Zeroes of Quadratic Polynomials PYQs

Practice CBSE Board exam questions on zeroes of quadratic polynomials $p(x) = ax^2 + bx + c$. Features middle-term splitting factorisation, graphical interpretation of zeroes from x-axis intercepts, and evaluating algebraic values such as $\alpha^2 + \beta^2$ and $\frac{1}{\alpha} + \frac{1}{\beta}$.

Class 10 Maths: Zeroes of Polynomial Previous Year Questions

Mastering the zeroes of polynomials is one of the highest-scoring topics in CBSE Class 10 Board Mathematics. Below are curated board exam questions with complete step-by-step marking scheme solutions.

CBSE 2020 • 1 Mark Question 1

If one zero of the quadratic polynomial $x^2 + 3x + k$ is 2, find the value of $k$.

Step-by-Step Solution:

Let $p(x) = x^2 + 3x + k$. Since $2$ is a zero of $p(x)$, we have $p(2) = 0$.
$(2)^2 + 3(2) + k = 0 \implies 4 + 6 + k = 0 \implies 10 + k = 0 \implies k = -10$.
Answer: The value of $k$ is -10.

CBSE 2019 • 3 Marks Question 2

Find the zeroes of the quadratic polynomial $6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients.

Step-by-Step Solution:

First, arrange the polynomial in standard descending powers: $p(x) = 6x^2 - 7x - 3$.
Split the middle term: product $= 6 \times (-3) = -18$, sum $= -7$. Factors are $-9$ and $+2$.
$6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3x + 1) = 0$.
Hence, the zeroes are $\alpha = \frac{3}{2}$ and $\beta = -\frac{1}{3}$.

Verification of Coefficients:
Here $a = 6, b = -7, c = -3$.
1. Sum of zeroes: $\alpha + \beta = \frac{3}{2} + \left(-\frac{1}{3}\right) = \frac{9 - 2}{6} = \frac{7}{6} = -\frac{b}{a} = -\frac{(-7)}{6} = \frac{7}{6}$. (Verified)
2. Product of zeroes: $\alpha\beta = \left(\frac{3}{2}\right)\left(-\frac{1}{3}\right) = -\frac{3}{6} = -\frac{1}{2} = \frac{c}{a} = \frac{-3}{6} = -\frac{1}{2}$. (Verified)

CBSE 2022 • 3 Marks Question 3

If $\alpha$ and $\beta$ are zeroes of the polynomial $x^2 - 5x + 6$, find the value of (i) $\alpha^2 + \beta^2$ and (ii) $\frac{1}{\alpha} + \frac{1}{\beta}$.

Step-by-Step Solution:

For $x^2 - 5x + 6$, $a = 1, b = -5, c = 6$.
Sum of zeroes $\alpha + \beta = -b/a = 5$. Product of zeroes $\alpha\beta = c/a = 6$.

(i) $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (5)^2 - 2(6) = 25 - 12 = 13$.
(ii) $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{5}{6}$.
Answers: (i) 13, (ii) 5/6.

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Zeroes of Quadratic Polynomials PYQs

Class 10 Maths: Zeroes of Polynomial Previous Year Questions

If one zero of the quadratic polynomial \(x^2 + 3x + k\) is 2, find the value of \(k\).

Find the zeroes of the quadratic polynomial \(6x^2 - 3 - 7x\) and verify the relationship between the zeroes and the coefficients.

If \(\alpha\) and \(\beta\) are zeroes of the polynomial \(x^2 - 5x + 6\), find the value of (i) \(\alpha^2 + \beta^2\) and (ii) \(\frac{1}{\alpha} + \frac{1}{\beta}\).