Class 10 Maths: Relationship between Zeroes and Coefficients of a Polynomial — Practice Questions with Answers

Explanation: The polynomial p(x) = x² - 3√2x + 4 can be factored by splitting the middle term into -2√2x and -√2x. This gives (x - 2√2)(x - √2) = 0, so the zeroes are 2√2 and √2.

Explanation: Factoring the polynomial p(y) = (1/3)(21y² - 11y - 2) gives (1/3)(7y+1)(3y-2). The zeroes are y = -1/7 and y = 2/3. There appears to be a typo in the options provided in the source document, as none match the correct answer. Option (a) is the closest if the signs were flipped.

Explanation: For the quadratic polynomial p(x) = x² - ax - b, the sum of zeroes α + β = -(-a)/1 = a, and the product of zeroes αβ = -b/1 = -b. Therefore, α + β + αβ = a - b.

Explanation: For the polynomial p(x) = kx² - 30x + 45k, the sum of zeroes α + β = -(-30)/k = 30/k, and the product of zeroes αβ = 45k/k = 45. Given α + β = αβ, we have 30/k = 45, which implies k = 30/45 = 2/3.

Explanation: For the polynomial 3x² + 6x + k, the sum of zeroes α + β = -6/3 = -2, and the product of zeroes αβ = k/3. Given α + β + αβ = -2/3, we substitute the values: -2 + k/3 = -2/3. This simplifies to k/3 = 2 - 2/3 = 4/3, so k = 4.

Explanation: If the zeroes of a quadratic polynomial are reciprocals of each other, their product is 1. For the polynomial ax² + bx + 2a/b, the product of zeroes is (constant term) / (coefficient of x²) = (2a/b) / a = 2/b. Setting the product to 1 gives 2/b = 1, which means b = 2.

Explanation: Let the polynomial be p(x) = x² - 5x + 4. If we add a constant k, the new polynomial is q(x) = p(x) + k. For 3 to be a zero, q(3) must be 0. So, (3)² - 5(3) + 4 + k = 0, which simplifies to 9 - 15 + 4 + k = 0, or -2 + k = 0. Therefore, k = 2.

Explanation: The expression 1/α + 1/β can be written as (α+β)/(αβ). For the polynomial x² + x - 1, the sum of zeroes (α+β) is -b/a = -1/1 = -1, and the product of zeroes (αβ) is c/a = -1/1 = -1. Therefore, the value is (-1)/(-1) = 1.

Explanation: For a quadratic polynomial ax² + bx + c, the sum of zeroes (α + β) is given by -b/a. In p(x) = x² - 1, a=1, b=0, and c=-1. Therefore, the sum of zeroes is -0/1 = 0.

Explanation: The expression 1/α + 1/β simplifies to (α + β) / (αβ). For the given polynomial, the sum of zeroes α + β = -(-3)/4 = 3/4, and the product of zeroes αβ = -7/4. Thus, the value is (3/4) / (-7/4) = -3/7.

Explanation: A polynomial with zeroes α and β is given by k(x-α)(x-β). For α=-1 and β=2, this becomes k(x+1)(x-2) = k(x² - x - 2). Assuming k=1, the polynomial is x² - x - 2.

Explanation: If -3 is a zero of the polynomial p(x), then p(-3) must be 0. Substituting x = -3 gives (k-1)(-3)² + k(-3) + 1 = 0, which simplifies to 9(k-1) - 3k + 1 = 0, leading to 6k = 8, or k = 4/3.

Explanation: The general form of a quadratic polynomial is k(x² - (sum of zeroes)x + (product of zeroes)). Substituting the given values, we get k(x² - (-5)x + 6) = k(x² + 5x + 6).

Explanation: The general form of a quadratic polynomial is k[x² - (sum of zeroes)x + (product of zeroes)]. Substituting the given sum (-5) and product (6) yields x² - (-5)x + 6, which simplifies to x² + 5x + 6.

Explanation: If 2 is a zero of p(x) = x² + 3x + k, then p(2) must be 0. Substituting x=2 gives (2)² + 3(2) + k = 0, which simplifies to 4 + 6 + k = 0, so k = -10.