Chapter 2: Polynomials
CBSE Official PYQs (2015-2026)
Q1. The graph of \(y = f(x)\) is given below. The number of zeroes of \(f(x)\) is :
Q2. If one zero of the quadratic polynomial \(x^2 + 3x + k\) is 2, then the value of \(k\) is :
Q3. If the sum of the zeroes of the quadratic polynomial \(3x^2 - kx + 6\) is 3, then the value of \(k\) is :
Q4. A quadratic polynomial, the sum and product of whose zeroes are \(-3\) and \(2\) respectively, is :
Q5. If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(2x^2 - x - 1\), then the value of \(\alpha^2 + \beta^2\) is :
Q6. The zeroes of the quadratic polynomial \(x^2 - 2x - 8\) are :
Q7. If one zero of the quadratic polynomial \((k-1)x^2 + kx + 1\) is \(-3\), then the value of \(k\) is :
Q8. The number of zeroes of a constant polynomial (non-zero) is :
Q9. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = x^2 - x - 4\), then the value of \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta\) is :
Q10. A quadratic polynomial whose zeroes are \(-3\) and \(4\) is :
Q11. If the zeroes of the quadratic polynomial \(x^2 + (a+1)x + b\) are \(2\) and \(-3\), then :
Q12. If one zero of the quadratic polynomial \(3x^2 + 8x + k\) is the reciprocal of the other, then the value of \(k\) is :
Q13. The sum and product of the zeroes of a quadratic polynomial are \(0\) and \(-\sqrt{2}\) respectively. The polynomial is :
Q14. If the sum of the zeroes of the quadratic polynomial \(ax^2 + 5x + 12\) is equal to their product, then the value of \(a\) is :
Q15. The common zeroes of the quadratic polynomials \(x^2 - 5x + 6\) and \(x^2 - 6x + 8\) is/are :
Q16. If \(\alpha\) and \(\beta\) are zeroes of the quadratic polynomial \(x^2 - p(x+1) - c\) such that \((\alpha + 1)(\beta + 1) = 0\), then the value of \(c\) is :
Q17. If the zeroes of the quadratic polynomial \(ax^2 + bx + c\) (\(c \neq 0\)) are equal, then :
Q18. If the product of the zeroes of the quadratic polynomial \(x^2 - 3kx + 2k^2 - 1\) is \(7\), then the value of \(k\) is (given \(k > 0\)) :
Q19. If the sum of zeroes of the quadratic polynomial \(2x^2 - (k+4)x + 9\) is \(3\), then the value of \(k\) is :
Q20. If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(f(x) = x^2 - 5x + k\) such that \(\alpha - \beta = 1\), then the value of \(k\) is :
Q21. The number of polynomials having zeroes as \(-2\) and \(5\) is :
Q22. If \(\alpha, \beta\) are the zeroes of the quadratic polynomial \(p(x) = 4x^2 - 5x - 1\), then the value of \(\alpha^2\beta + \alpha\beta^2\) is :
Q23. If one of the zeroes of a quadratic polynomial of the form \(x^2 + ax + b\) is the negative of the other, then it :
Q24. If the product of zeroes of the quadratic polynomial \(ax^2 - 6x - 6\) is \(4\), then the value of \(a\) is :
Q25. If the graph of a quadratic polynomial \(p(x)\) does not intersect the x-axis at any point, then the polynomial \(p(x)\) has :
Q26. If \(\alpha, \beta\) are the zeroes of the quadratic polynomial \(p(x) = 3x^2 - 5x + 2\), then the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is :
Q27. A quadratic polynomial whose sum of zeroes is 0 and one zero is 3 is :
Q28. If the zeroes of the quadratic polynomial \(x^2 + kx + k\) (\(k \neq 0\)) :
Q29. The zeroes of the quadratic polynomial \(2x^2 - 3x - 5\) are :
Q30. Find the zeroes of the quadratic polynomial \(4s^2 - 4s + 1\) and verify the relationship between the zeroes and the coefficients.
Q31. Find the zeroes of the quadratic polynomial \(4\sqrt{3}x^2 + 5x - 2\sqrt{3}\) and verify the relationship between the zeroes and coefficients.
Q32. Find the quadratic polynomial, sum of whose zeroes is \(2\sqrt{3}\) and their product is \(-9\).
Q33. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = x^2 - x - 2\), find a quadratic polynomial whose zeroes are \(2\alpha + 1\) and \(2\beta + 1\).
Q34. If the sum of the squares of the zeroes of the quadratic polynomial \(f(x) = x^2 - 8x + k\) is \(40\), find the value of \(k\).
Q35. If one zero of the quadratic polynomial \(2x^2 - 3x + p\) is 3, find the other zero of the polynomial. Also, find the value of \(p\).
Q36. Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial \(f(x) = ax^2 + bx + c\) (\(a \neq 0, c \neq 0\)).
Q37. If the zeroes of the quadratic polynomial \(x^2 - kx + 6\) are in the ratio \(2 : 3\), find the positive value of \(k\).
Q38. Find the zeroes of the quadratic polynomial \(6x^2 - 3 - 7x\) and verify the relationship between the zeroes and the coefficients.
Q39. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = 3x^2 - 6x + 4\), find the value of \(\left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) + 2\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) + 3\alpha\beta\).
Q40. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(f(x) = x^2 - 2x - 3\), find a quadratic polynomial whose zeroes are \(\frac{\alpha^2}{\beta}\) and \(\frac{\beta^2}{\alpha}\).
Q41. If the zeroes of the cubic polynomial \(x^3 - 3x^2 + x + 1\) are \(a-b\), \(a\), and \(a+b\), find the values of \(a\) and \(b\).
Q42. Find all the zeroes of the polynomial \(2x^4 - 3x^3 - 3x^2 + 6x - 2\), if you know that two of its zeroes are \(\sqrt{2}\) and \(-\sqrt{2}\).
Q43. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(y) = 5y^2 - 7y + 1\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).
Q44. If the sum of the zeroes of the quadratic polynomial \(f(t) = kt^2 + 2t + 3k\) is equal to their product, find the value of \(k\).
Q45. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = 2x^2 + 5x + k\) such that \(\alpha^2 + \beta^2 + \alpha\beta = \frac{21}{4}\), find the value of \(k\).
Q46. Prove that the quadratic polynomial \(x^2 + 2x + 5\) has no real zeroes.
Q47. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = ax^2 + bx + c\), evaluate \(\frac{1}{a\alpha + b} + \frac{1}{a\beta + b}\).
Q48. Case Study: Roller Coaster Underpass
The highway underpass or roller coaster tracks are modeled mathematically using quadratic parabolas. A roller coaster track is represented by a quadratic polynomial.
Based on the underpass path shown in the graph, answer the following questions:
- Find the number of zeroes of the polynomial represented by the roller coaster path in the graph. (1 Mark)
- If the track's parabolic path is represented by \(p(x) = x^2 - 2x - 3\), find its actual zeroes. (1 Mark)
- If one zero of a parabolic track represented by \(x^2 + (k-1)x + 12\) is 4, find the value of \(k\). Also, determine the second zero. (2 Marks)
1. 2 zeroes.
2. Zeroes are 3 and -1.
3. k = -6 and second zero is 3.
Q49. Case Study: Basketball Shot Path
A basketball player throws a ball towards the basket. The path of the ball through the air is in the shape of a parabola represented by a quadratic polynomial \(h(t) = -t^2 + 2t + 8\), where \(h(t)\) is the height of the ball (in metres) at time \(t\) (in seconds).
Based on the above information, answer the following questions:
- What is the maximum height reached by the ball? (1 Mark)
- At what time \(t\) does the basketball hit the ground? (2 Marks)
- Express the polynomial \(h(t) = -t^2 + 2t + 8\) in factorized form. (1 Mark)
1. Maximum height is 9 metres.
2. Ball hits the ground at t = 4 seconds.
3. \(h(t) = -(t - 4)(t + 2)\).
Q50. Case Study: Soccer Kick trajectory
During a soccer match, a goalkeeper kicks a soccer ball from the ground. The height \(y\) (in yards) of the soccer ball at any horizontal distance \(x\) (in yards) is given by the quadratic equation \(y = -\frac{1}{40}x^2 + x\).
Based on the soccer ball trajectory, answer the following:
- Find the horizontal distance covered by the ball when it hits the ground. (2 Marks)
- What is the maximum height attained by the ball? (2 Marks)
1. Horizontal distance = 40 yards.
2. Maximum height = 10 yards.
Q51. Case Study: Arch Bridge Design
An architectural firm designs an arch bridge over a river. The structural arch of the bridge represents a parabola and is modeled by the quadratic polynomial \(p(x) = -x^2 + 6x - 5\), where \(x\) is the horizontal distance (in decameters) from the left pillar.
Based on the arch bridge design, answer the following questions:
- Find the zeroes of the polynomial \(p(x) = -x^2 + 6x - 5\). What do these zeroes represent physically in the bridge design? (2 Marks)
- What is the height of the arch at a horizontal distance of 3 decameters from the left pillar? Is this the maximum height of the arch? Explain. (2 Marks)
1. Zeroes are 1 and 5. They represent the positions (in decameters) of the two pillars/ends of the arch on the ground.
2. Height at x = 3 decameters is 4 decameters. Yes, this is the max height because the vertex of the parabola is at x = 3.