Question 1

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

Accepted Answer

To find the value of $k$, we substitute the given zero $x = 2$ into the polynomial $p(x) = x^2 + 3x + k$ and equate it to zero. $p(2) = (2)^2 + 3(2) + k = 0$ $4 + 6 + k = 0$ $10 + k = 0$ $k = -10$
Therefore, the correct option is -10.

Question 2

The degree of a polynomial $p(x) = 4x^3 - 3x^2 + 5x - 7$ is:

Accepted Answer

The degree of a polynomial is the highest power of the variable $x$ in the polynomial. In the given polynomial $4x^3 - 3x^2 + 5x - 7$, the highest power of $x$ is 3. Therefore, the degree of the polynomial is 3.

Question 3

If the zeroes of the quadratic polynomial $ax^2 + bx + c$ are both positive, then:

Accepted Answer

Let the zeroes be $\alpha$ and $\beta$. Both are positive, so $\alpha + \beta > 0$ and $\alpha\beta > 0$. 
Sum of zeroes = $-b/a > 0$
Product of zeroes = $c/a > 0$
This implies that $a$, $b$, and $c$ must have signs such that $b/a 0$, meaning $a$ and $c$ have the same sign, and $b$ has the opposi

Question 4

The graph of $y = p(x)$ is given, where $p(x)$ is a polynomial. The number of zeroes of $p(x)$ is:

Accepted Answer

The number of zeroes of a polynomial $p(x)$ is equal to the number of times the graph of $y = p(x)$ intersects the x-axis.

Question 5

If $\alpha$ and $\beta$ are the zeroes of the polynomial $2x^2 - 5x + 7$, then the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is:

Accepted Answer

For the polynomial $2x^2 - 5x + 7$, $a = 2$, $b = -5$, $c = 7$.
Sum of zeroes: $\alpha + \beta = -\frac{b}{a} = \frac{5}{2}$
Product of zeroes: $\alpha\beta = \frac{c}{a} = \frac{7}{2}$
Now, $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{5/2}{7/2} = \frac{5}{7}$.

Polynomials — MP Board Class 10 Mathematics Important Questions & Revision Notes