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NCERT Solutions for Class 10 Maths Polynomials Exercise 2.4




In this page we have NCERT Solutions for Class 10 Maths Polynomials for EXERCISE 2.4 . Hope you like them and do not forget to like , social_share and comment at the end of the page.
Question 1.
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 - 5+ 2; 1/2, 1, -2
(ii) x3 - 4x2 + 5x - 2; 2, 1, 1
Answer
(i) p(x) = 2x3 + x2 - 5+ 2
Now for verification of zeroes, putting the given value in x.
P (1/2) = 2(1/2)3 + (1/2)2 - 5(1/2) + 2
= (2×1/8) + 1/4 - 5/2 + 2
= 1/4 + 1/4 - 5/2 + 2
= 1/2 - 5/2 + 2 = 0
P (1) = 2(1)3 + (1)2 - 5(1) + 2
= (2×1) + 1 - 5 + 2
= 2 + 1 - 5 + 2 = 0
P (-2) = 2(-2)3 + (-2)2 - 5(-2) + 2
= (2 × -8) + 4 + 10 + 2
= -16 + 16 = 0
Thus, 1/2, 1 and -2 are the zeroes of the given polynomial.
Comparing the given polynomial with ax3 + bx2 + c+ d, we get a=2, b=1, c=-5, d=2
Also, k1=1/2, k2=1 and k3=-2
NCERT Solutions for Question  based on Relationship between the zeros and Coefficent in Cubic poplynomial
Now,
-b/a = k1 + k2 +k3
⇒ 1/2 = 1/2 + 1 - 2
⇒ 1/2 = 1/2

c/a = k1k2+k2k3+k1k3
⇒ -5/2 = (1/2 × 1) + (1 × -2) + (-2 × 1/2)
⇒ -5/2 = 1/2 - 2 - 1
⇒ -5/2 = -5/2
-d/a = k1k2k3
⇒ -2/2 = (1/2 × 1 × -2)
⇒ -1 = 1
Thus, the relationship between zeroes and the coefficients are verified.
(ii)  p(x) = x3 - 4x2 + 5x - 2
Now for verification of zeroes, putting the given value in x.
p(2) = 23 - 4(2)2 + 5(2) - 2
= 8 - 16 + 10 - 2
= 0
p(1) = 13 - 4(1)2 + 5(1) - 2
= 1 - 4 + 5 - 2
= 0
p(1) = 13 - 4(1)2 + 5(1) - 2
= 1 - 4 + 5 - 2
= 0
Thus, 2, 1 and 1 are the zeroes of the given polynomial.
Comparing the given polynomial with ax3 + bx2 + c+ d, we get a=1, b=-4, c=5, d=-2
Also, k1=2, k2=1 and k3=1
NCERT Solutions for Exercise 2.4 Question  based on Relationship between the zeros and Coefficent in Cubic poplynomial
 
Now,
-b/a = k1 + k2 +k3
⇒ 4/1 = 2 + 1 + 1
⇒ 4 = 4

c/a = k1k2+k2k3+k1k3
⇒ 5/1 = (2 × 1) + (1 × 1) + (1 × 2)
⇒ 5 = 2 + 1 + 2
⇒ 5 = 5
-d/a = k1k2k3
⇒ 2/1 = (2 × 1 × 1)
⇒ 2 = 2
Thus, the relationship between zeroes and the coefficients are verified.

Question 2.
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.
Answer
Let the polynomial be ax3 + bx+ cx + d and the zeroes be k1, k2 and k3
Question based on Relationship between the zeros and Coefficent in Cubic poplynomial
Then, k1 + k2 +k3 = -(-2)/1 = 2 = -b/a
k1k2+k2k3+k1k3= -7 = -7/1 = c/a
k1k2k3 = -14 = -14/1 = -d/a
∴ a = 1, b = -2, c = -7 and d = 14
So, one cubic polynomial which satisfy the given conditions will be x3 - 2x2  - 7x + 14

Question 3.
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a–b, a, a+b, find a and b.
Answer
Since, (a - b), a, (a + b) are the zeroes of the polynomial x3 – 3x2 + x + 1.
Question based on Relationship between the zeros and Coefficent in Cubic poplynomialTherefore, sum of the zeroes = (a - b) + a + (a + b) = -(-3)/1 = 3
⇒ 3a = 3 ⇒ a =1
∴ Sum of the products of is zeroes taken two at a time = a(a - b) + a(a + b) + (a + b) (a - b) =1/1 = 1
a2 - ab + a2 + ab + a2 - b= 1
⇒ 3a2 - b2 =1
Putting the value of a,
⇒ 3(1)2 - b2 = 1
⇒ 3 - b2 = 1
⇒ b2 = 2
⇒ b = ±√2
Hence, a = 1 and b = ±√2

Question 4
If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2±√3, find other zeroes.
Answer
2+√3 and 2-√3 are two zeroes of the polynomial p(x) = x4 – 6x3 – 26x2 + 138x – 35.
Let x = 2±√3
So, x-2 = ±√3
On squaring, we get x2 - 4x + 4 = 3,
⇒ x2 - 4x + 1= 0
Now, dividing p(x) by x2 - 4x + 1
 
NCERT Solutions  for Class 10 Maths Polynomials EXERCISE 2.4 Question 4
p(x) = x4 - 6x3 - 26x2 + 138x - 35
= (x2 - 4x + 1) (x2 - 2x - 35)
= (x2 - 4x + 1) (x2 - 7x + 5x - 35)
= (x2 - 4x + 1) [x(x - 7) + 5 (x - 7)]
= (x2 - 4x + 1) (x + 5) (x - 7)
So (x + 5) and (x - 7) are other factors of p(x).
Therefore
 - 5 and 7 are other zeroes of the given polynomial.
 
Question 5.
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
 
Answer
On dividing x4 – 6x3 + 16x2 – 25x + 10 by x2 – 2x + k
NCERT Solutions  for Class 10 Maths Polynomials EXERCISE 2.4 Question 5
Remainder = (2k - 9)x - (8 - k)k + 10 
But the remainder is given as x+ a. 
On comparing their coefficients,
2k - 9 = 1
⇒ k = 10 
⇒ k = 5 and,
-(8-k)k + 10 = a
⇒ a = -(8 - 5)5 + 10 =- 15 + 10 = -5 
Hence, k = 5 and a = -5 
 
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Reference Books for class 10

Given below are the links of some of the reference books for class 10 math.

  1. Oswaal CBSE Question Bank Class 10 Hindi B, English Communication Science, Social Science & Maths (Set of 5 Books)
  2. Mathematics for Class 10 by R D Sharma
  3. Pearson IIT Foundation Maths Class 10
  4. Secondary School Mathematics for Class 10
  5. Xam Idea Complete Course Mathematics Class 10

You can use above books for extra knowledge and practicing different questions.


Class 10 Maths Class 10 Science

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