Class 10 Maths Chapter 2 HOTS Questions - Polynomials

Q1: Find the Quadratic polynomial whose sum and product of zeros are √2 + 1, 1/√2 + 1 .
Ans: sum = 2 √2
Product = 1
Q.P =
X² – (sum) x + Product
∴ x² – (2 √2 ) x + 1

Q2: If α,b are the zeros of the polynomial 2x² – 4x + 5 find the value of a) α² + β² b) (α - β)².
Ans: p (x) = 2 x² – 4 x + 5
α + β = -b/a = 4/2 = 2
αβ = c/a = 5/2
α² + β² = (α + β)² – 2αβ
Substitute then we get, α 2 + β2 = -1
(α - β)² = (α + β)² - 4 α β
Substitute, we get = (α - β)² = - 6

Q3: On dividing the polynomial 4x⁴ - 5x³ - 39x² - 46x – 2 by the polynomial g(x) the quotient is x² - 3x – 5 and the remainder is -5x + 8.Find the polynomial g(x).
Ans: p(x) = g (x) q (x) + r (x)
g(x) = p(x) - r(x)/q(x)
let p(x) = 4x⁴ – 5x³ – 39x² – 46x – 2
q(x) = x² – 3x – 5 and r (x) = -5x + 8
now p(x) – r(x) = 4x⁴ – 5x³ – 39x² – 41x - 10
when p(x) - r(x)/q(x) = 4x² + 7x + 2
∴ g(x) = 4x² + 7x + 2

Q4: If the squared difference of the zeros of the quadratic polynomial x² + px + 45 is equal to 144 , find the value of p.
Ans: Let two zeros are a and b where α > β
According given condition
(α - β)² = 144
Let p(x) = x² + px + 45
α + β = -b/a = -p/1 = -p
αβ = c/a = 45/1 = 45
now (a - β)² = 144
(α + β)² – 4 αβ = 144
(-p)² – 4 (45) = 144
Solving this we get p = ± 18

Q5: If α, β are the zeros of a Quadratic polynomial such that α + β = 24, α - β = 8. Find a Quadratic polynomial having α and β as its zeros.
Ans: α + β = 24
α - β = 8
_________
2α = 32
α = 32/2 = 16, ∴ α = 16
Work the same way to α+β = 24
So, β = 8
Q.P is x² – (sum) x + product
= x² – (16+8) x + 16 x 8
Solve this,
it is k (x² – 24x + 128)

Q6: Obtain all the zeros of the polynomial p(x) = 3x⁴ - 15x³ + 17x² +5x -6 if two zeroes are -1/√3 and 1/√3.
Ans: 3,2

Q7: If two zeros of the polynomial f(x) = x⁴ - 6x³ - 26x² + 138x – 35 are 2± √3.Find the other zeros.
Ans: Let the two zeros are 2 + √3 and 2 - √3
Sum of Zeros = 2 + √3 + 2 - √3
= 4
Product of Zeros = (2+ √3)(2 - √3)
= 4 – 3
= 1
Quadratic polynomial is x² – (sum) x + Product)


Q8: If α and β are the zeroes of the polynomial x² + 8x + 6 frame a quadratic polynomial whose zeroes are  
(a) 1/α and 1/β
(b) 1 + β/α, 1 + α/β.
Ans: Given polynomial x² + 8x + 6

Hence, the Required Quadratic polynomial f(x) is given by


Q9: If α & β are the zeroes of the polynomial 2x² - 4x + 5, then find the value of
(i) α² + β² 
(ii) 1/a + 1/ß
(iii) (α - β)² 
(iv) 1/α² + 1/β² 
(v) α³ + β³
Ans: 



Q10: If the ratios of the polynomial ax³ + 3bx² + 3cx + d are in AP, Prove that 2b³ - 3abc + a²d = 0
Ans: Let the zeros of the given polynomial be p, q, r. As the roots are in A.P., then it can be assumed as p−k, p, p + k, where k is a common difference.