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Important Questions for Class 10 Maths Chapter 2 Polynomials

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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, (Î± + Î²) = $\frac { -b }{ a }$ = 3 .....(given)
â‡’ $\frac { -(-k) }{ 3 }$ = 3
â‡’ k = 9

Question 2.
If Î± and Î² are the zeroes of the polynomial ax² + bx + c, find the value of Î±² + Î²². (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, (Î± + Î²) = 1 ...[Given]
â‡’ $\frac { -b }{ a }$ = 1
â‡’ $\frac { -\left( -2 \right) }{ { k }^{ 2 }-14 }$ = 1
â‡’ k² - 14 = 2
â‡’ k² = 16
â‡’ k = Â±4

Question 4.
If Î± and Î² are the zeroes of a polynomial such that Î± + Î² = -6 and Î±Î² = 5, then find the polynomial. (2016 D)
Solution:
Quadratic polynomial is x² - Sx + P = 0
â‡’ x² - (-6)x + 5 = 0
â‡’ 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 Î± and $\frac { 1 }{ \alpha }$ be the zeroes of P(x).
P(a) = ax² + bx + c ...(given)
Product of zeroes = $\frac { c }{ a }$
â‡’ Î± Ã— $\frac { 1 }{ \alpha }$ = $\frac { c }{ a }$
â‡’ 1 = $\frac { c }{ a }$
â‡’ a = c (Required condition)
Coefficient of x² = Constant term

Question 7.
Form a quadratic polynomial whose zeroes are 3 + âˆš2 and 3 - âˆš2. (2012)
Solution:
Sum of zeroes,
S = (3 + âˆš2) + (3 - âˆš2) = 6
Product of zeroes,
P = (3 + âˆš2) x (3 - âˆš2) = (3)² - (âˆš2)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 âˆš3 and $\frac { 1 }{ \surd 3 }$ respectively. (2014)
Solution:
Sum of zeroes, (S) = âˆš3
Product of zeroes, (P) = $\frac { 1 }{ \surd 3 }$
Quadratic polynomial = x² - Sx + P

Question 9.
Find a quadratic polynomial, the sum and product of whose zeroes are 0 and -âˆš2 respectively. (2015)
Solution:
Quadratic polynomial is
x² - (Sum of zeroes) x + (Product of zeroes)
= x² - (0)x + (-âˆš2)
= x² - âˆš2

Question 10.
Find the zeroes of the quadratic polynomial âˆš3 x² - 8x + 4âˆš3. (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
â‡’ x = 3 or x = $\frac { -1 }{ 2 }$
Since the zeroes of required polynomial is double of given polynomial.
Zeroes of the required polynomial are:
3 Ã— 2, ( $\frac { -1 }{ 2 }$ Ã— 2), i.e., 6, -1
Sum of zeroes, S = 6 + (-1) = 5
Product of zeroes, P = 6 Ã— (-1) = -6
Quadratic polynomial is x² - Sx + P
â‡’ 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 $\frac { 3+\surd 5 }{ 5 }$ and $\frac { 3-\surd 5 }{ 5 }$ . (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 = $\frac { -3 }{ 2 }$
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 $\frac { 1 }{ 2 }$ 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 $\frac { 1 }{ 2 }$ all are the zeroes of the given polynomial.

Question 19.
Show that $\frac { 1 }{ 2 }$ and $\frac { -3 }{ 2 }$ 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 (Î± + Î²) = S = -8 ...[Given]
Product of zeroes (Î±Î²) = 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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Important Questions for Class 10 Maths Chapter 2 Polynomials