4. Quadratic Equations

Quadratic Equations

4. Quadratic Equations

SECTION = A

Q1. The roots of the quadratic equation are :

(a) 3 , –

(b) – 3 ,

(c) 3 ,

(d) – 3 , –

Solution: (c) 3 ,

[ We have ,

So, or

Q2. 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 . ]

Q3. A quadratic equation has two distinct real roots , then

(a)

(b)

(c)

(d)

Solution: (a) .

Q4. A quadratic equation has coincident roots (two equal roots) , then [SEBA 2014]

(a)

(b)

(c)

(d)

Solution: (d) .

Q5. The quadratic equation whose roots are 1 and , then the equation is :

(a)

(b)

(c)

(d)

Solution: (d)

[ The quadratic equation,

]

Q6. 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 ,

]

Q7. Which of the following equations has 2 as a root ?

(a)

(b)

(c)

(d)

Solution: (c)

[ Given ,

]

Q8. A quadratic equation has no real roots , then

(a)

(b)

(c)

(d)

Solution: (b) .

Q9. Which of the following is not a quadratic equation ?

(a)

(b)

(c)

(d)

Solution: (d) .

Q10. If is a root of the equation , then the value of is :

(a) 2

(b) – 2

(c)

(d)

Solution: (a) 2

[ Given ,

]

Q11. If the roots of are reciprocal of each other, then

(a)

(b)

(c)

(d)

Solution: (d)

[ let and are two roots .

A/Q, ]

Q12. 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

]

Q13. 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 ]

Q14. The discriminant of the quadratic equation is [CBSE2020 basic]

(a) 12

(b) 84

(c)

(d) – 12

Solution: (d) – 12

[ We have , ;

Here , , ,

The discriminant ]

Q15. 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,

]

Q16. Which one of the following is not a quadratic equation ? [SEBA 2017]

(a)

(b)

(c)

(d)

Solution: (b) .

Q17. 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,

]

Q18. The roots of a quadratic equation are and , then the equation is : [SEBA 2013]

(a)

(b)

(c)

(d)

Solution: (b)

[ The quadratic equation,

]

Q19. Under what condition the roots of the quadratic equation will be real and unequal ? [ SEBA 2015]

(a)

(b)

(c)

(d)

Solution: (c) .

Q20. 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

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 ]

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 .

[Delhi2015]

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,

Section = II

Case study based questions are compulsory . Attempt any four sub parts of each question .

Each subpart carries 1 marks.

Q1. A quadratic equation in the variable is of the form , where are

real numbers and .

(a) Which of the following is a quadratic equation ?

(i) (ii)

(iii) (iv)

(b) The product of roots of quadratic equation is :

(i) (ii) (iii) (iv)

(c) If is a example of quadratic equation ,then the roots of given equation is :

(i) (ii) (iii) (iv)

(d) The nature of the quadratic equation is :

(i) coincident roots (ii) no real roots

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

Solution: (a) (iv)

[ We have ,

]

(b) (iii) [ The product of roots . ]

(c) (i) [ We have ,

]

(d) (ii) no real roots .

[ Here, , ,

The discriminant

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

Q3. Suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth . [ We assume be the breadth of the prayer hall ]

Answer the questions based upon this situation.

(a) What is the length of the prayer hall (in metres) ?

(i)

(ii)

(iii)

(iv)

(b) What is the quadratic equation of the above statement ?

(i)

(ii)

(iii)

(iv)

(c) What should be the length and breadth of the hall ( in metres) ?

(i) 24 , 12

(ii) 26 , 12

(iii) 25 , 12

(iv) 25 , 13

(d) In the given figure, if 13m decrease of length of the prayer hall and 13m increase of the breadth , then the shape of prayer hall is :

(i) Rectangle

(ii) Square

(iii) parallelogram

(iv) Rhombus

Solution: (a) (iii)

[ Given be the breadth of prayer hall , then the length of prayer hall ]

(b) (iv)

[ Given be the breadth of prayer hall , then the length of prayer hall is

A/Q ,

]

(c) (iii) 25 , 12

[ Given be the breadth of prayer hall , then the length of prayer hall is

A/Q ,

Thus, the breadth of prayer hall is 12 m and the length of prayer hall is m ]

(d) (ii) Rectangle .

SECTION = B

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 , 2016, 2014 F ]

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 :

[SEBA2020]

Solution: We have ,

Here , , ,

(impossible) or

Therefore, the value of is .

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) or

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 .

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 ,

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 taken to go upstream and downstream

A/Q,

(impossible)

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

Posted 5 years ago

4. Quadratic Equations

Quadratic Equations