Important Questions for Class 10 Maths Chapter 2 Polynomials

Important Questions for Class 10 Maths Chapter 2 Polynomials

Free PDF Download of CBSE Class 10 Maths Chapter 12 Areas Related to Circles 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 Areas Related to Circles MCQs with Answers to know their preparation level.

Important Questions for Class 10 Maths Chapter 1 Real Numbers

Question 1.
If the sum of zeroes of the quadratic polynomial 3x² - kx + 6 is 3, then find the value of k. (2012)
Solution:
Here a = 3, b = -k, c = 6
Sum of the zeroes, (α + β) = = 3 .....(given)
⇒ = 3
⇒ k = 9

Question 2.
If α and β are the zeroes of the polynomial ax² + bx + c, find the value of α² + β². (2013)
Solution:

Question 3.
If the sum of the zeroes of the polynomial p(x) = (k² - 14) x² - 2x - 12 is 1, then find the value of k. (2017 D)
Solution:
p(x) = (k² - 14) x² - 2x - 12
Here a = k² - 14, b = -2, c = -12
Sum of the zeroes, (α + β) = 1 ...[Given]
⇒ = 1
⇒ = 1
⇒ k² - 14 = 2
⇒ k² = 16
⇒ k = ±4

Question 4.
If α and β are the zeroes of a polynomial such that α + β = -6 and αβ = 5, then find the polynomial. (2016 D)
Solution:
Quadratic polynomial is x² - Sx + P = 0
⇒ x² - (-6)x + 5 = 0
⇒ x² + 6x + 5 = 0

Question 5.
A quadratic polynomial, whose zeroes are -4 and -5, is .... (2016 D)
Solution:
x² + 9x + 20 is the required polynomial.

Polynomials Class 10 Important Questions Short Answer-I (2 Marks)

Question 6.
Find the condition that zeroes of polynomial p(x) = ax² + bx + c are reciprocal of each other. (2017 OD)
Solution:
Let α and be the zeroes of P(x).
P(a) = ax² + bx + c ...(given)
Product of zeroes =
⇒ α × =
⇒ 1 =
⇒ a = c (Required condition)
Coefficient of x² = Constant term

Question 7.
Form a quadratic polynomial whose zeroes are 3 + √2 and 3 - √2. (2012)
Solution:
Sum of zeroes,
S = (3 + √2) + (3 - √2) = 6
Product of zeroes,
P = (3 + √2) x (3 - √2) = (3)² - (√2)² = 9 - 2 = 7
Quadratic polynomial = x² - Sx + P = x² - 6x + 7

Question 8.
Find a quadratic polynomial, the stun and product of whose zeroes are √3 and respectively. (2014)
Solution:
Sum of zeroes, (S) = √3
Product of zeroes, (P) =
Quadratic polynomial = x² - Sx + P

Question 9.
Find a quadratic polynomial, the sum and product of whose zeroes are 0 and -√2 respectively. (2015)
Solution:
Quadratic polynomial is
x² - (Sum of zeroes) x + (Product of zeroes)
= x² - (0)x + (-√2)
= x² - √2

Question 10.
Find the zeroes of the quadratic polynomial √3 x² - 8x + 4√3. (2013)
Solution:

Question 11.
If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² - 5x - 3, find the value of p and q. (2012)
Solution:
We have, 2x² - 5x - 3 = 0
= 2x² - 6x + x - 3
= 2x(x - 3) + 1(x - 3)
= (x - 3) (2x + 1)
Zeroes are:
x - 3 = 0 or 2x + 1 = 0
⇒ x = 3 or x =
Since the zeroes of required polynomial is double of given polynomial.
Zeroes of the required polynomial are:
3 × 2, ( × 2), i.e., 6, -1
Sum of zeroes, S = 6 + (-1) = 5
Product of zeroes, P = 6 × (-1) = -6
Quadratic polynomial is x² - Sx + P
⇒ x² - 5x - 6 ...(i)
Comparing (i) with x² + px + q
p = -5, q = -6

Question 12.
Can (x - 2) be the remainder on division of a polynomial p(x) by (2x + 3)? Justify your answer. (2016 OD)
Solution:
In case of division of a polynomial by another polynomial, the degree of the remainder (polynomial) is always less than that of the divisor. (x - 2) can not be the remainder when p(x) is divided by (2x + 3) as the degree is the same.

Question 13.
Find a quadratic polynomial whose zeroes are and . (2013)
Solution:

Question 14.
Find the quadratic polynomial whose zeroes are -2 and -5. Verify the relationship between zeroes and coefficients of the polynomial. (2013)
Solution:
Sum of zeroes, S = (-2) + (-5) = -7
Product of zeroes, P = (-2)(-5) = 10
Quadratic polynomial is x² - Sx + P = 0
= x² - (-7)x + 10
= x² + 7x + 10
Verification:
Here a = 1, b = 7, c = 10
Sum of zeroes = (-2) + (-5) = 7

Question 15.
Find the zeroes of the quadratic polynomial 3x² - 75 and verify the relationship between the zeroes and the coefficients. (2014)
Solution:
We have, 3x² - 75
= 3(x² - 25)
= 3(x² - 52)
= 3(x - 5)(x + 5)
Zeroes are:
x - 5 = 0 or x + 5 = 0
x = 5 or x = -5
Verification:
Here a = 3, b = 0, c = -75
Sum of the zeroes = 5 + (-5) = 0

Question 16.
Find the zeroes of p(x) = 2x² - x - 6 and verify the relationship of zeroes with these co-efficients. (2017 OD)
Solution:
p(x) = 2x² - x - 6 ...[Given]
= 2x² - 4x + 3x - 6
= 2x (x - 2) + 3 (x - 2)
= (x - 2) (2x + 3)
Zeroes are:
x - 2 = 0 or 2x + 3 = 0
x = 2 or x =
Verification:
Here a = 2, b = -1, c = -6

Question 17.
What must be subtracted from the polynomial f(x) = x⁴ + 2x³ - 13x² - 12x + 21 so that the resulting polynomial is exactly divisible by x² - 4x + 3? (2012, 2017 D)
Solution:

(2x - 3) should be subtracted from x⁴ + 2x³ - 13x² - 12x + 21.

Polynomials Class 10 Important Questions Short Answer-II (3 Marks)

Question 18.
Verify whether 2, 3 and are the zeroes of the polynomial p(x) = 2x³ - 11x² + 17x - 6. (2012, 2017 D)
Solution:
p(x) = 2x³ - 11x² + 17x - 6
When x = 2,
p(2) = 2(2)³ - 11(2)² + 17(2) - 6 = 16 - 44 + 34 - 6 = 0
When x = 3, p(3) = 2(3)³ - 11(3)² + 17(3) - 6 = 54 - 99 + 51 - 6 = 0

Yes, x = 2, 3 and all are the zeroes of the given polynomial.

Question 19.
Show that and are the zeroes of the polynomial 4x² + 4x - 3 and verify the relationship between zeroes and co-efficients of polynomial. (2013)
Solution:
Let P(x) = 4x² + 4x - 3

Question 20.
Find a quadratic polynomial, the sum and product of whose zeroes are -8 and 12 respectively. Hence find the zeroes. (2014)
Solution:
Let Sum of zeroes (α + β) = S = -8 ...[Given]
Product of zeroes (αβ) = P = 12 ...[Given]
Quadratic polynomial is x² - Sx + P
= x² - (-8)x + 12
= x² + 8x + 12
= x² + 6x + 2x + 12
= x(x + 6) + 2(x + 6)
= (x + 2)(x + 6)
Zeroes are:
x + 2 = 0 or x + 6 = 0
x = -2 or x = -6

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