Problem Set 2 Quadratic Equations Class 10th Mathematics Part 1 MHB Solution

Problem Set 2

1. Which one is the quadratic equation? Choose the correct answer for the following…
2. Out of the following equations which one is not a quadratic equation? Choose the…
3. The roots of x^2 + kx + k = 0 are real and equal, find k. Choose the correct answer…
4. For root 2x^2 - 5x + root 2 = 0 find the value of the discriminant. Choose the correct…
5. Which of the following quadratic equations has roots 3, 5? Choose the correct answer…
6. Out of the following equations, find the equation having the sum of its roots -5.…
7. root 5m^2 - root 5m + root 5 = 0 which of the following statement is true for this…
8. One of the roots of equation x^2 + mx - 5 = 0 is 2; find m. Choose the correct answer…
9. Which of the following equations is quadratic? (1) x^2 + 2x + 11 = 0 (2) x^2 - 2x + 5 =…
10. 2y^2 - y + 2 = 0 Find the value of discriminant for each of the following equation.…
11. 5m^2 - m = 0 Find the value of discriminant for each of the following equation.…
12. root 5x^2 - x - root 5 = 0 Find the value of discriminant for each of the following…
13. One of the roots of quadratic equation 2x^2 + kx - 2 = 0 is -2, find k.…
14. 10 and -10 Two roots of quadratic equations are given ; frame the equation.…
15. 1-3√5 and 1 + 3√5 Two roots of quadratic equations are given ; frame the equation.…
16. 0 and 7 Two roots of quadratic equations are given ; frame the equation.…
17. 3x^2 - 5x + 7 = 0 Determine the nature of roots for each of the quadratic equation.…
18. root 3x^2 + root 2x-2 root 3 = 0 Determine the nature of roots for each of the…
19. m^2 - 2m + 1 = 0 Determine the nature of roots for each of the quadratic equation.…
20. 1/x+5 = 1/x^2 Solve the following quadratic equation.
21. x^2 - 3x/10 - 1/10 = 0 Solve the following quadratic equation.
22. (2x + 3)^2 = 25 Solve the following quadratic equation.
23. m^2 + 5m + 5 = 0 Solve the following quadratic equation.
24. 5m^2 + 2m + 1 = 0 Solve the following quadratic equation.
25. x^2 - 4x - 3 = 0 Solve the following quadratic equation.
26. Find m if (m-12)x^2 + 2 (m - 12)x + 2 = 0 has real and equal roots.…
27. The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find…
28. Find quadratic equation such that its roots are square of sum of the roots and square…
29. Mukund possesses ₹50 more than what Sagar possesses. The product of the amount they…
30. The difference between squares of two numbers is 120. The square of smaller number is…
31. Ranjana wants to distribute 540 oranges among some students. If 30 students were more…
32. Mr. Dinesh owns an agricultural farm at village Talvel. The length of the farm is 10…
33. A tank fills completely in 2 hours if both the taps are open. If only one of the taps…

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, 

∴ 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.

2. 

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

3. 

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.