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Polynomials is chapter 2 of NCERT exemplar Class 10 Maths. This chapter in class 10 is the extension of what we have studied in class 9, polynomials. NCERT exemplar Class 10 Maths solutions chapter 2 provides a detailed understanding of Polynomials. A polynomial is an expression that consists of variables, coefficients, with various mathematical operations like addition, subtraction, multiplication, division, and a non-negative integer exponent. The solutions of NCERT exemplar Class 10 Maths chapter 2 important questions are a result of the extensive efforts of subject matter experts and are an excellent source to prepare for NCERT Class 10 Maths. Class 10 Maths chapter 2 mcq are also available in this article.
These NCERT exemplar Class 10 Maths chapter 2 solutions help you to understand the concepts of polynomials with plenty of questions. The CBSE Syllabus Class 10 Maths is the reference for NCERT exemplar Class 10 Maths solutions chapter 2.
Polynomials class 10 MCQ solutions are given below:-
Class 10 Maths chapter 2 exemplar solutions Exercise: 2.1 Page number: 9-10 Total questions: 11 |
Question:1
If one of the zeroes of the quadratic polynomial
(A)
(B)
(C)
(D)
Answer. [A]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.
Let
If –3 is one of the zeroes of p(x) then p(–3) = 0
put x = –3 in p(x)
Question:2
A quadratic polynomial, whose zeroes are –3 and 4, is
(A)
(B)
(C)
(D)
Answer. [C]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is polynomial of degree 2.
(A)
put x = –3 and put x = 4
(B)
put x = –3 and put x = 4
(C)
put x = –3 and put x = 4
(D)
put x = –3 and put x = 4
If –3, 4 is zeros of a polynomial p(x) then p(–3) = p(4) = 0
Here only (C) option satisfy p(–3) = p(4) = 0
Question:3
If the zeroes of the quadratic polynomial
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
Answer. [D]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is a polynomial of degree 2.
If 2 and –3 are the zero of
Add equations (1) and (2)
put a = 0 in (1)
2(0) + b = –6
Hence a = 0, b = –6.
Question:4
The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
Answer. [D]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Let the polynomial is
We know that sum of zeroes
Multiplication of zeroes
From equations (1) and (2), it is clear that
a = 1, b = –3, c = –10
Put values of a, b, and c in equation (*)
But we can multiply or divide equations. (3) by any real number except 0, and the zeroes remain the same.
Hence, there are an infinite number of polynomials that exist with zeroes 2 and 5.
Hence, the answer is more than 3.
Question:5
Given that one of the zeroes of the cubic polynomial
(A)
(B)
(C) 0
(D)
Answer. [B]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is a polynomial of degree 3.
Here the given cubic polynomial is
Let three zeroes are
We know that
Put
Hence, the product of the other two zeroes is
Question:6
If one of the zeroes of the cubic polynomial
(A) b – a + 1
(B) b – a – 1
(C) a – b + 1
(D) a – b –1
Answer. [A]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s) and a quadratic polynomial is a polynomial of degree 3.
Here the given cubic polynomial is
Let
put x = –1 in
We know that
Here, a = 1, b = a, c = b, d = c
So,
Hence, the product of the other two is b – a + 1.
Question:7
The zeroes of the quadratic polynomial
(A) both positive
(B) both negative
(C) one positive and one negative
(D) both equal
Answer. [B]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is a polynomial of degree 2.
Here, the given quadratic polynomial is
The value of both the zeroes is negative.
Question:8
The zeroes of the quadratic polynomial
(A) cannot both be positive
(B) cannot both be negative
(C) are always unequal
(D) are always equal
Answer. [A]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is a polynomial of degree 2.
Here, the given quadratic polynomial is
Hence, the value of k is either less than 0 or greater than 4.
If the value of k is less than 0, only one zero is positive.
If the value of k is greater than 4, only one zero is positive.
Hence both the zeroes can not be positive.
Question:9
If the zeroes of the quadratic polynomial
(A) c and a have opposite signs
(B) c and b have opposite signs
(C) c and a have the same sign
(D) c and b have the same sign
Answer. [C]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s), and a quadratic polynomial is a polynomial of degree 2.
Here the given polynomial is
If both the zeroes are equal, then
(A) c and a have opposite signs
If c and a have opposite sign, then R.H.S. of equation (1) is negative, but L.H.S. is always positive. So (A) is not a correct one.
(B) c and b have opposite sign
If c is negative and b is positive L.H.S. is positive but R.H.S. of eqn. (1) is negative. Hence, (B) is not the correct one.
(C) c and a have the same sign
If c and a have same sign R.H.S. of eqn. (1) is positive and L.H.S. is always positive hence it is a correct one.
(D) c and b have same sign
If c and b both have negative sign then R.H.S. of eqn. (1) is negative and L.H.S. is positive. So this is not correct.
Only one option i.e. (c) is correct one.
Question:10
If one of the zeroes of a quadratic polynomial of the form
(A) has no linear term, and the constant term is negative.
(B) has no linear term and the constant term is positive.
(C) can have a linear term but the constant term is negative.
(D) can have a linear term, but the constant term is positive.
Answer. [A]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.
Here the given quadratic polynomial is
a = 1, b = a, c = b
Let x1, x2 are the zeroes of the equation (1)
According to the question:
sum of zeroes =
Product of zeroes
Put the value of a and b in (1)
Hence, it has no linear term, and the constant term is negative.
Question:11
Which of the following is not the graph of a quadratic polynomial?
Answer:
Answer. [D]
Solution. Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s) and a quadratic polynomial is polynomial of degree 2.
If p(x) is a quadratic polynomial, then there are almost 2 values of x exists called roots.
Hence, the graph of p(x) cuts the x-axis almost 2 times.
But here we see that in option (D) the curve cuts the x-axis at 3 times
Hence, graph (D) is not a graph of a quadratic polynomial.
Class 10 Maths chapter 2 exemplar solutions Exercise: 2.2 Page number: 11-12 Total questions: 2 |
Question:1
Answer:
(i) Answer. [false]
Solution.
Let divisor of a polynomial in x of degree 5 is =
Quotient =
Dividend =
According to division algorithm if one polynomial p(x) is divided by the other polynomial
i,e. Dividend = Divisor × Quotient + Remainder
Here it is of degree seven but given dividend is of degree six.
Therefore
Hence, given statement is false.
(ii) Here dividend is
and divisor is
According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) ¹ 0 then the relation among p(x), g(x) quotient q(x) and remainder r(x) is given by
where degree of r(x) < degree of g(x).
i.e. Dividend = Devisor × Quotient + Remainder
Here degree of divisor is greater than degree of dividend therefore.
According to division algorithm theorem
That is remainder =
Quotient = 0
(iii) Division algorithm theorem :- According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) is then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by
p(x) = g(x) × q(x) + r(x)
where degree of r(x) < degree of g(x)
i.e. Dividend = Division × Quotient + Remainder
In the given statement it is given that on division of a polynomial p(x) by a polynomial g(x), the quotient is zero.
The given condition is possible only when degree of divisor is greater than degree of dividend
i.e. degree of g(x) > degree of p(x).
(iv)
Division algorithm theorem :- According to division algorithm if one polynomial p(x) is divided by the other polynomial g(x) is then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by
p(x) = g(x) × q(x) + r(x) ….(1)
where degree of r(x) < degree of g(x)
According to given statement on division of a non-zero polynomial p(x) by a polynomial g(x) then remainder r(x) is zero then equation (1) becomes
p(x) = g(x) × q(x) ….(2)
From equation (2) we can say g(x) is a factor of p(x) and degree of g(x) may be less than or equal to p(x).
(v)
Answer. [false]
Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contains different powers of the same variable(s)
Quadratic polynomial : when the degree of polynomial is two then the polynomial is called quadratic polynomial.
Let
It is given that zeroes of p(x) has equal and we know that when any polynomial having equal zeroes than their discriminate is equal to zero
i.e.
here, a = 1, b = k, c = k
Hence, the given quadratic polynomial
Hence, the given statement is not correct.
Question:2
(i) Answer. [false]
Polynomial: It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Quadratic polynomial: when the degree of polynomial is two then the polynomial is called quadratic polynomial.
If a quadratic polynomial is
Sum of zeroes
Product of zeroes
Here, two possibilities can occurs:
b > 0 and a < 0, c < 0
OR b < 0 and a > 0, c > 0
Here we conclude that a, b and c all have nor same sign is given statement is false.
(ii) Answer. [False]
Polynomial: It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Quadratic polynomial : when the degree of polynomial is two then the polynomial is called quadratic polynomial.
We know that the roots of a quadratic polynomial is almost 2, Hence the graph of a quadratic polynomial intersects the x-axis at 2 point, 1 point or 0 point.
For example :
Here, only one value of x exist, which is –2.
Hence, the graph of the quadratic polynomial
Hence, we can say that if the graph of a polynomial intersect the x-axis at only one point can be a quadratic polynomial.
Hence, the given statement is false.
(iii)
Answer. [True]
Polynomial: It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Quadratic polynomial : when the degree of polynomial is two then the polynomial is called quadratic polynomial.
Let us take an example :-
It is a cubic polynomial
If we find its roots then x = 1, 2
Hence, there are only 2 roots of cubic polynomial
In other words we can say that the graph of this cubic polynomial intersect x-axis at two points x = 1, 2.
Hence we can say that if the graph of a polynomial intersect the x-axis at exactly two points, it need not to be a quadratic polynomial it may be a polynomial of higher degree.
Hence the given statement is true.
(iv)
Answer. [True]
Polynomial: It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Cubic polynomial: when the degree of polynomial is three then the polynomial is called cubic polynomial.
Let
It is given that two of the given zeroes have value zero.
i.e.
Let
Here, we conclude that if two zeroes of a cubic polynomial are zero than the polynomial does not have linear and constant terms.
Hence, given statement is true.
(v)
Answer. [True]
Polynomial : It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Cubic polynomial : when the degree of polynomial is three then the polynomial is called cubic polynomial.
Let the standard equation of cubic polynomial is:
Let
It is given that all the zeroes of a cubic polynomial are negative
i.e
Sum of zeroes
It is given that zeroes are negative then
That is
Sum of the products of two zeroes at a time
Replace
That is
Product of all zeroes
Replace
That is
From equations (1), (2) and (3) we conclude that all the coefficient and the constant term of the polynomial have the same sign.
Hence, the given statement is true.
(vi)
Answer. [False]
Polynomial: It is an expression of more than two algebraic terms, especially then sum of several terms that contains different powers of the same variable(s).
Cubic polynomial: when the degree of polynomial is three then the polynomial is called cubic polynomial.
The given cubic polynomial is
Let a, b and g are the roots of the given polynomial
Sum of zeroes
We know that when all zeroes of a given polynomial are positive then their sum is also positive
But here a is negative
Sum of the product of two zeroes at a time
Also, here b is negative
Product of all zeroes
Also, c is negative
Hence, if all three zeroes of a cubic polynomial
Hence given statement is false.
(vii)
Answer. [False]
Polynomial: It is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Quadratic polynomial: when the degree of polynomial is two then the polynomial is called quadratic polynomial.
Let
Here it is gives that zeroes of p(x) are equal and we know that when any polynomial having equal zeroes then their discriminate will be equal to zero.
i.e. d = 0
Here, a = k, b = 1, c = k
Class 10 Maths chapter 2 exemplar solutions Exercise: 2.3 Page number: 12-13 Total questions: 10 |
Question:1
Answer.
Zeroes : zeroes of polynomial are the value(s) that makes it equal to 0.
x – 1 = 0 or 4x + 1 = 0
x = 1 or 4x = –1
Hence, 1,
We know that
Sum of the zeroes
Here,
which is equal to
Product of zeroes
Here
Question:2
Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that make it equal to 0.
Hence,
a = 3, b = 4, c = –4
Sum of the zeroes
Here,
Product of the zeroes
Here,
Question:3
Answer.
Zeroes : zeroes of the polynomial are the values(s) the makes it equal to 0
t + 1 = 0 or 5t + 7 = 0
t = –1 or
Hence,
Sum of zeroes
Here,
Product of zeroes
Here,
Question:4
Answer.
Zeroes : zeroes of the polynomial area the value(s) that makes it equal to 0.
t = 0 or t – 5 = 0 or t + 3 = 0
t = 5 or t = –3
Hence, 0, 5, –3 are the zeroes of the polynomial
Sum of zeroes
Here,
Product of zeroes
Here,
If
Hence proved.
Question:5
Answer.
Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.
Multiply by 4
Hence,
Here,
Sum of zeroes
Here,
Product of zeroes
Here,
Question:6
Answer.
Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.
Hence,
Here, a = 4, b =
Sum of zeroes
Here,
Product of zeroes
Here,
Question:7
Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that make it equal to 0.
Hence,
Here, a = 2,
Sum of zeroes
Here,
Product of zeroes
Here,
Question:8
Answer.
Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.
Hence,
Here, a = 1, b =
Sum of zeroes
Here,
Product of zeroes
Here,
Question:9
Answer.
Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.
Multiply by 2
Hence,
Here, a = 2, b =
Sum of zeroes
Here,
Product of zeroes
Here,
Question:10
Answer.
Solution. Zeroes : zeroes of the polynomial are the value(s) that makes it equal to 0.
Multiply by 3
Hence,
Here, a = 21, b = –11, c = –2
Sum of zeroes
Here,
Product of zeroes
Here,
Class 10 Maths chapter 2 solutions Exercise: 2.4 Page number: 14-15 Total questions: 6 |
Question:1
For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
(i)
(ii)
(iii)
(iv)
(i) Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that makes it equal to 0.
Here, sum of zeroes
product of zeroes
Let p(x) is the required polynomial
p(x) = x2 – (sum of zeroes)x + (product of zeroes)
Multiply by 3
Hence,
(ii) Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that makes it equal to 0.
Here, sum of zeroes
Product of zeroes
Let p(x) is the required polynomial
p(x) = x2 – (sum of zeroes)x + (product of zeroes)
Multiply by 16 we get
Hence,
=
=
(iii) Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that makes it equal to 0.
Here, sum of zeroes
Product of zeroes = –9
Let p(x) is required polynomial
p(x) = x2 – (sum of zeroes)x + (product of zeroes)
Hence,
=
=
=
(iv) Answer.
Solution. Zeroes: zeroes of the polynomial are the value(s) that makes it equal to 0.
Here sum of zeroes
Product of zeroes
Let p(x) is the required polynomial
p(x) = x2 – (sum of zeroes)x + (product of zeros)
Multiplying by
Hence,
=
=
Question:2
Answer. [5, 2, –1]
Solution. Here the given cubic polynomial is
A = 1, B = –6, C = 3, D = 10
Given that,
We know that,
a = 5, –1
Put a = 5 in (1) put a = –1 in (1)
5 + b = 2 –1 + b = 2
b = –3 b = 3
Hence, value of a = 5, b = –3 and a = –1, b = 3
put a = 5, b = –3 and we get zeroes
a = 5
a + b = 5 – 3 = 2
a + 2b = 5 + 2(–3) = –1
Hence, zeroes are 5, 2, –1.
Question:3
Given that
Answer.
Given cubic polynomial is
If
Hence,
Question:4
Find k so that
Answer:
The given polynomial is
Here,
Use division algorithm
Here,
It is given that
By comparing L.H.S. and R.H.S.
7k + 21 = 0
k = – 3 ….(1)
k = –1, –3 …..(2)
From (1) and (2)
k = –3
Hence,
Put k = –3
x + 3 = 0 x – 1 = 0 x – 2 = 0 2x + 1 = 0
x = –3 x = 1 x = 2 2x = – 1
Here zeroes of
Zeroes of
Question:5
Given that
Answer:
Answer.
Solution.
The given cubic polynomial is
Hence
Use divided algorithm
a = 1, b =
Hence,
Question:6
Answer: [2, 1]
Solution.
Given :-
Hence, q(x) is a factor of p(x) use divided algorithm
Since
By comparing
– 1 – a = 0 b – 2a = 0
a = – 1 2a = b
put a = –1
2(–1) = b
b = –2
Hence, a = –1, b = –2
x – 2 = 0 x – 1 = 0
x = 2 x = 1
Hence, 2, 1 are the zeroes of p(x), which are not the zeroes of q(x).
Here you can find the complete solutions for NCERT exemplar of maths and science.
These Class 10 Maths NCERT exemplar Chapter 2 solutions provide a basic knowledge of polynomials, which has great importance in higher classes.
The questions based on polynomials can be practiced in a better way, along with these solutions.
The NCERT exemplar Class 10 Maths chapter 2 solution polynomials has a good amount of problems for practice and is sufficient for a student.
NCERT Exemplar Class 10 Maths Solutions Chapter 2 pdf downloads are available through online tools for the students to access this content in an offline version, so that no breaks in continuity are faced while practising NCERT Exemplar Class 10 Maths Chapter 2.
Here are the subject-wise links for the NCERT solutions of class 10:
Given below are the subject-wise NCERT Notes of class 10 :
Here are some useful links for NCERT books and NCERT syllabus for class 10:
Given below are the subject-wise exemplar solutions of class 10 NCERT:
The chapter of Polynomials is important for Board examinations as it holds around 5-6% weightage of the whole paper.
The chapter on Polynomials serves as a key topic for JEE Advanced and clarity of concepts beginning from lower classes such as Class 10 can help the students to build a strong foundation for competitive exams as well as higher classes.
NCERT exemplar Class 10 Maths solutions chapter 2 delivers very descriptive solutions to the student and can be referred while facing any issues in solving the problems of exemplar.
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Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.
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Hello Aspirant, Hope your doing great, your question was incomplete and regarding what exam your asking.
Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.
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Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.
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According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.
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