Chapter 4: Quadratic Equations– Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 4: Quadratic Equations Important Questions with Solutions and Answers

Chapter 4: Quadratic Equations

SECTION = A

Question: The discriminant of the quadratic equation 8x² + 2x – 3 = 0, is : [2024 B]

(A) 100 (B) 92 (C) – 92 (D) 96

Solution: (a) 100

[ We have,

Here,

]

Question: The roots of the quadratic equation x² + x – p (p + 1) = 0 are : [2024 S]

(a) p, p + 1 (b) – p, p + 1 (c) – p, – (p + 1) (d) p, – ( p + 1)

Solution: (d) p , – (p+1)

We have,

or ]

Question: If the roots of quadratic equation are real and equal , then value of k is : [2024 Basic]

(a) (b) (c) (d)

Solution: (b)

[ We have,

Here,

]

Question: The discriminant of the quadratic equation is [2022 B]

(a) 1 (b) 49 (c) 7 (d) 19

Solution: (b) 49

[ Here,

]

Question: The roots of the quadratic equation are :

(a) 3 , – (b) – 3 , (c) 3 , (d) – 3 , –

Solution: (c) 3 ,

[ We have ,

So, or

Question: The nature of quadratic equation are :

(a) two distinct real roots (b) two equal roots (c) no real roots (d) none of these .

Solution: (b) two equal roots

[ We have,

Here, , ,

The discriminant

Hence, the given quadratic equation has two equal real roots . ]

Question: A quadratic equation has two distinct real roots , then

(a) (b) (c) (d)

Solution: (a) .

Question: A quadratic equation has coincident roots (two equal roots) , then [SEBA 2014]

(a) (b) (c) (d)

Solution: (d) .

Question: The quadratic equation whose roots are 1 and , then the equation is :

(a) (b) (c) (d)

Solution: (d)

[ The quadratic equation,

]

Question: Which constant should be added and subtracted to solve the quadratic equation by the method of completing the square ?

(a) (b) (c) (d)

Solution: (a)

[ We have ,

]

Question: Which of the following equations has 2 as a root ?

(a) (b) (c) (d)

Solution: (c)

[ Given ,

]

Question: A quadratic equation has no real roots , then

(a) (b) (c) (d)

Solution: (b) .

Question: Which of the following is not a quadratic equation ?

(a) (b) (c) (d)

Solution: (d) .

Question: If is a root of the equation , then the value of is :

(a) 2 (b) – 2 (c) (d)

Solution: (a) 2

[ Given ,

]

Question: If the roots of are reciprocal of each other, then

(a) (b) (c) (d)

Solution: (d)

[ let and are two roots .

A/Q, ]

Question: If the roots of the equation are in the ratio 3 : 2 , then is :

(a) – 5 (b) + 5 (c) (d) 6

Solution: (c)

[ let and are two roots .

Given ,

and

]

Question: Which of the following equations has the sum of its roots as 3 ?

(a) (b) (c) (d)

Solution: (b) .

[ We have, the sum of roots ]

Question: The discriminant of the quadratic equation is [CBSE2020 basic]

(a) 12 (b) 84 (c) (d) – 12

Solution: (d) – 12

[ We have , ;

Here , , ,

The discriminant ]

Question: The value of for which the quadratic equation has equal roots , is

(a) 4 (b) (c) – 4 (d) 0

Solution: (b)

[ We have,

A/Q,

]

Question: Which one of the following is not a quadratic equation ? [SEBA 2017]

(a) (b)

Solution: (b) .

Question: One root of a quadratic equation is 2 and sum of the two roots is 0 , the equation is : [SEBA 2016]

(a) (b) (c) (d)

Solution: (b)

[ Here,

and

The quadratic equation,

]

Question: The roots of a quadratic equation are and , then the equation is : [SEBA 2013]

(a) (b)

Solution: (b)

[ The quadratic equation,

]

Question: Under what condition the roots of the quadratic equation will be real and unequal ? [ SEBA 2015]

(a) (b) (c) (d)

Solution: (c) .

Question: The roots of the quadratic equation is :

(a) – 5 , – 1 (b) 2 , 3 (c) 6 , – 1 (d) – 2 , – 3

Solution: (a) – 5 , – 1

[ We have ,

]

Fill in the blanks

Question: If – 5 is a root of the quadratic equation , then the value of is .

Solution: 7

[ Given,

]

Question: If the quadratic equation has equal roots , then the value of is .

Solution:

[ We have, and let be the root of quadratic equation.

A/Q,

and

]

Question: If and are the roots of equation and , then is equal to .

Solution: – 24

[ We have,

A/Q,

and

From , we get

]

Question: Given and are roots of quadratic equation, if and , then the equation is .

Solution: .

[ Given , and

The quadratic equation,

]

Question: If the quadratic equation has two equal real roots , then is .

Solution:

[ We have,

]

Question: The discriminant of the quadratic equation is .

Solution: 0

[ We have,

Here, , ,

The discriminant ]

Answers following the question

Question: Find the roots of the quadratic equation. [2022 S]

Solution: We have,

Therefore, the roots of the given quadratic equation are

Question: If is one root of the quadratic equation, then find the value of . [CBSE2018]

Solution: Given ,

Question: Find the nature of roots of quadratic equation [CBSE 2019]

Solution: We have ,

Here, , ,

The discriminant

Therefore, the given equation has no real roots .

Question: Find the roots of the quadratic equation

Solution: We have ,

Question: If the quadratic equation has two equal roots, then find the value of p . [2015] .

Solution: We have,

Here,

A/Q,

(Impossible) or

Therefore, the value of p is 3 .

Question: Check whether the equation is a quadratic equation :

Solution: We have,

is a quadratic equation.

Question: Find the roots of the quadratic equation :

Solution: We have,

Question: If is a solution of the quadratic equation , find the value of . [Delhi2015]

Solution: Given,

We have ,

Therefore, the value of k is

Question: If and are the roots of , then find the value of .

Solution: We have ,

Now,

SECTION = B

Question: Find the values of k for which the quadratic equation has real and equal roots . [2001C , 2013]

Solution: We have,

Here,

A/Q,

Therefore, the value of k is .

Question: Write the nature of roots of quadratic equation :

Solution: We have ,

Here, , ,

The discriminant

Thus , the given equation has two distinct real roots .

Question: Find the roots of the quadratic equation using the quadratic formula .

Solution: We have ,

Here , , ,

Using the quadratic formula ,

Thus, the roots are .

Question: Solve for : [CBSE , 2016, 2014 F ]

Solution: We have ,

Question: Check whether the equation is a quadratic equation :

Solution: We have,

It is of the form of . So, the given equation is a quadratic equation .

Question: Solve the quadratic equation for . [Delhi 2014]

Solution: We have,

Question: In a class test, the sum of Shefali’s marks in Mathematics and English is 30 . Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210 . Find her marks in the two subjects .

Solution : Let, be the marks in Mathematics of Shefali and her English marks will be .

A/Q ,

Therefore, the marks obtained by Shefali is 12 or 18 and 13 or 17 respectively .

Question: Find the roots of the quadratic equation .

Solution: We have,

Therefore , the roots are .

Question: Find the roots of the quadratic equation

Solution: We have ,

Thus, the roots of the quadratic equation are .

Question: Solve for :

Solution: We have ,

Here , , ,

Using the quadratic formula ,

Question: Find the roots the equation: ;

Solution: We have,

;

Here, , ,

Using the quadratic formula ,

Thus , the roots are and .

Question: Find the value of for the following quadratic equation , so that it has two equal roots : [SEBA2020]

Solution: We have ,

Here , , ,

(impossible) or

Therefore, the value of is .

SECTION = C

Question: The sum of two numbers is 18 and the sum of their reciprocals is . Find the numbers. [2024 S]

Solution: Let , x be the number and the other number be 18 – x .

A/Q,

Therefore, the two number are 8 or 9 and 9 or 8 .

Question: Solve for x : [2024 S]

Solution: We have,

Therefore,

Question: Solve the equation , for . [Delhi 2014 , CBSE 2013]

Solution: We have,

Thus , the value of are 1 or – 2 .

Question: Solve for :

Solution: We have ,

Therefore , the value of are and .

Question: Find that non-zero value of , for which the quadratic equation has equal roots. Hence find the roots of the equation. [Delhi2015 , CBSE2002C]

Solution: We have,

Here , , ;

(impossible) or

Thus, the equation of the roots are and .

Question: Find the roots of the equation by the method of completing the square .

Solution: We have,

Question: Solve the following quadratic equation by applying the quadratic formula :

Solution: Here, , ,

Applying the quadratic formula,

Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them .

Solution: Here , , ,

Hence, the given quadratic equation has two equal real roots and the roots are exist .

Applying quadratic formula,

Question: Sum of the areas of two squares is 468 . If the difference of their perimeters is 24 m , find the sides of the two squares . [SEBA 2016]

Solution: let and are the side of two square respectively.

A/Q ,

and

(impossible) or

Putting in , we get

Therefore, m and m are the side of two square respectively.

Question: Sum of the areas of two squares is 544 . If the difference of their perimeters is 32 m , find the sides of the two squares . [CBSE 2020]

Solution: let and are the side of two square respectively.

A/Q ,

and

(impossible) or

Putting in , we get

Therefore, m and m are the side of two square respectively.

Question: Find two consecutive odd positive integers, sum of whose squares is 290 .

Solution: let the two consecutive odd positive integers are and respectively .

A/Q ,

Thus , the two consecutive odd integers are 11 and 13 .

SECTION = D

Question: 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 . [2025 S]

Solution : Let (in km/hrs) be the speed of the train .

A/Q,

(Impossible) or

Therefore, the speed of the train is 32 Km/h .

Question: If – 5 is a root of the quadratic equation and the quadratic equation has equal roots, then find the values of p and k . [2002 , 2009 ]

Solution: given ,

We have,

Again,

Here,

A/Q,

Question: An aeroplane left 50 minutes later than its schedule time , and in order to reach the destination , 1250 km away, in time , it had to increase its speed by 250 km/hr from its usual speed . Find its usual speed . [2010]

Solution: Let x be the usual speed of the aeroplane (in km/h) and the increase speed be (x+250) km/h .

A/Q,

(impossible) or

So, the usual speed of the aeroplane is 500 km/hr.

Question: A shopkeeper buys books of the Rs 80 .If he had bought 4 more books for the same amount , each book would have cost Rs 1 less . Find the number of books he bought . [2012 ]

Solution: Let the number of books he bought be x .

The cost per book

And new number of books

the new cost per book

A/Q,

(impossible) or

Therefore, the shopkeeper originally bought 16 books.

Question: Solve for x : [2013 ]

Solution: We have,

Therefore,

Question: Find in terms of , and ; [CBSE 2016]

Solution: We have ,

Question: If the roots of the equation are equal, then prove that that

Solution: Given, the equation is

Here , , ;

A/Q ,

Proved.

Question: 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 total journey , what is its first speed ? .

Solution: let, (in km/h) be the speed of the first train.

A/Q ,

Therefore , the speed of the first train is 36 km/h.

Question: If the equation has equal roots , show that . [CBSE 2018]

Solution: Given, the equation is

; and

proved.

Question: Find the roots of the equations : [2025 S C]

Solution: We have ,

Therefore , the roots of the equations are 1 and 2 .

Question: The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Solution: let, be the shorter side of a rectangular field and the longer side will be ( m

Therefore, the diagonal of a rectangular field is m .

A/Q ,

and (Impossible)

Thus, the shorter side of a rectangular field is 90 m and the longer side is

Question: If the roots of the equation are equal, prove that .

Solution: We have ,

Here, ; ;

A/Q ,

Proved.

Question: A motor boat whose sped is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot . Find the speed of the stream . [CBSE 2018]

Solution: let (in km/h) be the speed of the stream .

So, the speed of the boat upstream km/h and the speed of the boat downstream km/h

The taken to go upstream = and downstream .

A/Q,

(impossible)

Thus, the speed of the stream is 6 km/h .

Posted 5 years ago

Chapter 4: Quadratic Equations– Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 4: Quadratic Equations Important Questions with Solutions and Answers

Chapter 4: Quadratic Equations

Fill in the blanks

Answers following the question

SECTION = B

SECTION = C

SECTION = D

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