Important question for Class 10 Quadratic question Board exam

Important question for Class 10 Quadratic question Board exam 2025. Do all these question and get full markes in question of quadratic equation. These question cover all types of question

Q 1:- Which of the following are quadratic equation?

(i) $2x^2 - 7x = 0$

(ii) $x + \dfrac{3}{x} = x^2$

(iii) $x^2 + \dfrac{1}{x^2} = 2$

(iv) $x^2 + 2\sqrt{x} - 3 = 0$

(v) $3x^2 - 4x + 2 = 2x^2 - 2x + 4$

(vi) $(2x + 1)(3x + 1) = 6(x - 1)(x - 2)$

(vii) $(x + \dfrac{1}{x}) = 3(x + \dfrac{1}{x}) + 4$

Q 2:- In each of the following, find the value of k for which the given is a solution of the given equation:

(i) $7x^2 + kx - 3 = 0, x = 2/3$

(ii) $x^2 - x(a + b) + k = 0,x = a$

(iii) $kx^2 + \sqrt{2}x - 4 = 0, x = \sqrt{2}$

Q 3:- If x = 2/3 and x = -3 are the root of the equation $ax^2 + 7x + b = 0$ , find the value of a and b.

Q 4:- Solve the quadratic equation by factorization method:

(i) $x^2 - 8x + 16 = 0$

(ii) $5x^ - 3x - 2 = 0$

(iii) $\dfrac{x}{x + 1}+ \dfrac{x + 1}{x} = \dfrac{34}{15},x \neq 0, x \neq -1$

(iv) $\dfrac{x + 3}{x - 2} - \dfrac{1 - x}{x} = \dfrac{17}{4}$

(v) $\dfrac{x - 1}{x - 2} + \dfrac{x - 3}{x - 4} = 3\dfrac{1}{3}, x \neq2, 4$

Nature of root

5. Determine the nature of the roots of the following quadratic equations:

(i) $2x^2 + x - 1 = 0$

(ii) $2x^2 + 5x + 5 = 0$

(iii) $3x^2 - 2\sqrt{6}x + 2 = 0$

6. Find the values of k for which the roots are real and equal in each of the following equations:

(i) $kx^2 + 4x + 1 = 0$

(ii) $x^2 - 2(5 + 2k)x + 3(7 + 10 k) = 0$

(iii) $(k + 1)x^2 - 2(3k+ 1)x + 8k+1 = 0$

(iv) $(2k + 1)x^2 + 2(k + 1)x + (k + 5) = 0$

Q 7 :- If the roots of the equation $(b - c)x^2 + (c -a)x + (a - b) = 0$ are equal, then prove that $2b = a + c$ .

Q 8:- The sum of two numbers is 15. If the sum of their reciprocas is $\dfrac{3}{10}$ , find the numbers.

Q 9:- The sum of the squares of two consecutive natural numbers is 313. Find the numbers.

Q 10:- A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

Q 11:- A two digit number is such that the product of the digits is 14. When 45 is added to the number, then the digits are reversed. Find the number

Q 12:- If the sum of first n even natural numbers is 420, find the velue of n.

Q 13:- The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2\dfrac{16}{21}$ , find the fraction

Q 14:- A fast train takes 3 hours less than a slow train for a journey of 600km. If the speed of the slow train is 10 km/h less than that of the fast train, find the speeds of the two trains.

Q 15:- In a flight of 600 km, a aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time of flight increased by 30 minutes. Find the duration of flight.

Q 16:- One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Find their present ages.

Q 17:- The product of Ramu’s age(in years) five years ago with his age(in years) 9 years later is 15. Find Ramu’s present age.

Q 18:- Seven years ago Varun’s age was five times the square of Swati’s age. Three year hence Swati’s age will be two fifth of Varun’s age. Find their present ages.

Q 19:- A farmer wishes to grow a 100 squre meter rectangular vegetable garden. Since he has with the only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side-fence. Find the dimensions of his garden.

Q 20:- A chess board contains 64 equal squares and the area of each square is 6.25 $cm^2$ . A border round the board is 2 cm wide. Find the length of the side of the chess board.

Q 21:- Some students planned a picnic. The budget for food was Rs 480. But eight of these failed to go and thus the cost of food for each mamber increased by Rs. 10. How many students attended the picnic ?

Nature of root

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