)^2 = 0$
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Solving a Quadratic Equation by Factorisation
9 $x^2 - px + 6 = 0$
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Nature of Roots and the Discriminant
10 $\sqrt{3}x^2 - kx + 2\sqrt{3} = 0$
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Nature of Roots and the Discriminant
11 $x^2 - k\sqrt{3}\,x + 2 = 0$
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Nature of Roots and the Discriminant
14 $3x^2 - 7x + m = 0$
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Nature of Roots and the Discriminant
21 $(p - q)x^2 + (q - r)x + (r - p) = 0$
Nature of Roots and the Discriminant
29 $Find two consecutive negative integers, sum of whose squares is 481.$
Solving a Quadratic Equation by Factorisation
32 $Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.$
Solving a Quadratic Equation by Factorisation
32 (a) $A person on tour has ₹ 5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹ 180. Find the original duration of the tour and daily expense.$
Solving a Quadratic Equation by Factorisation
32 $By selling an article for ₹ 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.$
Solving a Quadratic Equation by Factorisation
33 (a) $A person on a tour has ₹ 4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by ₹ 70. Find the original duration of the tour.$
Solving a Quadratic Equation by Factorisation
33 (A) $A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains.$
Solving a Quadratic Equation by Factorisation
32 (b) $The total cost of certain piece of cloth was ₹ 2,100. During special sale time, the shopkeeper offered 2 m extra cloth for free thus reducing the price of cloth per metre by ₹ 120. What was the original per metre price of cloth and its length ?$
Solving a Quadratic Equation by Factorisation
34 (a) $In a flight of 600 km, an aircraft slowed down its speed due to bad weather. Its average speed for the trip reduced by 200 km/h from its usual speed and time of flight increased by 30 minutes. Find the scheduled duration of the flight.$
Solving a Quadratic Equation by Factorisation
33 (B) $640\,\text{m}^2$
Solving a Quadratic Equation by Factorisation
33 (b) $600\,\text{cm}^2$
Solving a Quadratic Equation by Factorisation
34 (b) $Two pipes are used to fill a swimming pool. If the pipe of the larger diameter is used for 4 hours and the pipe of the smaller diameter for 9 hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool, separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.$
Solving a Quadratic Equation by Factorisation
35 $8\dfrac{8}{9}$
Solving a Quadratic Equation by Factorisation
35 (a) $In a class test, the sum of Anamika’s marks obtained in Maths and Science is 30. Had she got 2 marks more in Maths and 3 marks less in Science, the product of the marks would have been 210. Find the marks she got in the two subjects.$
Solving a Quadratic Equation by Factorisation
35 (b) $The length of hypotenuse (in cm) of a right-angled triangle is 6 cm more than twice the length of its shortest side. If the length of its third side is 6 cm less than thrice the length of its shortest side, find the dimensions of the triangle.$
Solving a Quadratic Equation by Factorisation

2025

31 q

1 $\dfrac{1}{a}$
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Identifying and Forming Quadratic Equations
3 $Which of the following quadratic equations has real and equal roots ?$
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Nature of Roots and the Discriminant
3 $ax^2 + x + a = 0$
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Nature of Roots and the Discriminant
3 $\dfrac{1}{7}$
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Identifying and Forming Quadratic Equations
4 $Which of the following equations is a quadratic equation ?$
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Identifying and Forming Quadratic Equations
5 $x^2 + bx + b = 0$
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Nature of Roots and the Discriminant
5 $Which of the following quadratic equations has real and distinct roots ?$
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Nature of Roots and the Discriminant
5 $\dfrac{x}{12} - \dfrac{3}{x} = 0$
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Solving a Quadratic Equation by Factorisation
8 $Which of the following equations is a quadratic equation ?$
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Identifying and Forming Quadratic Equations
12 $Which of the following equations does not have a real root ?$
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Nature of Roots and the Discriminant
13 $Which of the following equation is a quadratic equation ?$
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Identifying and Forming Quadratic Equations
14 $bx^2 + ax + c = 0;\ b \neq 0$
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Nature of Roots and the Discriminant
24 (a) $4x^2 + kx + 1 = 0$
Nature of Roots and the Discriminant
32 (a) $The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.$
Solving a Quadratic Equation by Factorisation
32 (a) $A 2-digit number is seven times the sum of its digits and two (2) more than 5 times the product of its digits. Find the number.$
Solving a Quadratic Equation by Factorisation
32 (a) $There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter. An entry gate is to be constructed at a point P on the boundary of the park such that distance of P from A is 35 m more than the distance of P from B. Find distance of point P from A and B respectively.$
Solving a Quadratic Equation by Factorisation
32 (b) $(p + 4)x^2 - (p + 1)x + 1 = 0$
Nature of Roots and the Discriminant
32 (b) $\dfrac{x - 2}{x - 3} + \dfrac{x - 4}{x - 5} = \dfrac{10}{3}$
Solving a Quadratic Equation by Factorisation
32 (b) $x^2 - 2(p + 1)x + p^2 = 0$
Nature of Roots and the Discriminant
33 (a) $52\,\text{cm}^2$
Solving a Quadratic Equation by Factorisation
33 (b) $2\dfrac{1}{2}$
Solving a Quadratic Equation by Factorisation
34 (a) $The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.$
Solving a Quadratic Equation by Factorisation
35 (a) $A two-digit number is such that the product of its digits is 12. When 36 is added to this number, the digits interchange their places. Find the number.$
Solving a Quadratic Equation by Factorisation
34 (b) $A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.$
Solving a Quadratic Equation by Factorisation
35 (a) $1\dfrac{9}{20}$
Solving a Quadratic Equation by Factorisation
35 (b) $A student scored a total of 32 marks in class tests in Mathematics and Science. Had he scored 2 marks less in Science and 4 marks more in Mathematics, the product of his marks would have been 253. Find his marks in the two subjects.$
Solving a Quadratic Equation by Factorisation
35 (b) $A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.$
Solving a Quadratic Equation by Factorisation
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions :
36 (i) $Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway = x m.$
Identifying and Forming Quadratic Equations
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions :
36 (ii) (a) $Solve the quadratic equation to find the width of the walkway ‘x’.$
Solving a Quadratic Equation by Factorisation
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions :
36 (ii) (b) $If the cost of paving the walkway at the rate of ₹ 50 per square metre is ₹ 12,000, calculate the area of the walkway.$
Identifying and Forming Quadratic Equations
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions :
36 (iii) $Find the perimeter of the lawn.$
Identifying and Forming Quadratic Equations

2024

26 q

3 $ax^2 + bx + c = 0, a \neq 0$
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Nature of Roots and the Discriminant
4 $3x^2 - 2x + c = 0$
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Nature of Roots and the Discriminant
4 $x^2 + x + 1 = 0$
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Nature of Roots and the Discriminant
6 $ax^2 + bx + c = 0$
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Nature of Roots and the Discriminant
8 $4x^2 - 5x + 4 = 0$
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Nature of Roots and the Discriminant
13 $x = 2$
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Solving a Quadratic Equation by Factorisation
17 $y = 1$
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Solving a Quadratic Equation by Factorisation
28 $Three consecutive integers are such that sum of the square of second and product of other two is 161. Find the three integers.$
Solving a Quadratic Equation by Factorisation
30 $A dealer sells an article for ₹ 75 and gains as much percent as the cost price of the article. Find the cost price of the article.$
Solving a Quadratic Equation by Factorisation
31 $In a 2-digit number, the digit at the unit’s place is 5 less than the digit at the ten’s place. The product of the digits is 36. Find the number.$
Solving a Quadratic Equation by Factorisation
32 (a) $(k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0$
Nature of Roots and the Discriminant
32 (a) $In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the flight.$
Solving a Quadratic Equation by Factorisation
32 (a) $(k + 1)x^2 - 2(3k + 1)x + (8k + 1) = 0$
Nature of Roots and the Discriminant
32 (b) $2\dfrac{16}{21}$
Solving a Quadratic Equation by Factorisation
32 (b) $The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages.$
Solving a Quadratic Equation by Factorisation
32 (b) $A 2-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.$
Solving a Quadratic Equation by Factorisation
34 (a) $A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train.$
Solving a Quadratic Equation by Factorisation
35 (a) $A 2-digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number.$
Solving a Quadratic Equation by Factorisation
34 (b) $(c + 1)x^2 - 6(c + 1)x + 3(c + 9) = 0$
Nature of Roots and the Discriminant
35 (b) $\text{cm}^2$
Solving a Quadratic Equation by Factorisation
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
36 (i) $Assuming the original length of each side of a tile be x units, make a quadratic equation from the above information.$
Identifying and Forming Quadratic Equations
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
36 (ii) $Write the corresponding quadratic equation in standard form.$
Identifying and Forming Quadratic Equations
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
36 (iii) (a) $Find the value of x, the length of side of a tile by factorisation.$
Solving a Quadratic Equation by Factorisation
A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial $h = 25t - 5t^2$ . Observe the graph of the polynomial and answer the following questions :
36 (iii) (a) $After throwing upward, how much time did the ball take to reach to the height of 30 m ?$
Solving a Quadratic Equation by Factorisation
A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.
36 (iii) (b) $Solve the quadratic equation for x, using quadratic formula.$
Nature of Roots and the Discriminant
A ball is thrown in the air so that t seconds after it is thrown, its height h metre above its starting point is given by the polynomial $h = 25t - 5t^2$ . Observe the graph of the polynomial and answer the following questions :
36 (iii) (b) $Find the two different values of t when the height of the ball was 20 m.$
Solving a Quadratic Equation by Factorisation

2023

18 q

2 $x^2 + 3x - 10 = 0$
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Solving a Quadratic Equation by Factorisation
5 $x = 0{\cdot}3$
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Solving a Quadratic Equation by Factorisation
5 $(2 + \sqrt{3})$
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Identifying and Forming Quadratic Equations
8 $2x^2 + kx - 4 = 0$
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Nature of Roots and the Discriminant
9 $ax^2 + bx + c = 0$
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Nature of Roots and the Discriminant
10 $(3 - \sqrt{2})$
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Identifying and Forming Quadratic Equations
16 $x^2 - (p + q)\,x + k = 0$
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Solving a Quadratic Equation by Factorisation
In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option out of the following :
19 $5 + \sqrt{7}$
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Nature of Roots and the Discriminant
23 (B) $4x^2 - 5 = 0$
Nature of Roots and the Discriminant
27 (A) $A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.$
Solving a Quadratic Equation by Factorisation
29 (A) $\dfrac{3}{10}$
Solving a Quadratic Equation by Factorisation
31 $px(x - 2) + 6 = 0$
Nature of Roots and the Discriminant
34 (a) $A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ?$
Solving a Quadratic Equation by Factorisation
34 (b) $\dfrac{15}{8}$
Solving a Quadratic Equation by Factorisation
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions :
36 (I) $Write an algebraic equation depicting the above information.$
Identifying and Forming Quadratic Equations
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions :
36 (II) $Write the corresponding quadratic equation in standard form.$
Identifying and Forming Quadratic Equations
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions :
36 (III) $What should be the new dimensions of the enlarged photo ?$
Solving a Quadratic Equation by Factorisation
While designing the school year book, a teacher asked the student that the length and width of a particular photo is increased by $x$ units each to double the area of the photo. The original photo is 18 cm long and 12 cm wide. Based on the above information, answer the following questions :
36 (III) OR $x$
Nature of Roots and the Discriminant

2022

13 q

1 $x^2 + 2\sqrt{2}x - 6 = 0$
Solving a Quadratic Equation by Factorisation
1 (a) $x^2 - 2ax - (4b^2 - a^2) = 0$
Solving a Quadratic Equation by Factorisation
1 (b) $(1 + a^2)x^2 + 2abx + (b^2 - c^2) = 0$
Nature of Roots and the Discriminant
3 (b) $x = -2$
Solving a Quadratic Equation by Factorisation
3 (a) $(m - 1)x^2 + 2(m - 1)x + 1 = 0$
Nature of Roots and the Discriminant
3 (b) $\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0$
Solving a Quadratic Equation by Factorisation
5 $The product of Rehan’s age (in years) 5 years ago and his age 7 years from now, is one more than twice his present age. Find his present age.$
Solving a Quadratic Equation by Factorisation
8 (a) $p(x - 4)(x - 2) + (x - 1)^2 = 0$
Nature of Roots and the Discriminant
8 (b) $Had Aarush scored 8 more marks in a Mathematics test, out of 35 marks, 7 times these marks would have been 4 less than square of his actual marks. How many marks did he get in the test ?$
Solving a Quadratic Equation by Factorisation
12 (a) $A 2-digit number is such that the product of its digits is 24. If 18 is subtracted from the number, the digits interchange their places. Find the number.$
Solving a Quadratic Equation by Factorisation
12 (b) $The difference of the squares of two numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.$
Solving a Quadratic Equation by Factorisation
In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width $x$ m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m.
13 (a) $x$
Identifying and Forming Quadratic Equations
In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width $x$ m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m.
13 (b) $Find the width of the sidewalk around the pool.$
Solving a Quadratic Equation by Factorisation

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2022*, 2024*, 2025 (* = numbers or wording changed that year)
35 (a) $A two-digit number is such that the product of its digits is 12. When 36 is added to this number, the digits interchange their places. Find the number.$
Question 2
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
34 (a) $In a flight of 600 km, an aircraft slowed down its speed due to bad weather. Its average speed for the trip reduced by 200 km/h from its usual speed and time of flight increased by 30 minutes. Find the scheduled duration of the flight.$
Question 3
Asked 2 times: 2023*, 2026 (* = numbers or wording changed that year)
35 $8\dfrac{8}{9}$
Question 4
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
33 (B) $640\,\text{m}^2$
Question 5
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
35 (a) $In a class test, the sum of Anamika’s marks obtained in Maths and Science is 30. Had she got 2 marks more in Maths and 3 marks less in Science, the product of the marks would have been 210. Find the marks she got in the two subjects.$
Question 6
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
35 (b) $The length of hypotenuse (in cm) of a right-angled triangle is 6 cm more than twice the length of its shortest side. If the length of its third side is 6 cm less than thrice the length of its shortest side, find the dimensions of the triangle.$
Question 7
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
32 $By selling an article for ₹ 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.$
Question 8
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
34 (b) $A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.$
Question 9 (case study, 4 parts)
Asked 2 times: 2022*, 2025 (* = numbers or wording changed that year)
A garden designer is planning a rectangular lawn that is to be surrounded by a uniform walkway. The total area of the lawn and the walkway is 360 square metres. The width of the walkway is same on all sides. The dimensions of the lawn itself are 12 metres by 10 metres. Based on the information given above, answer the following questions :
36 (i) $Formulate the quadratic equation representing the total area of the lawn and the walkway, taking width of walkway = x m.$
36 (ii) (a) $Solve the quadratic equation to find the width of the walkway ‘x’.$
36 (ii) (b) $If the cost of paving the walkway at the rate of ₹ 50 per square metre is ₹ 12,000, calculate the area of the walkway.$
36 (iii) $Find the perimeter of the lawn.$
Question 10
Asked 2 times: 2023, 2024
3 $ax^2 + bx + c = 0, a \neq 0$
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4.2 Quadratic Equations

Solved Examples

Exercise 4.1

4.3 Solution of a Quadratic Equation by Factorisation

Solved Examples

Exercise 4.2

4.4 Nature of Roots

Solved Examples

Exercise 4.3

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9 (case study, 4 parts)

Question 10