Class 10 Mathematics: Quadratic Equations important questions

Quadratic Equations is a chapter of Class 10 Mathematics for the Andhra Pradesh State Board. These 16 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.

1. Practice · 1 marks

   1. Find the discriminant of x² − 5x + 6 = 0.

   D = b² − 4ac = 25 − 24 = 1.

2. Very Important · 8 marks

   2. Find the roots of x² + 3x − 18 = 0 by completing the square.

   x² + 3x = 18 x² + 3x + (3/2)² = 18 + 9/4 (x + 3/2)² = 81/4 x + 3/2 = ±9/2 x = 9/2 − 3/2 = 3, or x = −9/2 − 3/2 = −6. Roots: 3 and −6.

3. Practice · 1 marks

   3. If x = 1/x, then the roots are A) 1 B) −1 C) both 1 and −1 D) none

   C) x = 1/x gives x² = 1, so x = 1 or x = −1.

4. Important · 2 marks

   4. Check whether these are quadratic equations: i) (x − 2)² + 1 = 2x − 3 ii) x(x + 1) + 8 = (x + 2)(x − 2)

   i) x² − 4x + 4 + 1 = 2x − 3 gives x² − 6x + 8 = 0. Degree 2, so it is a quadratic equation. ii) x² + x + 8 = x² − 4 gives x + 12 = 0. Degree 1, so it is not a quadratic equation.

5. Important · 4 marks

   5. Find two numbers whose sum is 27 and product is 182.

   Let the numbers be x and 27 − x. x(27 − x) = 182, so x² − 27x + 182 = 0. (x − 13)(x − 14) = 0, so x = 13 or 14. The numbers are 13 and 14.

6. Practice · 1 marks

   6. Form a quadratic equation which has 1 as one of its roots.

   For example, (x − 1)(x − 2) = 0, that is x² − 3x + 2 = 0. Any equation (x − 1)(x − k) = 0 works.

7. Important · 2 marks

   7. Find the value of k if both roots of 2x² + kx + 3 = 0 are equal.

   For equal roots, b² − 4ac = 0: k² − 4(2)(3) = 0, so k² = 24 and k = ±2√6.

8. Important · 4 marks

   8. Find the discriminant of 3x² − 2x + 1/3 = 0 and hence the nature of its roots. Find the roots if they are real.

   a = 3, b = −2, c = 1/3. D = b² − 4ac = 4 − 4 × 3 × (1/3) = 4 − 4 = 0. Since D = 0, the roots are real and equal: x = −b/(2a) = 2/6 = 1/3. Roots: 1/3 and 1/3.

9. Important · 2 marks

   9. State the nature of the roots of 2x² − 5x + 1 = 0.

   D = b² − 4ac = 25 − 8 = 17. Since D > 0 (and 17 is not a perfect square), the equation has two distinct real roots, which are irrational.

10. Important · 4 marks

    10. State the conditions under which a quadratic equation ax² + bx + c = 0 has i) two distinct real roots ii) two equal roots.

    i) Two distinct real roots when the discriminant b² − 4ac > 0. ii) Two equal real roots when b² − 4ac = 0. (If b² − 4ac < 0, there are no real roots.)

11. Practice · 1 marks

    11. Find the value of k if the equation x² + kx + 6 = 0 has equal roots.

    For equal roots, the discriminant b² − 4ac = 0. k² − 4(1)(6) = 0, so k² = 24 and k = ±2√6.

12. Important · 2 marks

    12. Rahul believes that the quadratic equation x² + 5x + 6 = 0 has both roots positive. Do you agree with him? Justify your answer.

    No. x² + 5x + 6 = (x + 2)(x + 3) = 0, so the roots are x = −2 and x = −3. Both roots are negative. (Also: sum of roots = −5 is negative while product = 6 is positive, which means both roots are negative.)

13. Important · 4 marks

    13. The speed of a boat in still water is 11 km/h. It can go 12 km upstream and return downstream to the original point in 2 hours 45 minutes. Find the speed of the stream.

    Let the speed of the stream be x km/h. Upstream speed = 11 − x, downstream speed = 11 + x. Total time = 2 h 45 min = 11/4 h. 12/(11 − x) + 12/(11 + x) = 11/4 12(11 + x + 11 − x)/(121 − x²) = 11/4 264/(121 − x²) = 11/4 1056 = 11(121 − x²) 96 = 121 − x² x² = 25, so x = 5 (speed cannot be negative). The speed of the stream is 5 km/h.

14. Practice · 1 marks

    14. Write any quadratic equation whose roots are reciprocals of each other.

    Take roots 2 and 1/2. Sum = 5/2, product = 1. Equation: x² − (5/2)x + 1 = 0, or 2x² − 5x + 2 = 0. (Any equation ax² + bx + a = 0 works, because the product of roots c/a = 1.)

15. Important · 2 marks

    15. Student 1 says the length of a rectangular park should be twice its breadth. Student 2 says the area of the park should be 20 m². Use both ideas to find the length and breadth of the park.

    Let the breadth be b m. Then the length is 2b m. Area: 2b × b = 20, so b² = 10 and b = √10 ≈ 3.16 m. Length = 2√10 ≈ 6.32 m.

16. Important · 4 marks

    16. Find two consecutive positive integers the sum of whose squares is 365.

    Let the integers be x and x + 1. x² + (x + 1)² = 365 2x² + 2x + 1 = 365 x² + x − 182 = 0 (x + 14)(x − 13) = 0, so x = 13 (x must be positive). The integers are 13 and 14. (Check: 169 + 196 = 365.)