Summary (What we have discussed)
Hello! Are you using the NCERT textbook to study Class 10 Maths Chapter 1: “Real Numbers“? The Fundamental Theorem of Arithmetic, Euclid’s Division Lemma, and the distinction between rational and irrational numbers are just a few of the fascinating questions that are covered in this page.
During your studies, you will discover things such as:
- How to determine a number’s LCM (Least Common Multiple) and HCF (Highest Common Factor).
- How to demonstrate why certain numbers, such as √2 and √3, are irrational.
- How to determine if a decimal is non-terminating or terminating.
| Class 10 | MATHS |
| Chapter | POLYNOMIALS |
| NCERT MATHS (ENGLISH) | Long Answer Type Questions |
| Medium | English |
| Academic Year | 2023-2024 |
Question 1
lf -3 is one of the zeroes of the polynomial (k−1)x2+kx+1, find the value of k.
Question 2
A quadratic polynomial, whose zeroes are-3 and 4, is
Question 3
If the zeroes of the quadratic polynomial x2+(a+1)x+b are 2 and -3, then find the values of a and b
Question 4
The number of polynomials having zeroes as -2 and 5 is
Question 5
If one of the zeroes of the cubic polynomial ax3+bx2+cx+d is zero, then the product of other two zeroes is
Question 6
If one of the zeroes of the cubic polynomial x3+ax2+bx+c is −1,then find the product of other two zeroes.
Question 7
The zeroes of the quadratic polynomial x2+99x+127 are
Question 8
The zeroes of the quadratic polynomial x2+kx+k where k≠0, are : (1)cannot both be positive (2)cannot both be negative (3)are always unequal (4)are always equal
Question 9
If the zeroes of the quadratic polynomial ax2+bx+c, where c≠0, are equal, then
Question 10
If one of the zeroes of a quadratic polynomial of the form x2+ax+b is the negative of the other, then it
Question 11
Which of the following is not the graph of a quadratic polynomial?
Question 12
Answer the following and justify.
(i) can x2−1 be the quotient on division of x6+2x3+x−1 by a polynomial in x of degree 5?
(ii) What will the quotient and remainder be on division of ax2+bx+c by px3+qx2+rx+s,p≠0?
(iii) If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, what is the relation between the degree of p(x) and g(x)?
(iv) 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) Can the quadratic polynomial x2+kx+k have equal zeroes for some odd integer k>1?
Question 13
Are the following statements ‘True’ or ‘False’? Justify your answer.
(i) If the zeroes of a quadratic polynomial ax2+bx+c 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 ve 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 x3+ax2−bx+c are positive, then atleast one of a,b and c is non-negative.
(vii) The only value of k for which the quadratic polynomial kx2+x+k has equal zeroes is 1/2.
Question 14
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(i) 4x2−3x−1
Question 15
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(ii) 3x2+4x−4
Question 16
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(iii) 5t2+12t+7.
Question 17
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (iv) t3−2t2−15t.
Question 18
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(v) 2x2+(7/2)x+3/4
Question 19
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (vi) 4x2+5√2x−3.
Question 20
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(vii) 2s2−(1+2√2)s+√2
Question 21
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(viii) v2+4√3v−15
Question 22
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(ix) y2+(3√5/2)y−5
Question 23
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(x) 7y2−(11/3)y−2/3
Question 24
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) −8/3,4/3 (ii) −3/2√5,−1/2
Question 25
Given that zeroes of cubic polynomial x3−6x2+3x+10 are of the form a,a+b,a+2b for some real numbers a and b, find the values of a and b as well as zeroes of the given polynomial.
Question 26
If √2 is a zero of p(x)=6x3+√2x2−10x−4√2, find the remaining zeros
Question 27
The value of k, if x2+2x+k is a factor of 2x4+x3−14x2+5x+6 is
Question 28
If x−√5 is a factor of the cubic polynomial x3−3√5x2+13x−3√5, then find all the zeroes of the polynomial.
Question 29
For which values of a and b, the zeroes of q(x)=x3+2x2+a are also the zeroes of the polynomial p(x)=x5−x4−4x3+3x2+3x+b? Which zeores of p(x) are not the zeroes of q(x)?
| Class 10 | MATHS |
| Chapter | POLYNOMIALS |
| NCERT MATHS (ENGLISH) | Long Answer Type Questions |
| Medium | English |
| Academic Year | 2023-2024 |
Question 1
lf -3 is one of the zeroes of the polynomial (k−1)x2+kx+1, find the value of k.
Question 2
A quadratic polynomial, whose zeroes are-3 and 4, is
Question 3
If the zeroes of the quadratic polynomial x2+(a+1)x+b are 2 and -3, then find the values of a and b
Question 4
The number of polynomials having zeroes as -2 and 5 is
Question 5
If one of the zeroes of the cubic polynomial ax3+bx2+cx+d is zero, then the product of other two zeroes is
Question 6
If one of the zeroes of the cubic polynomial x3+ax2+bx+c is −1,then find the product of other two zeroes.
Question 7
The zeroes of the quadratic polynomial x2+99x+127 are
Question 8
The zeroes of the quadratic polynomial x2+kx+k where k≠0, are : (1)cannot both be positive (2)cannot both be negative (3)are always unequal (4)are always equal
Question 9
If the zeroes of the quadratic polynomial ax2+bx+c, where c≠0, are equal, then
Question 10
If one of the zeroes of a quadratic polynomial of the form x2+ax+b is the negative of the other, then it
Question 11
Which of the following is not the graph of a quadratic polynomial?
Question 12
Answer the following and justify.
(i) can x2−1 be the quotient on division of x6+2x3+x−1 by a polynomial in x of degree 5?
(ii) What will the quotient and remainder be on division of ax2+bx+c by px3+qx2+rx+s,p≠0?
(iii) If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, what is the relation between the degree of p(x) and g(x)?
(iv) 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) Can the quadratic polynomial x2+kx+k have equal zeroes for some odd integer k>1?
Question 13
Are the following statements ‘True’ or ‘False’? Justify your answer.
(i) If the zeroes of a quadratic polynomial ax2+bx+c 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 ve 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 x3+ax2−bx+c are positive, then atleast one of a,b and c is non-negative.
(vii) The only value of k for which the quadratic polynomial kx2+x+k has equal zeroes is 1/2.
Question 14
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(i) 4x2−3x−1
Question 15
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(ii) 3x2+4x−4
Question 16
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(iii) 5t2+12t+7.
Question 17
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (iv) t3−2t2−15t.
Question 18
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(v) 2x2+(7/2)x+3/4
Question 19
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials (vi) 4x2+5√2x−3.
Question 20
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(vii) 2s2−(1+2√2)s+√2
Question 21
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(viii) v2+4√3v−15
Question 22
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(ix) y2+(3√5/2)y−5
Question 23
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(x) 7y2−(11/3)y−2/3
Question 24
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) −8/3,4/3 (ii) −3/2√5,−1/2
Question 25
Given that zeroes of cubic polynomial x3−6x2+3x+10 are of the form a,a+b,a+2b for some real numbers a and b, find the values of a and b as well as zeroes of the given polynomial.
Question 26
If √2 is a zero of p(x)=6x3+√2x2−10x−4√2, find the remaining zeros
Question 27
The value of k, if x2+2x+k is a factor of 2x4+x3−14x2+5x+6 is
Question 28
If x−√5 is a factor of the cubic polynomial x3−3√5x2+13x−3√5, then find all the zeroes of the polynomial.
Question 29
For which values of a and b, the zeroes of q(x)=x3+2x2+a are also the zeroes of the polynomial p(x)=x5−x4−4x3+3x2+3x+b? Which zeores of p(x) are not the zeroes of q(x)?