RD Sharma Solutions Class 10 Polynomials Exercise 2.2

RD Sharma Class 10 Solutions Chapter 2 Ex 2.2 PDF Free Download

EXERCISE – 2.2

  1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeros and coefficients in each of the following cases:

    (i) f(x) = 2x3+x2-5x+2; ½, 1,-2

    (ii) g(x) = x3-4x2+5x-2; 2,1,1

    Solutions:

    • f(x) = 2x3+x2-5x+2; ½, 1,-2——————-(i)

    Let x = ½ in equation (i),

    class 10 maths chapter 2 polynomials exercise 2.2-1

    Let x = 1 in equation (i),

    f(1) = 2(1)3+(1)2-5(1)+2

    = 2 + 1 – 5 + 2 = 0

    Let x = -2 in equation (i),

    f(-2) = 2(-2)3+(-2)2-5(-2)+2

    = -16 + 4 + 10 + 2 = -16 + 16 = 0

    Now,

    class 10 maths chapter 2 polynomials exercise 2.2-2
    class 10 maths chapter 2 polynomials exercise 2.2-3

    Hence, verified.

    • g(x) = x3-4x2+5x-2; 2,1,1——————-(ii)

    Let x = 2 in equation (ii),

    g(2) = 23-4(2)2+5(2)-2

    = 8 – 16 + 10 – 2 = 18 – 18 = 0

    Let x = 1 in equation (ii),

    g(1) = 13-4(1)2+5(1)-2=0

    class 10 maths chapter 2 polynomials exercise 2.2-4

    Hence, verified.

  2. Find a cubic polynomial with the sum, sum of the product of its zeroes is taken two at a time, and product of its zeroes as 3, -1 and -3 respectively.

    Solutions:

    A cubic polynomial is of the form ax3+bx2+cx+d.

    class 10 maths chapter 2 polynomials exercise 2.2-5

    k is any non-zero real numbers.

  3. If the zeroes of the polynomial f(x) = 2x3 – 15 x2 + 37 x – 30, find them.

    Solutions:

    class 10 maths chapter 2 polynomials exercise 2.2-6

  4. Find the condition that the zeroes of the polynomial f( x ) = x3 + 3px2 + 3qx + r may be in A.P.

    Solutions:

    Given that, f( x ) = x3 + 3px2 + 3qx + r

    Here, a=1, b=3p, c=3q, d=r

    Let the zeroes of the polynomials be = a – d, a, a + d

    Now,

    The sum of zeroes = -b/a

    a + a – d + a + d = -3p

    3a = -3p

    a = -p

    We know that, a is the zero of the polynomial f(x),

    ∴, f(a) = 0

    class 10 maths chapter 2 polynomials exercise 2.2-7

  5. If zeroes of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 – 3abc + a2d = 0.

    Solutions:

    Given that,

    f(x) = ax3 + 3bx2 + 3cx + d

    Here, a=1, b=3p, c=3q, d=r

    Let the zeroes of the polynomials be a – d, a, a + d.

    Then,

    The sum of zeroes = -b/a

    a + a – d + a + d = -3b/a

    class 10 maths chapter 2 polynomials exercise 2.2-8

  6. If the zeroes of the polynomial f(x) = x3 – 12x2 + 39x + k are in A.P., find the value of k.

    Solutions:

    f(x) = x3 – 12x2 + 39x + k

    Let the zeroes of the polynomial f(x) be a-d, a, a+d be .

    Sum of the zeroes = 12

    3a = 12

    a = 4

    Now,

    f(a) = 0

    class 10 maths chapter 2 polynomials exercise 2.2-8

    64 – 192 + 156 + k = 0

    k = -28

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