Q1. If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 2x + 3k$ such that $\alpha + \beta = \alpha\beta$, then the value of $k$ is:
Q2. If one zero of the polynomial $3x^2 + 8x + k$ is the reciprocal of the other, then the value of $k$ is:
Q3. If $\alpha, \beta$ are zeroes of $x^2 + 5x + 5$, then the value of $\alpha^{-1} + \beta^{-1}$ is:
Q4. Assertion (A): The polynomial whose zeroes are 2 and -3 is $x^2 + x - 6$. Reason (R): A quadratic polynomial with zeroes $\alpha$ and $\beta$ is given by $k[x^2 - (\alpha + \beta)x + \alpha\beta]$.
Q5. Find the zeroes of the quadratic polynomial $4\sqrt{3}x^2 + 5x - 2\sqrt{3}$ and verify the relationship between the zeroes and the coefficients.
Solve on white paper.
Q6. If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - x - 2$, find a quadratic polynomial whose zeroes are $2\alpha + 1$ and $2\beta + 1$.
Q7. If the sum of the squares of the zeroes of the polynomial $x^2 - 8x + k$ is 40, find the value of $k$.
Q8. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - 5x + k$ such that $\alpha - \beta = 1$, find the value of $k$.
Q9. Find a quadratic polynomial whose zeroes are the reciprocals of the zeroes of the polynomial $ax^2 + bx + c, a \neq 0, c \neq 0$.
Q10. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - 3x - 2$, find a quadratic polynomial whose zeroes are $2\alpha + 3\beta$ and $3\alpha + 2\beta$.
Q11. Case Study:
A highway underpass is parabolic in shape. The curve of the underpass is represented by the polynomial $p(x) = x^2 - 2x - 8$.
(i) Find the zeroes of the polynomial $p(x)$.
(ii) Verify that the sum of zeroes is equal to $-\frac{b}{a}$.
(iii) If the graph is shifted such that the zeroes become 0 and 4, find the new polynomial.