CBSE Class 10 Mathematics Polynomials Worksheet Set 03

Question. Choose and write the correct option in each of the following questions.
(i) If 5 is a zero of the quadratic polynomial, \( x^2 - kx - 15 \) then the value of k is
(a) 2
(b) -2
(c) 4
(d) -4
Answer: (a) 2

 

Question. The number of polynomials having zeros 1 and -2 is
(a) 1
(b) 2
(c) 3
(d) more than 3
Answer: (d) more than 3

 

Question. A quadratic polynomial, whose zeros are -3 and 4, is
(a) \( x^2 - x + 12 \)
(b) \( x^2 + x + 12 \)
(c) \( \frac{x^2}{2} - \frac{x}{2} - 6 \)
(d) \( 2x^2 + 2x - 24 \)
Answer: (c) \( \frac{x^2}{2} - \frac{x}{2} - 6 \)

 

Question. The zeros of the quadratic polynomial \( x^2 + kx + k, k \neq 0 \)
(a) cannot be positive
(b) cannot be negative
(c) are always equal
(d) are always unequal
Answer: (a) cannot be positive

 

Question. If the graph of a polynomial intersects the x-axis at exactly two points, then it
(a) cannot be a linear or a cubic polynomial
(b) can be a quadratic polynomial only
(c) can be a cubic or a quadratic polynomial
(d) can be a linear or a quadratic polynomial
Answer: (c) can be a cubic or a quadratic polynomial

 

Question. If 1 is a zero of the polynomial \( p(x) = ax^2 - 3(a - 1)x - 1 \), then find the value of a.
Answer: 1

 

Question. If one root of the polynomial \( p(y) = 7y^2 + 14y + m \) is reciprocal of other, then find the value of m.
Answer: 7

 

Question. Find the other zero of the quadratic polynomial \( y^2 + 7y - 60 \) if one zero is -12.
Answer: 5

 

Question. Find the quadratic polynomial whose zeros are - 3 and - 5.
Answer: \( x^2 + 8x + 15 \)

 

Question. What number should be added to the polynomial \( x^2 + 7x - 35 \) so that 3 is the zero of the polynomial?
Answer: 5

 

Question. If one zero of the quadratic polynomial \( p(x) = x^2 + 4kx - 25 \) is negative of the other, find the value of k.
Answer: 0

 

Question. If m and n are the zeros of the polynomial \( 3x^2 + 11x - 4 \), find the value of \( \frac{m}{n} + \frac{n}{m} \).  
Answer: \( \frac{-145}{12} \)

 

Question. Find the zeros of the polynomial \( 5y^2 - 11y + 2 \).
Answer: \( \frac{1}{5}, 2 \)

 

Question. If one of the zeros of the quadratic polynomial \( (k - 2) x^2 - 2x - (k + 5) \) is 4, find the value of k.
Answer: 3

 

Question. If \( \alpha, \beta \) are the zeros of the polynomial \( x^2 + x - 6 \), find the value of \( \frac{1}{\alpha^2} + \frac{1}{\beta^2} \).
Answer: \( \frac{13}{36} \)

 

Question. If \( \alpha, \beta \) are the two zeros of the polynomial \( f(y) = y^2 - 8y + a \) and \( \alpha^2 + \beta^2 = 40 \), find the value of a.
Answer: 12

 

Question. If the sum of the zeros of the quadratic polynomial \( f(x) = kx^2 + 2x + 3k \) is equal to their product, find the value of k.
Answer: \( \frac{-2}{3} \)

 

Question. Find the zeros of the following polynomials and verify the relationship between the zeros and the coefficients of the polynomials.
(i) \( x^2 + \frac{1}{6}x - 2 \)
(ii) \( \sqrt{3}x^2 - 11x + 6\sqrt{3} \)
(iii) \( a(x^2 + 1) - x(a^2 + 1) \)
Answer: (i) \( (\frac{4}{3}, \frac{-3}{2}) \) (ii) \( \frac{2}{\sqrt{3}}, 3\sqrt{3} \) (iii) \( a, \frac{1}{a} \)

 

Question. If \( (x - 2) \) is a factor of \( x^3 + ax^2 + bx + 16 \) and \( b = 4a \), find the values of a and b.
Answer: \( a = -2, b = -8 \)

 

Question. If \( \alpha \) and \( \beta \) are zeros of polynomial \( f(x) = 2x^2 + 11x + 5 \), then find
(i) \( \alpha^4 + \beta^4 \)
(ii) \( \frac{1}{\alpha} + \frac{1}{\beta} - 2\alpha\beta \)
Answer: (i) \( \frac{10001}{16} \) (ii) \( \frac{-36}{5} \)

 

Question. Verify that the numbers given alongside the cubic polynomials below are their zeros. Also verify the relationship between the zeros and the coefficients.
(i) \( x^3 - 2x^2 - 5x + 6; -2, 1, 3 \)
(ii) \( 2x^3 + 7x^2 + 2x - 3; -3, -1, \frac{1}{2} \)
Answer: Proved

 

Question. If the zeroes of the polynomial \( x^3 - 3x^2 + x + 1 \) are \( a - b, a, a + b \), find a and b. 
Answer: \( a = 1, b = \pm\sqrt{2} \)

SECTION A

Question. Choose and write the correct option in the following questions.
(i) A quadratic polynomial with 3 and 2 as the sum and product of its zeros respectively is
(a) \( x^2 + 3x - 2 \)
(b) \( x^2 - 3x + 2 \)
(c) \( x^2 - 2x + 3 \)
(d) \( x^2 - 2x - 3 \)
Answer: (b) \( x^2 - 3x + 2 \)

 

Question. The zeros of the quadratic polynomial \( x^2 + 99x + 127 \) are
(a) both positive
(b) both negative
(c) one positive and one negative
(d) both equal
Answer: (b) both negative

 

Question. If the product of the zeros of the polynomial \( p(x) = ax^3 - 6x^2 + 11x - 6 \) is 4, then a equals
(a) \( -\frac{3}{2} \)
(b) \( \frac{2}{3} \)
(c) \( -\frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (a) \( -\frac{3}{2} \)

 

Question. Solve the following questions.
(i) If \( \alpha \) and \( \beta \) are zeros of a polynomial \( x^2 - 4\sqrt{3}x + 3 \), then find the value of \( \alpha + \beta - \alpha\beta \).
Answer: \( 4\sqrt{3} - 3 \)

 

Question. If the square of difference of the zeros of the quadratic polynomial \( x^2 + px + 45 \) is equal to 144. Find out the value of p.
Answer: \( \pm 18 \)

SECTION B

 

Question. If p and q are the zeros of the polynomial \( f(x) = 2x^2 - 7x + 3 \), find the value of \( p^2 + q^2 \).
Answer: \( \frac{37}{4} \)

 

Question. If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial \( f(x) = 3x^2 - 5x - 2 \), then evaluate \( \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} \).
Answer: \( - \frac{215}{18} \)

 

Question. If \( \alpha, \beta \) are the zeros of the quadratic polynomial \( p(y) = y^2 - 4y + 3 \), find the value of \( \alpha^4\beta^3 + \alpha^3\beta^4 \).
Answer: 108

 

Question. Find the cubic polynomial with the sum, sum of the products of its zeros taken two at a time, and the product of its zeros as -3, -8 and 2 respectively.
Answer: \( x^3 + 3x^2 - 8x - 2 \)

 

Question. If \( \frac{2}{3} \) and -3 are the zeros of the polynomial \( ax^2 + 7x + b \), then find the values of a and b.
Answer: \( a = 3, b = -6 \)

 

Question. If \( \alpha \) and \( \beta \) are the zeros of the polynomial \( 2x^2 - 3x + 1 \), then find the value of (i) \( \alpha^2\beta + \alpha\beta^2 \) (ii) \( \alpha^2 + \beta^2 \).
Answer: (i) \( \frac{3}{4} \) (ii) \( \frac{5}{4} \)

 

Question. Find a quadratic polynomial whose zeros are
\( \frac{5 + \sqrt{2}}{5 - \sqrt{2}}, \frac{5 - \sqrt{2}}{5 + \sqrt{2}} \)
Answer: \( 23x^2 - 54x + 23 \)

 

Question. If the sum of squares of the zeroes of the quadratic polynomial \( x^2 - 8x + k \) be 40, find k.
Answer: 12

 

Question. If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial \( f(x) = 3x^2 - 7x - 6 \), find a polynomial whose zeros are (i) \( \alpha^2 \) and \( \beta^2 \) (ii) \( 2\alpha + 3\beta \) and \( 3\alpha + 2\beta \).
Answer: (i) \( \frac{1}{9}(9x^2 - 85x + 36) \) (ii) \( \frac{1}{3}(3x^2 - 35x + 92) \)

 

Question. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:
\( x^2 - (2a + b)x + 2ab \)
Answer: \( 2a, b \)

 

Please click on below link to download CBSE Class 10 Mathematics Polynomials Worksheet Set C