## Chapter 2 Polynomials R.D. Sharma Solutions for Class 10th Math Exercise 2.1

Exercise 2.1

1. Find the zeroes of each of the following quadratic polynomials and verify the relationship between the zeroes and their co efficient:
(i) f(x) = x2 - 2x - 8
(ii) g(s) = 4s2 - 4x + 1
(iii) h(t) = t2 - 15
(iv) p(x) = x2 +2√2x + 6
(v) q(x) = √3x2 + 10x + 7√3
(vi) f(x) = x2 - (√3+1)x + √3
(vii) g(x) = a(x2 +1) - x(a2 +1)
(viii) 6x2 - 3 - 7x

Solution

2. For each of the following , find a quadratic polynomial whose sum and product respectively of the zeroes are as give . Also, find the zeroes of these polynomials by factorization.
(i) -8/3, 4/3
(ii) 21/8, 5/16
(iii) -2√3, -9
(iv) -3/2√5, -1/2

Solution

3. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 – 5x + 4, find the value of 1/Î± + 1/ Î² - 2 Î±Î².

Solution

4. If Î± and Î² are the zeros of the quadratic polynomial p(y) = 5y2 - 7y + 1, find the value of 1/Î± + 1/Î² - 2Î±Î² .

Solution

5. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 - x - 4, find the value of 1/Î± + 1/ Î² - Î±Î² .

Solution

6. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 + x - 2, find the value of 1/Î± - 1/ Î² .

Solution

7. If one zeros of the quadratic polynomial f(x) = 4x2 - 8kx - 9 is negative of the other , find the value of k .

Solution

8. If the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k. product, find the value of k.

Solution

9. If Î± and Î² are the zeros of the quadratic polynomial p(x) = 4x2 - 5x - 1, find the value of Î±2Î² + Î±Î²2.

Solution

10. If Î± and Î² are the zeros of the quadratic polynomial f(t) = t2 - 4t + 3 , find the value of Î±4Î²3 + Î±3Î²4.

Solution

11. If Î± and Î² are the zeros of the quadratic polynomial f(x) = 6x2 + x - 2 , find the value of Î±/Î² + Î²/Î±.

Solution

12. If Î± and Î² are the zeros of the quadratic polynomial p(s) = 3s2 – 6s + 4, find the value Î±/Î² + Î²/Î± + 2 (1/Î± + 1/Î²) + 3Î±Î² .

Solution

13. If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.

Solution

14. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 – px + q , prove that Î±2/Î²2 + Î²2/ Î±2 = p4/q2 – 4p2/q + 2

Solution

15. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 – p(x+1) – c, show that (Î± + 1) (Î²+1) = 1 – c .

Solution

16. If Î± and Î² are the zeros of a quadratic polynomial such that Î± + Î² = 24 and Î± − Î² = 8, find a quadratic polynomial have Î± and Î² as its zeros.

Solution

17. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are 2 Î±/Î² and 2 Î²/Î±

Solution

18. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeroes are 1/2Î± + Î² + 1/2Î² + Î± .

Solution

19. If Î± and Î² are the zeros of the polynomial f(x) = x2 + px + q , form a polynomial whose zeroes are (Î±+ Î²)2 and (Î± – Î²)2

Solution

20. If Î± and Î² are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are (i) Î±+2, Î²+2 (ii) Î±–1/Î±+1, Î²–1/ Î²+1

Solution

21. If Î± and Î² are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, the evaluate :-

Solution