Vardaan Learning Institute

1. If the graph of a polynomial $y = p(x)$ intersects the X-axis at 3 distinct points, then the number of zeroes of the polynomial is:

1. 0
2. 1
3. 2
4. 3

2. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $x^2 - 5x + 6$, then the value of $\alpha + \beta$ is:

1. 5
2. -5
3. 6
4. -6

3. A quadratic polynomial whose zeroes are -3 and 4 is:

1. $x^2 - x + 12$
2. $x^2 + x + 12$
3. $x^2 - x - 12$
4. $2x^2 + 2x - 24$

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

1. 10
2. -10
3. 5
4. -5

5. The shape of the graph of a quadratic polynomial $ax^2 + bx + c$ (where $a \neq 0$) is a:

1. Straight Line
2. Circle
3. Parabola
4. Ellipse

6. If the zeroes of the quadratic polynomial $x^2 + (a + 1)x + b$ are 2 and -3, then:

1. a = -7, b = -1
2. a = 5, b = -1
3. a = 2, b = -6
4. a = 0, b = -6

7. If the sums of the zeroes of the polynomial $p(x) = 2x^2 - kx + 5$ is 3, then the value of k is:

1. 6
2. -6
3. 3
4. -3

8. If $\alpha, \beta$ are the zeroes of $x^2 + 5x + 5$, then value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is:

1. 1
2. -1
3. 5
4. -5

9. The number of polynomials having zeroes as -2 and 5 is:

1. 1
2. 2
3. 3
4. More than 3

10. The zeroes of the quadratic polynomial $x^2 + 99x + 127$ are:

1. both positive
2. both negative
3. one positive and one negative
4. both equal

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

12. Find a quadratic polynomial, the sum and product of whose zeroes are $\frac{1}{4}$ and $-1$ respectively.

13. If the zeroes of the polynomial $x^2 + px + q$ are double in value to the zeroes of $2x^2 - 5x - 3$, find the values of p and q.

14. If one zero of the polynomial $(k - 1)x^2 + kx + 1$ is -3, find the value of k.

15. 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 $k$.

16. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = 2x^2 - 4x + 5$, form a quadratic polynomial whose zeroes are $1/\alpha$ and $1/\beta$.

17. If the squared difference of the zeroes of the quadratic polynomial $f(x) = x^2 + px + 45$ is equal to 144, find the value of $p$.

18. Find the values of $a$ and $b$ if the sum and product of the zeroes of the polynomial $2x^2 - (a + 3)x + (2b - 1)$ are 4 and 3 respectively.

19. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $x^2 - bx + c$, find the value of $\alpha^2 + \beta^2$ in terms of b and c.

20. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - 4x + 1$, find a quadratic polynomial whose zeroes are $\frac{\alpha^2}{\beta}$ and $\frac{\beta^2}{\alpha}$.

21. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(s) = 3s^2 - 6s + 4$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} + 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) + 3\alpha\beta$.

22. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = ax^2 + bx + c$, then evaluate the following in terms of a, b, and c:
(i) $\alpha^2 - \beta^2$
(ii) $\alpha^3 + \beta^3$
(iii) $\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}$

23. Case Study: The Rainbow Bridge.
A suspension bridge has a main cable shaped like a parabola. The graph of a parabola can be represented by a quadratic polynomial. Consider a bridge represented by the polynomial $P(x) = -x^2 + 2x + 8$.

(i) Does the parabola open upwards or downwards? Give a reason. (1 Mark)
(ii) Find the zeroes of the polynomial representing the bridge. (1 Mark)
(iii) What is the value of the polynomial at x = 0? (1 Mark)
(iv) Verify if x = 4 is a zero of the polynomial. (1 Mark)