4. Quadratic Equations

SEBA Class 10 Maths Chapter 4. Quadratic Equations

Chapter 4. Quadratic Equations

Class 10 Maths Chapter 4. Quadratic Equations Multiple Choice Questions and Solutions :

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

(a)

(b)

(c)

(d)

Solution: (b)

[ The quadratic equation,

]

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

(a)

(b)

(c)

(d)

Solution: (d) .

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

(a)

(b)

(c)

(d)

Solution: (c) .

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 the quadratic equation are :

(a) 3 ,

(b) – 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

(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 : Under what condition will be a quadratic equation ? [SEBA 2018]

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

Solution : (d)

[If is a quadratic equation , then and are real numbers . ]

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

(a)

(b)

(c)

(d)

Solution: (d)

[ The quadratic equation,

]

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

(a) 0 only (b) 4 (c) 8 only (d) 0 , 8

Solution : (d) 0 , 8

[We have,

Here,

A/Q,

or ]

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) .

[We have,

]

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

[from (i)]

]

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)

(c)

(d)

Solution: (b) .

[We have,

is not a quadratic equation. ]

Question: The roots of the quadratic equation is :

(a) – 5 , – 1

(b) 2 , 3

(d) – 2 , – 3

Solution: (a) – 5 , – 1

[ We have ,

]

Question : Which of the following is a quadratic equation ?

(a)

(b)

(c)

(d)

Solution : (d)

[ We have ,

is a quadratic equation ]

Question : The product of roots of quadratic equation is :

(a) (ii) (iii) (iv)

Solution : (c)

[ The product of roots . ]

Question : If is a example of quadratic equation ,then the roots of given equation is :

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

Solution : (a)

[ We have ,

So, or

]

Question: The nature of the quadratic equation is :

(i) coincident roots (ii) no real roots

(iii) two distinct roots (iv) two equal roots

Solution : (d) (ii) no real roots .

[ Here, , ,

The discriminant

So, there are no real roots for the given equation . ]

Class 10 Quadratic Equations Fill in the blanks

Q1. If – 5 is a root of the quadratic equation , then the value of is .

Solution: 7

[ Given,

]

Q2. 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

]

Q3. If and are the roots of equation and , then is equal to .

Solution: – 24

[ We have,

A/Q,

and

From , we get

]

Q4. Given and are roots of quadratic equation, if and , then the equation is .

Solution: .

[ Given , and

The quadratic equation,

]

Q5. If the quadratic equation has two equal real roots , then is .

Solution:

[ We have,

]

Q6. The discriminant of the quadratic equation is .

Solution: 0

[ We have,

Here, , ,

The discriminant ]

Class 10 Quadratic Equations Answers following the question

Q1. If is one root of the quadratic equation, then find the value of . [CBSE2018]

Solution: Given ,

Q2. Find the nature of roots of quadratic equation [CBSE 2019]

Solution: We have ,

Here, , ,

The discriminant

Therefore, the given equation has no real roots .

Q3. Find the roots of the quadratic equation

Solution: We have ,

Q4. Check whether the equation is a quadratic equation :

Solution: We have,

is a quadratic equation.

Q5. Find the roots of the quadratic equation :

Solution: We have,

Q6. If is a solution of the quadratic equation , find the value of . [DELHI 2015]

Solution: Given,

We have ,

Q7. If and are the roots of , then find the value of .

Solution: Since , and are the roots of .

and

Now,

Class 10 Quadratic Equations 2 Marks Questions and Answers

Q1. Write the nature of roots of quadratic equation : .

Solution: We have ,

Here, , ,

The discriminant

Thus , the given equation has two distinct real roots .

Q2. Find the roots of the quadratic equation using the quadratic formula .

Solution: We have ,

Here , , ,

Using the quadratic formula ,

Thus, the roots are and .

Q3. Solve for : [CBSE 2014F , 2016]

Solution: We have ,

Q4. 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 .

Q5. Solve the quadratic equation for . [Delhi 2014]

Solution: We have,

Q6. If and are roots of the quadratic equation , find the values of and . [CBSE 2016]

Solution: Since, and are roots of

The sum of roots

and

Product of roots

Q7. Find the roots of the quadratic equation .

Solution: We have,

Therefore , the roots are and .

Q8. Find the roots of the quadratic equation

Solution: We have ,

Thus, the roots of the quadratic equation are and .

Q9. Solve for :

Solution: We have ,

Here , , ,

Using the quadratic formula ,

Q10. Find the roots the equation: ;

Solution: We have,

;

Here, , ,

Using the quadratic formula ,

Thus , the roots are and .

Q11. Find the value of for the following quadratic equation , so that it has two equal roots : .[SEBA 2020]

Solution: We have ,

Here , , ,

(impossible) or

Therefore, the value of is .

Class 10 Quadratic Equations 3 Marks Questions and Answers

SECTION = C

Q1. Solve the equation , for . [Delhi 2014 , CBSE 2013]

Solution: We have,

Thus , the value of are 1 or – 2 .

Q2. Solve for :

Solution: We have ,

Therefore , the value of are and .

Q3. 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 .

Q4. Find the roots of the equation by the method of completing the square .

Solution: We have,

Q5. Solve the following quadratic equation by applying the quadratic formula :

Solution: Here, , ,

Applying the quadratic formula,

Q12. 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,

Q8. 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)

Putting in , we get

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

Q9. 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.

Q10. 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 .

Class 10 Quadratic Equations 4 Marks Questions and Answers

SECTION = D

Q1. Find in terms of , and ; [CBSE 2016]

Solution: We have ,

Q2.If the roots of the equation are equal, then prove that that

Solution: Given, the equation is

Here , , ;

A/Q ,

Proved.

Q3. 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, 3

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

Q4. If the equation has equal roots , show that . [CBSE 2018]

Solution: Given, the equation is

; and

proved.

Q5. Find the roots of the equations :

Solution: We have ,

= 0

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

Q6. 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

Q7. If the roots of the equation are equal, prove that

Solution: We have ,

Here, ; ;

A/Q ,

Proved.

Q8. 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 time taken to go upstream

and the time taken go downstream

A/Q,

(impossible)

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