NCERT book Solutions for Class 10 Maths Polynomials Exercise 2.2
NCERT book Solutions for Class 10 Maths Chapter 2 Exercise 2.2
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Exercise 2.2 on page 33 . Hope you like them and do not forget to like , social_share
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Question 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x2 – 2x – 8
(ii) 4s2 – 4s + 1
(iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u
(v)t2 – 15
(vi) 3x2 – x – 4 Answer
(i) x2 – 2x – 8
= x2 -4x+ 2x – 8
= (x - 4) (x + 2)
Therefore, the zeroes of x2 – 2x – 8 are 4 and -2.
(ii) 4s2 – 4s + 1
From (a-b)2 = a2 -2ab + b2
= (2s-1)2
Therefore, the zeroes of 4s2 - 4s + 1 are 1/2 and 1/2.
(iii) 6x2 – 3 – 7x
= 6x2 – 7x – 3
= 6x2 -9x +2x -3
= (3x + 1) (2x - 3)
Therefore, the zeroes of 6x2 - 3 - 7x are -1/3 and 3/2.
(iv) 4u2 + 8u
= 4u2 + 8u
= 4u(u + 2)
Therefore, the zeroes of 4u2 + 8u are 0 and - 2.
(v) t2 – 15
From (a2 -b2) =(a-b) (a+b)
= (t - √15) (t + √15)
Therefore, the zeroes of t2 - 15 are √15 and -√15.
(vi) 3x2 – x – 4
=3x2 – 4x+3x – 4
= (3x - 4) (x + 1)
Therefore, the zeroes of 3x2 – x – 4 are 4/3 and -1. Verification of the relationship between the zeroes
S. No
Sum of zeroes=-(Coefficient of x)/Coefficient of x2
Product of zeroes= Constant term/Coefficient of x2.
i)
4 + (-2) = 2 = -(-2)/1
4 × (-2) = -8 = -8/1
ii)
1/2 + 1/2 = 1 = -(-4)/4
1/2 × 1/2 = 1/4
iii)
-1/3 + 3/2 = 7/6 = -(-7)/6
-1/3 × 3/2 = -1/2 = -3/6
iv)
0 + (-2) = -2 = -(8)/4
0 × (-2) = 0 = 0/4
v)
√15 + -√15 = 0 = -0/1
(√15) (-√15) = -15 = -15/1
vi)
4/3 + (-1) = 1/3 = -(-1)/3
4/3 × (-1) = -4/3
Question 2
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4, -1
(ii) √2, 1/3
(iii) 0, √5
(iv) 1,1
(v) -1/4 ,1/4
(vi) 4,1 Answer
(i) 1/4 , -1
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = 1/4 = -b/a
pq = -1 = -4/4 = c/a
Now we have two method to find the quadratic polynomial Method -1
The polynomial can be written as
k[x2 - (sum of roots)x + (products of roots)]
So,
$k[x^2 - (\frac {1}{4})x -1]$
Taking k=4
$4x^2 -x -4$ Method -2
If we take a= 4, then b = -1, c= -4
Therefore, the quadratic polynomial is 4x2 - x -4.
We could choose any of the method to get the polynomial
(ii) √2 , 1/3
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = √2 = 3√2/3 = -b/a
pq = 1/3 = c/a
If a = 3, then b = -3√2, c = 1
Therefore, the quadratic polynomial is 3x2 -3√2x +1.
(iii) 0, √5
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = 0 = 0/1 = -b/a
pq = √5 = √5/1 = c/a
If a = 1, then b = 0, c = √5
Therefore, the quadratic polynomial is x2 + √5.
(iv) 1, 1
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = 1 = 1/1 = -b/a
pq = 1 = 1/1 = c/a
If a = 1, then b = -1, c = 1
Therefore, the quadratic polynomial is x2 - x +1.
(v) -1/4 ,1/4
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = -1/4 = -b/a
pq = 1/4 = c/a
If a = 4, then b = 1, c = 1
Therefore, the quadratic polynomial is 4x2 + x +1.
(vi) 4,1
Let the polynomial be ax2 + bx + c, and its zeroes be p and q
p + q = 4 = 4/1 = -b/a
pq = 1 = 1/1 = c/a
If a = 1, then b = -4, c = 1
Therefore, the quadratic polynomial is x2 - 4x +1.
