Quadratic Equations Solution of MHB Class 10 Mathematics Part 1

Decide which of the following are quadratic equations.

(1) x² + 5x – 2 = 0

(2) y² = 5y - 10

(3)

(4)

(5) (m + 2)(m–5) = 0

(6) m³ + 3m² – 2 = 3 m³

Answer:

1. is a quadractic equation because it is the form of and it has degree 2.

∴ it is a quadratic equation because it is the form of and it has degree 2.

⇒

∴ it is not a quadratic equation because it is not in the form of and it does not have degree 2.

∴ it is a quadratic equation because it is the form of and it has degree 2.

⇒

∴ it is a quadratic equation because it is the form of and it has degree 2.

⇒

∴ it is not a quadratic equation because it is not in the form of and it does not have degree 2.

Question 3.

Write the following equations in the form ax² + bx + c = 0, then write the values of a, b, c for each equation.

(1) 2y = 10 – y²

(2) (x-1)² = 2x + 3

(3) x² + 5x = - (3-x)

(4) 3m² = 2m² - 9

(5) P(3 + 6p) = - 5

(6) x² – 9 = 13

Answer:

(1)

⇒

y² + 2y - 10 = 0;

a = 1, b = 2, c = -10

(2)

⇒

⇒ x² - 4x - 2 = 0;

a = 1, b = -4, c = -2

(3) ⇒

⇒

⇒ x² + 4x + 3 = 0;

a = 1, b = 4, c = 3

(4) ⇒

a = 1, b = 0, c = 9

(5) ⇒

;

a = 6, b = 3, c = 5

(6) ⇒

= 0

a = 1, b = 0, c = -22

Question 4.

Determine whether the values given against each of the quadratic equation are the roots of the equation.

(1) x² + 4x – 5 = 0, x = 1, -1

(2)

Answer:

Put

⇒

Put

⇒

∴ is a root of the equation and is not a root of the equation.

Put , ⇒ ⇒

Put , ⇒

∴ is not root of the equation and is a root of the equation.

Question 5.

Find k if x = 3 is a root of equation kx² – 10x + 3 = 0.

Answer:

Put

⇒

Question 6.

One of the roots of equation 5m² + 2m + k = 0 is Complete the following activity to find the value of 'k'.

Answer:

is a root of quadratic equation

∴ Put in the equation.

⇒

Practice Set 2.2

Question 1.

Solve the following quadratic equation by factorization.

x² – 15x + 54 = 0

Answer:

⇒

Hence, are roots of the equation.

Question 2.

Solve the following quadratic equation by factorization.

x² + x – 20 = 0

Answer:

⇒

Hence, are roots of the equation.

Question 3.

Solve the following quadratic equation by factorization.
2y² + 27y + 13 = 0

Answer:

⇒ = 0

⇒

⇒ (2y + 1) (y + 13) = 0

Hence, are roots of the equation.

Question 4.

Solve the following quadratic equation by factorization.

5m² = 22m + 15

Answer:

⇒

∴ Hence, are roots of the equation.

Question 5.

Solve the following quadratic equation by factorization.

Answer:

⇒

Question 6.

Solve the following quadratic equation by factorization.

Answer:

⇒

Hence, are roots of the equation.

Question 7.

Solve the following quadratic equation by factorization.

to solve this quadratic equation by factorization, complete the following activity.

Answer:

∴

Question 8.

Solve the following quadratic equation by factorization.

Answer:

⇒

Question 9.

Solve the following quadratic equation by factorization.

2m (m-24) = 50

Answer:

⇒

Hence, are roots of the equation.

Question 10.

Solve the following quadratic equation by factorization.

25m² = 9

Answer:

⇒

Hence, are roots of the equation.

Question 11.

Solve the following quadratic equation by factorization.

7m² = 21m

Answer:

⇒

Hence, are roots of the equation.

Question 12.

Solve the following quadratic equation by factorization.

m² - 11 = 0

Answer:

⇒

Hence , m = ±11 are roots of the equation.

Practice Set 2.3

Question 1.

Solve the following quadratic equation by completing the square method.

x² + x – 20 = 0

Answer:

⇒

Question 2.

Solve the following quadratic equation by completing the square method.

x² + 2x – 5 = 0

Answer:

⇒

Question 3.

Solve the following quadratic equation by completing the square method.

m² – 5m = -3

Answer:

⇒ (Adding and Subtracting )

⇒

_⇒

Question 4.

Solve the following quadratic equation by completing the square method.

9y² – 12y + 2 = 0

Answer:

(3y)² - 2 × 3y × 4 + (4)² - (4)² + 2 = 0

(3y)² - 2 × 3y × 4 + (4)² - 16 + 2 = 0

(3y - 4)² - 14 = 0

(3y - 4)² = 14

3y - 14 = ±√14

3y = 14 ± √14

y = (14 ± √14)/3

Question 5.

Solve the following quadratic equation by completing the square method.

2y² + 9y + 10 = 0

Answer:

Steps involved in solving quadratic equation by completing the square method are -
1. Making the first variable free of coefficient
Dividing by the coefficient of 2, we get,

⇒
2. The coefficient of linear variable(variable with degree 1) is then squared and then added and subtracted from the equation.

⇒
3. Take out the terms following the formula (a + b)² = a² + b² + 2 a b

⇒

Question 6.

Solve the following quadratic equation by completing the square method.

5x² = 4x + 7 = 0

Answer:

⇒

⇒ (Adding and Subtracting )

⇒

Practice Set 2.4

Question 1.

Compare the given quadratic equations to the general form and write values of a, b, c.

(1) x² – 7x + 5 = 0

(2) 2m² = 5m – 5

(3) y² = 7y

Answer:

(1)

a = 1, b = -7, c = 5

(2)

a = 2, b = -5, c = 5

(3)

a = 1, b = -7, c = 0

Question 2.

Solve using formula.

x² + 6x + 5 = 0

Answer:

⇒

Question 3.

Solve using formula.

x² – 3x – 2 = 0

Answer:

⇒

Question 4.

Solve using formula.

3m² + 2m – 7 = 0

Answer:

⇒

Question 5.

Solve using formula.

5m² – 4m – 2 = 0

Answer:

⇒

Question 6.

Solve using formula.

Answer:

⇒

Question 7.

Solve using formula.

5x² + 13x + 8 = 0

Answer:

⇒

Question 8.

With the help of the flow chart given below solve the equation using the formula.

Answer:

⇒

Practice Set 2.5

Question 1.

Activity: Fill in the gaps and complete.

(1)

(2)

(3) If α, β are roots of quadratic equation,

Answer:

(1) Roots are distinct and real when b² - 4ac = 5, not real when b² - 4ac = -5.

(2) x² + 7x + 5 = 0

(3)

Question 2.

Find the value of discriminant.

x² + 7x – 1 = 0

Answer:

⇒

Question 3.

Find the value of discriminant.

2y² – 5y + 10 = 0

Answer:

⇒

Question 4.

Find the value of discriminant.

Answer:

⇒

Question 5.

Determine the nature of roots of the following quadratic equation.

x² – 4x + 4 = 0

Answer:

⇒

∴

Question 6.

Determine the nature of roots of the following quadratic equation.

2y² – 7y + 2 = 0

Answer:

⇒

∴

Question 7.

Determine the nature of roots of the following quadratic equation.

m² + 2m + 9 = 0

Answer:

⇒

∴

Question 8.

Form the quadratic equation from the roots given below.

(1) 0 and 4

(2) 3 and -10

(3)

(4)

Answer:

(1) Let

∴

(2) Let

∴

(3) Let

∴

(4) Let

∴

Question 9.

Sum of the roots of a quadratic equation is double their product. Find k if equation is

x² – 4kx + k + 3 = 0

Answer:

According to question

⇒

Question 10.

α, β are roots of y² – 2y – 7 = 0 find,

(1) α² + β²

(2) α³ + β³

Answer:

(1).

⇒

(2).

⇒

Question 11.

The roots of each of the following quadratic equation are real and equal, find k.

3y² + ky + 12 = 0

Answer:

⇒

∴

⇒

∴

Question 12.

The roots of each of the following quadratic equation are real and equal, find k.

kx (x-2) + 6 = 0

Answer:

⇒

∴

⇒

∴

Practice Set 2.6

Question 1.

Product of Pragati’s age 2 years ago and 3 years hence is 84. Find her present age.

Answer:

Let her present age be x

According to question,

⇒

Question 2.

The sum of squares of two consecutive natural numbers is 244; find the numbers.

Answer:

Let the two consecutive natural numbers be x and x + 2. Then,

⇒

No.s cannot be negative, ∴ numbers are 10 and 12

Question 3.

In the orange garden of Mr. Madhusudan there are 150 orange trees. The number of trees in each row is 5 more than that in each column. Find the number of trees in each row and each column with the help of following flow chart.

Answer:

Let the number of columns be x

∴ rows = x + 5

⇒

Hence, columns cannot be negative. ∴ columns are 10

and rows are 15.

Question 4.

Vivek is older than Kishor by 5 years. The sum of the reciprocals of their ages is 1/6. Find their present ages.

Answer:

Let Kishor’s present age be x. Then, vivek’s age = x + 5

∴

⇒

Hence, age cannot be negative. ∴ age od Kishor is 10

and age of Vivek is 15.

Question 5.

Suyash scored 10 marks more in second test than that in the first. 5 times the score of the second test is the same as square of the score in the first test. Find his score in the first test.

Answer:

Let the score of first test be x. Then, second test score = x + 10.

∴

⇒

Hence, score of first test is 10 as marks are not negative.

Question 6.

Mr. Kasam runs a small business of making earthen pots. He makes certain number of pots on daily basis. Production cost of each pot is ₹40 more than 10 times total number of pots, he makes in one day. If production cost of all pots per day is ` 600, find production cost of one pot and number of pots he makes per day.

Answer:

Let the number of pots made by Mr. Kasam each day be x. Then, production cost of each pot =

∴ total cost =

⇒

Hence number of pots made cannot be negative. ∴ number of pots he made each day = 6

Cost of one pot =

Question 7.

Pratik takes 8 hours to travel 36 km downstream and return to the same spot. The speed of boat in still water is 12 km. per hour. Find the speed of water current.

Answer:

Let the speed of water current be x.

∴

⇒

Speed od water current is 6km/hr

Question 8.

Pintu takes 6 days more than those of Nishu to complete certain work. If they work together they finish it in 4 days. How many days would it take to complete the work if they work alone.

Answer:

Suppose Nishu alone takes x days to finish work. Then , Pintu alone can finish in (x + 6)days.

⇒ Nishu’s one day work + Pintu’s one day work =

(Nishu + Pintu)’s one day work =

∴

⇒

x = -4 is not possible, as no of days can't be negative.

Nishu will take 6 days alone and Pintu takes 12 days alone.

Question 9.

If 460 is divided by a natural number, quotient is 6 more than five times the divisor and remainder is 1. Find quotient and diviser.

Answer:

Let the divisor be x. Then, Quotient be 6 + 5x

Now according to question,

⇒

∴ divisor = 9 and quotient =

∴Divisor = 9, quotient = 51

Question 10.

In the adjoining fig. □ABCD is a trapezium AB||CD and its area is 33 cm². From the information given in the figure find the lengths of all sides of the □ABCD. Fill in the empty boxes to get the solution.

Answer:

□ ABCD is a trapezium.

AB||CD

A(□ABCD) =

∴

But length is never negative.

∴

AB = 7 cm, CD = 15 cm, AD = BC = 5 cm.

Problem Set 2

Question 1.

Choose the correct answer for the following question.

Which one is the quadratic equation?
A.

B. x (x + 5) = 2

C. n-1 = 2n

D.

Answer:

In option A hence , it is not a quadratic equation.

In Option B , it is a quadratic equation.

In Option C, it is not a quadratic equation.

In Option D , hence, it is not a quadratic equation.

Question 2.

Choose the correct answer for the following question.

Out of the following equations which one is not a quadratic equation?
A. x² + 4x = 11 + x²

B. x² = 4x

C. 5x² = 90

D. 2x – x² = x² + 5

Answer:

⇒

In all other options highest degree of equation is 2, which also the degree of quadratic equation. But in Option A, degree of polynomial is 1

Question 3.

Choose the correct answer for the following question.

The roots of x² + kx + k = 0 are real and equal, find k.
A. 0

B. 4

C. 0 or 4

D. 2

Answer:

equation has real and equal roots.

∴

⇒

k = 0 or

∴ k = 0 or 4

Question 4.

Choose the correct answer for the following question.

For find the value of the discriminant.
A. -5

B. 17

C. 2

D.

Answer:

⇒

Question 5.

Choose the correct answer for the following question.

Which of the following quadratic equations has roots 3, 5?
A. x² – 15x + 8 = 0

B. x² – 8x + 15 = 0

C. x² + 3x + 5 = 0

D. x² + 8x - 15 = 0

Answer:

In option A,

⇒

In option B

⇒

In option c,

⇒

In option d

Question 6.

Choose the correct answer for the following question.

Out of the following equations, find the equation having the sum of its roots -5.
A. 3x² – 15x + 3 = 0

B. x² – 5x + 3 = 0

C. x² + 3x - 5 = 0

D. 3x² + 15x + 3 = 0

Answer:

Sum of the roots i.e.

∴ in option A,

∴ in option B,

∴ in option A,

Question 7.

Choose the correct answer for the following question.

which of the following statement is true for this given equation?
A. Real and uneual roots

B. Real and equal roots

C. Roots are not real

D. Three roots.

Answer:

⇒

∴

Question 8.

Choose the correct answer for the following question.

One of the roots of equation x² + mx – 5 = 0 is 2; find m.
A. -2

B.

C.

D. 2

Answer:

, Put value of x = 2

Question 9.

Which of the following equations is quadratic?

(1) x² + 2x + 11 = 0

(2) x² – 2x + 5 = x²

(3) (x + 2)² = 2x²

Answer:

1. is a quadractic equation because it is the form of and it has degree 2.

∴ it is not a quadratic equation because it is not in the form of and it doesn’t have degree 2.

is a quadractic equation because it is the form of and it has degree 2.

Question 10.

Find the value of discriminant for each of the following equation.

2y² – y + 2 = 0

Answer:

⇒

Question 11.

Find the value of discriminant for each of the following equation.

5m² – m = 0

Answer:

⇒

Question 12.

Find the value of discriminant for each of the following equation.

Answer:

⇒

Question 13.

One of the roots of quadratic equation 2x² + kx – 2 = 0 is -2, find k.

Answer:

⇒

Question 14.

Two roots of quadratic equations are given ; frame the equation.

10 and -10

Answer:

Let α = 10 and β = -10

∴ α + β = 10 - 10
= 0

α β = 10(-10)
= - 100

∴

⇒ x² - 0(x) - 100 = 0

⇒ x² - 100 = 0

Question 15.

Two roots of quadratic equations are given ; frame the equation.

1–3√5 and 1 + 3√5

Answer:

Let

∴

Question 16.

Two roots of quadratic equations are given ; frame the equation.

0 and 7

Answer:

: Let

∴

Question 17.

Determine the nature of roots for each of the quadratic equation.

3x² – 5x + 7 = 0

Answer:

⇒

∴

Question 18.

Determine the nature of roots for each of the quadratic equation.

Answer:

⇒

∴

Question 19.

Determine the nature of roots for each of the quadratic equation.

m² – 2m + 1 = 0

Answer:

⇒

∴

Question 20.

Solve the following quadratic equation.

Answer:

⇒

Question 21.

Solve the following quadratic equation.

Answer:

⇒

Question 22.

Solve the following quadratic equation.

(2x + 3)² = 25

Answer:

⇒

Question 23.

Solve the following quadratic equation.

m² + 5m + 5 = 0

Answer:

⇒

Question 24.

Solve the following quadratic equation.

5m² + 2m + 1 = 0

Answer:

⇒

Hence , roots are not real.

Question 25.

Solve the following quadratic equation.

x² – 4x – 3 = 0

Answer:

⇒

Question 26.

Find m if (m-12)x² + 2 (m - 12)x + 2 = 0 has real and equal roots.

Answer:

⇒

∴

⇒

m = 12 or m = 14

Question 27.

The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find the equation.

Answer:

⇒

∵

⇒

Question 28.

Find quadratic equation such that its roots are square of sum of the roots and square of difference of the roots of equation

2x² + 2(p + q)x + p² + q² = 0

Answer:

Let’s assume roots are m and n.

So, we want the equation whose roots would be

So, the sum of the roots of our desired equation would be and product of the roots would be

What we know from given equation are:

and

the sum and product are:

and

Our desired equation would be

So, x² - 4pqx - (p² - q²)² = 0 is our desired equation

Question 29.

Mukund possesses ₹50 more than what Sagar possesses. The product of the amount they have is 15,000. Find the amount each one has.

Answer:

Let Sagar has x amount

Mukund’s amount =

⇒

Splitting the middle term we get:-

⇒ x² - 100x + 150x - 15000 = 0

⇒ x(x - 100) + 150(x - 100)

⇒ (x - 100)(x+150)

∴ x = (-150) , 100

x = 100 as money cant be negative therefore we ignore (-150)

∴ Sagar has 100Rs and Mukund has 150Rs

Question 30.

The difference between squares of two numbers is 120. The square of smaller number is twice the greater number. Find the numbers.

Answer:

Let the two numbers be a and b, such that, .

As per the given conditions,

The difference of the square of the two numbers is 120.

The square of smaller number is 2 times the larger number.

Put the value of from eq. II in Eq. I, it gives

⇒

12 and or 12 and -

Question 31.

Ranjana wants to distribute 540 oranges among some students. If 30 students were more each would get 3 oranges less. Find the number of students.

Answer:

Total oranges = 540

Initial student =

Initial orange for 1 student =

∵

⇒

⇒ (∵

∴ number of students = 60 students.

Question 32.

Mr. Dinesh owns an agricultural farm at village Talvel. The length of the farm is 10 meter more than twice the breadth. In order to harvest rain water, he dug a square shaped pond inside the farm. The side of pond is 1/3 of the breadth of the farm. The area of the farm is 20 times the area of the pond. Find the length and breadth of the farm and of the pond.

Answer:

Let the breadth of the farm be x.

∴

According to the question,

⇒

∴

Breadth 45 m. length 100 m, side of the pond 15 m.

Question 33.

A tank fills completely in 2 hours if both the taps are open. If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank. How much time does each tap take to fill the tank completely?

Answer:

Let the time taken by larger tap alone be x hr. Then ,

Time taken by smaller tap be x + 3 hr

In an hour, the larger tap can fill tank.

∴ In an hour, the larger tap can fill tank.

Two taps together can fill a tank in 2 hr.

But in an hour, taps fill in of the tank.

∴

⇒

For larger tap 3 hours and for smaller tap 6 hours.

Quadratic Equations

Class 10th Mathematics Part 1 MHB Solution

Practice Set 2.1

Practice Set 2.2

Practice Set 2.3

Practice Set 2.4

Practice Set 2.5

Practice Set 2.6

Problem Set 2

Class 10^th Mathematics Part 1 MHB Solution