Q1. If one zero of the quadratic polynomial $x^2 + 3x + k$ is 2, then the value of $k$ is:
Q2. The quadratic polynomial whose zeros are -3 and 4 is:
Q3. The number of polynomials having zeros as -2 and 5 is:
Q4. Assertion (A): $x^2 + 4x + 5$ has two real zeroes. Reason (R): A quadratic polynomial can have at most two zeroes.
Q5. Find the zeroes of the quadratic polynomial $6x^2 - 3 - 7x$ and verify the relationship between the zeroes and the coefficients.
Solve on white paper.
Q6. If $\alpha$ and $\beta$ are zeroes of $x^2 - x - 4$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta$.
Q7. Find a quadratic polynomial, the sum and product of whose zeroes are $\sqrt{2}$ and $-\frac{3}{2}$ respectively.
Q8. If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $5x^2 + 5x + 1$, find the value of $\alpha^2 + \beta^2$.
Q9. If the sum of the zeroes of the polynomial $p(x) = (k^2 - 14)x^2 - 2x - 12$ is 1, find the value of $k$.
Q10. If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $f(x) = x^2 - px + q$, prove that $\frac{\alpha^2}{\beta^2} + \frac{\beta^2}{\alpha^2} = \frac{p^4}{q^2} - \frac{4p^2}{q} + 2$.
Q11. Case Study:
The graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola. The shape of the parabola depends on the value of 'a'. If $a > 0$, it opens upwards, and if $a < 0$, it opens downwards. Consider the polynomial $p(x)=x^2 - 8x + 12$.
(i) What is the shape of the graph of $p(x)$?
(ii) Find the zeroes of the polynomial $p(x)$.
(iii) Find the value of the polynomial at $x = 0$.