Are the following statements ‘True’ or ‘False’? Justify your answers.
(i)If the zeroes of a quadratic polynomial are both positive, then a, b and c all have the same sign
(ii) If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
(iii) If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
(iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
(v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
(vi) If all three zeroes of a cubic polynomial are positive, then at least one of a, b and c is non-negative.
(vii) The only value of k for which the quadratic polynomial has equal zeros is.
(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 then
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 :
(a quadratic polynomial)
Here only one value of x exist which is –2.
Hence the graph of the quadratic polynomial intersect the x-axis at x = –2.
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 exist.
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 be the zeroes of a cubic polynomial.
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 and be the roots of p(x)
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
…….(1)
That is
Sum of the products of two zeroes at a time
Replace
…..(2)
That is
Product of all zeroes
Replace
……(3)
That is
From equation (1), (2) and (3) we conclude that all the coefficient and the constant term of the polynomial have the same sign.
Hence, 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 are positive then a, b and c must be negative.
Hence given statement is false.
(vii)
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 cubic 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
(Q d = b2 – 4ac)
Here, a = k, b = 1, c = k
When and then the given quadratic polynomial has equal zeroes.
View Full Answer(1)(i)Answer the following and justify:
Can be the quotient on division of by a polynomial in x of degree 5?
(ii)Answer the following and justify:
What will the quotient and remainder be on division of
(iii)Answer the following and justify:
If on division of a polynomial p (x) by a polynomial g (x), the quotients zero, what is the relation between the degrees of p (x) and g (x)?
(iv) Answer the following and justify:
If on division of a non-zero polynomial p (x) by a polynomial g (x), the remainder is zero, what is the relation between the degrees of p (x) and g (x)?
(v) Answer the following and justify:
Can the quadratic polynomial have equal zeroes for some odd integer k > 1?
(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 , then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by
i,e. Dividend = Divisor × Quotient + Remainder
Remainder
Remainder
Here it is of degree seven but given dividend is of degree six.
Therefore can not be the quotient of because division algorithm is not satisfied.
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 is the remainder and quotient will be zero.
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 have equal zeroes only when the values of k will be 0 and 4.
Hence, given statement is not correct.
View Full Answer(1)
Write whether the following statements are True or False. Justify your answer.
i. A binomial can have at most two terms
ii. Every polynomial is a binomial.
iii. A binomial may have degree 5.
iv. Zero of a polynomial is always 0
v.A polynomial cannot have more than one zero.
vi. The degree of the sum of two polynomials each of degree 5 is always 5
i. False
Solution :- Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign + or –.
Binomial necessarily means consisting of two terms only. These terms should not be like terms.
For example: is a binomial
x + 2x is not a binomial as these are like terms.
So the given statement is false, as a binomial has exactly two terms (not at most two terms).
ii. False
Solution:- Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign + or –.
Binomial necessarily means consisting of two terms only. These terms should not be like terms.
For example: is a binomial
x + 2x is not a binomial as these are like terms.
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)
Its degree is always a whole number.
For example: etc.
Because a binomial has exactly two terms but a polynomial can be monomial (single term), binomial (two terms), trinomial (three terms) etc.
The given statement is False.
iii. True
Solution:- Binomial: A binomial is an expression that has two numbers, terms or letters joined by the sign + or –.
Binomial necessarily means consisting of two terms only. These terms should not be like terms.
For example: is a binomial
x + 2x is not a binomial as these are like terms.
Degree of polynomial: Degree of polynomial is the highest power of the polynomial’s monomials with non-zero coefficient.
For any binomial of the form , we can see that the degree is 5.
So, a binomial may have degree 5.
Therefore the given statement is True.
iv. False
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)
Its degree is always a whole number.
For example: etc.
We know that for finding the zero of a polynomial, we need to find a value of x for which the polynomial will be zero
i.e., p(x)=0
Let us consider an example:
p(x) = x - 2
Now to find the zero of this polynomial, we have:
x - 2 =0
x = 2 (which is not zero)
Hence the given statement is false because zero of a polynomial can be any real number.
v. False
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).
Its degree is always a whole number.
For example: etc.
We know that for finding the zero of a polynomial, we need to find a value of x for which the polynomial will be zero
i.e., p(x)=0
So, a polynomial can have any number of zeroes. It depends upon the degree of polynomial.
The given statement is False.
vi. False
Solution :- Degree of polynomial: Degree of polynomial is the highest power of the polynomial’s monomials with non-zero coefficient.
The degree of the sum of two polynomials may be less than or equal to 5.
For example: and are two polynomials of degree 5 but the degree of the sum of the two polynomials is 3.
Hence the given statement is False.
View Full Answer(1)Which of the following expressions are polynomials? Justify your answer
(i) 8
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(i, ii, iv, vii)
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)
Its degree is always a whole number.
For example: etc.
(i) Here 8 is a polynomial because it can also be written as i.e., multiply by .
(ii) is also a polynomial having degree two.
(iii) is not a polynomial because its exponent is in fraction.
(iv) can be written as and it is a polynomial having degree two.
(v) is not polynomial because it has negative exponent.
(vi) is not a polynomial because it have negative exponent.
(vii) is a polynomial of degree three.
(viii) is not a polynomial because it have negative exponent.
View Full Answer(1)
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8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed of the water. Pratap consumed the remaining water.
(i) How much water did Vidya drink?
(ii) What fraction of the total quantity of water did Pratap drink?
Given
Total water = 5 litre.
i) The amount of water vidya consumed :
Hence vidya consumed 2 liters of water from the bottle.
ii) The amount of water Pratap consumed :
Hence, Pratap consumed 3 liters of water from the bottle.
View Full Answer(1)7. Find:
As we know that of is equivalent to multiply,
As we know that of is equivalent to multiply,
As we know that of is equivalent to multiplication, so
As we know that of is equivalent to multiplication, so
View Full Answer(1)6. Multiply and express as a mixed fraction :
On Multiplying, we get
Converting This into Mixed Fraction,
On Multiplying, we get
Converting This into Mixed Fraction,
On multiplying, we get
Converting it into a mixed fraction, we get
On multiplying, we get
Converting it into a mixed fraction,
Multiplying them, we get
Now, converting the result fraction we got to mixed fraction,
On multiplying, we get
Converting this into a mixed fraction, we get
View Full Answer(1)
4. Shade:
of the circles in box (a) of the triangles in box (b)
of the squares in box (c).
1) In figure a there are 12 circles: half of 12 = 6
2) In figure b there are 9 triangles: 2/3 of 9 = 6
3) In figure c there are 15 triangles: 3/5 of 15 = 9
View Full Answer(1)5. Find:
On Multiplying we get,
On multiplying, we get
On Multiplying, we get
On multiplying, we get
On multiplying, we get
On multiplying, we get
On Multiplying, we get,
View Full Answer(1)3. Multiply and reduce to lowest form and convert into a mixed fraction:
On Multiplying, we get
On Multiplying, we get
On Multiplying, we get
On Multiplying, we get
On Multiplying, we get
On Multiplying, we get
On Multiplying, we get
Converting this into a mixed fraction, we get
On Multiplying, we get
On multiplying, we get
Converting this into a mixed fraction,
.
On multiplying, we get,
.
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