Polynomials Class 10 Maths NCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials are part of NCERT Solutions for Class 10 Maths. Here we have given Maths NCERT Solutions Class 10 Chapter 2 Polynomials.

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.1

Question 1:
The graphs of y = p(x) are given below for some polynomials p(x). Find the number of zeroes of p(x) in each case.



Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.2

Question 1.
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and their coefficients:
(i) x² – 2x – 8
(ii) 4s² – 4s + 1
(iii) 6x² – 3 – 7x
(iv) 4u² + 8u
(v) t² – 15
(vi) 3x² – x – 4
Solution:

Question 2.
Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively:

Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.3

Question 1.
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2
(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x
(iii) p(x) = x⁴– 5x + 6, g(x) = 2 – x^2
Solution:

Question 2.
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t² – 3, 2t⁴ + 3t³ – 2t²– 9t – 12
(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2
(iii) x² + 3x + 1, x⁵ – 4x³ + x² + 3x + 1
Solution:

Question 3.
Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are  and (sqrt { frac { 5 }{ 3 } }) and –(sqrt { frac { 5 }{ 3 } })
Solution:

Question 4.
On dividing x³ – 3x² + x + 2bya polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).
Solution:

Question 5.
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and:
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.4

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 – 5x + 2;  (frac { 1 }{ 4 }), 1, -2
(ii) x³ – 4x² + 5x – 2; 2, 1, 1
Solution:

Question 2.
Find a cubic polynomial with the sum, some of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.
Solution:

Question 3.
If the zeroes of the polynomial x³ – 3x² + x + 1 are a-b, a, a + b, find a and b.
Solution:

Question 4.
If two zeroes of the polynomial x⁴ – 6x³ – 26x² + 138x – 35 are 2 ± √3, find other zeroes.
Solution:

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.
Solution:

Class 10 Maths Polynomials Mind Map

Polynomial

An algebraic expression f(x) of the form
f(x) = a₀ + a₁x + a₂x² +…. + a_nxⁿ,
where a₀, a₁, ……., a_n are real numbers and all indices of variables are non-negative integers is called polynomial in variable x.
(i) The highest power of x is called degree of the polynomial.
(ii) a₀, a₁x,….,a_nxⁿ are terms of the polynomial.
(iii) a₀, a₁ ,….a_n are co-efficients of the polynomial.

Standard Forms of Linear, Quadratic and Cubic Polynomials

(i) Linear Polynomial:
ax + b, where a, b are real numbers and a ≠ 0.
(ii) Quadratic Polynomial:
ax² + bx + c, where a, b, c are real numbers & a ≠ 0.
(iii) Cubic Polynomials:
ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.

Value of a Polynomial

The value of a polynomial f(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by/(a).

Zero(es)/Root(s) of Polynomial

x = r is a zero of a polynomial p(x) if p(r) = 0.

Geometrical Meaning of Zeroes of a Polynomial

Zero(es) of a polynomial is/are the x-coordinate of the point(s) where graph y = fix) intersects the x-axis.
(i) Linear polynomial: Graph of linear polynomial is a straight line and has exactly one zero.
(ii) Quadratic polynomial: Graph of quadratic polynomial is always a parabola and this polynomial can have atmost two zeroes.
(iii) Cubic polynomial: Cubic polynomial can have atmost three zeroes.

Cases of Quadratic Polynomial

Case-I : If a quadratic polynomial P(x) = ax² + bx + c has two zeroes, then its graph will intersect the x-axis at two distinct points A and B as shown in the figure.

Case-II: If a quadratic polynomial P(x) = ax² + bx + c has only one zero, then its graph will touch the x-axis at only one point A as shown in the figure.

Case-III : If a quadratic polynomial P(x) = ax² + bx + c has no zero, then its graph will not intersect /touch the x-axis at any point as shown in the figure.

Relationship between Zeroes and Coefficients of a Polynomial
(i) Zero of a linear polynomial ax + b is x = (-frac{b}{a})
(ii) If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then
α + β = (-frac{b}{a})

(iii) If α, β and γ are zeroes of the cubic polynomial ax³ + bx² + cx + d then

Division Algorithm

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x)
where either r(x) = 0 or degree of r(x) < degree of g(x)

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Free PDF Download of CBSE Class 10 Maths Chapter 5 Arithmetic Progressions Multiple Choice Questions with Answers. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 10 Maths Arithmetic ProgressionsMCQs with Answers to know their preparation level.

Class 10 Maths MCQs Chapter 5 Arithmetic Progressions

1. The n^th term of an A.P. is given by a_n = 3 + 4n. The common difference is
(a) 7
(b) 3
(c) 4
(d) 1

2. If p, q, r and s are in A.P. then r – q is
(a) s – p
(b) s – q
(c) s – r
(d) none of these

3. If the sum of three numbers in an A.P. is 9 and their product is 24, then numbers are
(a) 2, 4, 6
(b) 1, 5, 3
(c) 2, 8, 4
(d) 2, 3, 4

4. The (n – 1)^th term of an A.P. is given by 7,12,17, 22,… is
(a) 5n + 2
(b) 5n + 3
(c) 5n – 5
(d) 5n – 3

5. The n^th term of an A.P. 5, 2, -1, -4, -7 … is
(a) 2n + 5
(b) 2n – 5
(c) 8 – 3n
(d) 3n – 8

6. The 10^th term from the end of the A.P. -5, -10, -15,…, -1000 is
(a) -955
(b) -945
(c) -950
(d) -965

7. Find the sum of 12 terms of an A.P. whose nth term is given by a_n = 3n + 4
(a) 262
(b) 272
(c) 282
(d) 292

8. The sum of all two digit odd numbers is
(a) 2575
(b) 2475
(c) 2524
(d) 2425

9. The sum of first n odd natural numbers is
(a) 2n²
(b) 2n + 1
(c) 2n – 1
(d) n²

10. If (p + q)^th term of an A.P. is m and (p – q)^tn term is n, then pth term is

11. If a, b, c are in A.P. then (frac{a-b}{b-c}) is equal to

12. The number of multiples lie between n and n² which are divisible by n is
(a) n + 1
(b) n
(c) n – 1
(d) n – 2

13. If a, b, c, d, e are in A.P., then the value of a – 4b + 6c – 4d + e is
(a) 0
(b) 1
(c) -1.
(d) 2

14. The next term of the sequence

15. n^th term of the sequence a, a + d, a + 2d,… is
(a) a + nd
(b) a – (n – 1)d
(c) a + (n – 1)d
(d) n + nd

16. The 10th term from the end of the A.P. 4, 9,14, …, 254 is
(a) 209
(b) 205
(c) 214
(d) 213

17. If 2x, x + 10, 3x + 2 are in A.P., then x is equal to
(a) 0
(b) 2
(c) 4
(d) 6

18. The sum of all odd integers between 2 and 100 divisible by 3 is
(a) 17
(b) 867
(c) 876
(d) 786

19. If the numbers a, b, c, d, e form an A.P., then the value of a – 4b + 6c – 4d + e is
(a) 0
(b) 1
(c) -1
(d) 2

20. If 7 times the 7^th term of an A.P. is equal to 11 times its 11^th term, then 18^th term is
(a) 18
(b) 9
(c) 77
(d) 0

We hope the given MCQ Questions for Class 10 Maths Arithmetic Progressions with Answers will help you. If you have any query regarding CBSE Class 10 Maths Chapter 5 Arithmetic Progressions Multiple Choice Questions with Answers, drop a comment below and we will get back to you at the earliest.

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Polynomials Class 10 Extra Questions Maths Chapter 2

Extra Questions for Class 10 Maths Chapter 2 Polynomials. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.

You can also download Maths NCERT Solutions Class 10 to help you to revise complete syllabus and score more marks in your examinations.

Extra Questions for Class 10 Maths

NCERT Solutions for Class 10 Maths

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