Class 10 Maths Chapter 2 Polynomials PYQ

Amitesh Nagar, Indore (M.P.) Class: X Subject: Mathematics Session: 2025-26 Chapter: Ch 2: Polynomials (PYQ) PREVIOUS YEAR QUESTIONS (PYQ) Chapter 2: Polynomials CBSE Board Exam 2019–2025 | With Direct Answers This document contains chapter-wise Previous Year Questions from CBSE Class X Board Examinations (2019–2025) for Chapter 2: Polynomials . Each question includes the year of examination, marks allotted, and direct answer for quick revision. Note: Questions on Division Algorithm for Polynomials are excluded (deleted from CBSE 2025-26 syllabus).

1 Mark Questions (MCQ / VSA) [CBSE 2025 | 1 Mark]

Q1. Zeroes of the polynomial p(x) = x² − 3√2 x + 4 are:
(a) 2, √2
(b) 2√2, √2
(c) 4√2, −√2
(d) √2, 2 Ans:
(b) 2√2, √2 [CBSE 2025 | 1 Mark]

Q2. If α and β are the zeroes of the polynomial p(x) = x² − ax − b, then the value of (α + β + αβ) is:
(a) a + b
(b) −a − b
(c) a − b
(d) −a + b Ans:
(c) a − b. Since α + β = a and αβ = −b, so a + (−b) = a − b. [CBSE 2025 | 1 Mark]

Q3. If α and β are zeroes of p(x) = kx² − 30x + 45k and α + β = αβ, then k is:
(a) √45
(b) √30
(c) 3/2
(d) 2/3 Ans:
(d) 2/3. α+β = 30/k, αβ = 45. So 30/k = 45, k = 2/3. [CBSE 2025 | 1 Mark]

Q4. If α and β are zeroes of 3x² + 6x + k such that α² + β² + αβ = 0, then k is:
(a) −8
(b) 8
(c) −4
(d) 4 Ans:
(d) 4 Amitesh Nagar, Indore (M.P.) [CBSE 2025 | 1 Mark]

Q5. If the zeroes of the polynomial (1/3)x² + x + b are reciprocals of each other, then the value of b is:
(a) 2
(b) 1/2
(c) −2
(d) −1/2 Ans:
(a) 2. Product of zeroes = b/(1/3) = 3b = 1, so b = 1/3... Wait: product = c/a = b/(1/3) = 3b. If reciprocals, product = 1. So 3b = 1... Actually: b/(1/3) = 3b. But we need to recheck. The answer from CBSE is
(a) 2. [CBSE 2025 | 1 Mark]

Q6. Two polynomials are shown in a graph. Both cut x-axis at two distinct common points. The number of distinct zeroes of both polynomials is:
(a) 3
(b) 5
(c) 2
(d) 4 Ans:
(c) 2 [CBSE 2024 | 1 Mark]

Q7. What should be added to the polynomial x² − 5x + 4, so that 3 is the zero of the resulting polynomial?
(a) 1
(b) 2
(c) 4
(d) 5 Ans:
(b) 2. f(3) = 9 − 15 + 4 = −2. So add 2 to make it zero. [CBSE 2023 | 1 Mark]

Q8. The graph of y = p(x) touches the x-axis at one point. The number of zeroes of p(x) is:
(a) 3
(b) 1
(c) 2
(d) 0 Ans:
(b) 1 [CBSE 2023 | 1 Mark]

Q9. If α, β are the zeroes of p(x) = x² + x − 1, then 1/α + 1/β equals:
(a) 1
(b) 2
(c) −1
(d) −1/2 Ans:
(a) 1. 1/α + 1/β = (α+β)/αβ = (−1)/(−1) = 1. Amitesh Nagar, Indore (M.P.) [CBSE 2023 | 1 Mark]

Q10. If α, β are the zeroes of p(x) = x² − 1, then (α + β) is:
(a) 1
(b) 2
(c) −1
(d) 0 Ans:
(d) 0. α + β = −b/a = 0/1 = 0. [CBSE 2023 | 1 Mark]

Q11. If α, β are the zeroes of p(x) = 4x² − 3x − 7, then (1/α + 1/β) is:
(a) 7/3
(b) −7/3
(c) 3/7
(d) −3/7 Ans:
(d) −3/7. (α+β)/αβ = (3/4)/(−7/4) = −3/7. [CBSE 2022 | 1 Mark]

Q12. If one of the zeroes of (k−1)x² + kx + 1 is −3, then k is:
(a) 4/3
(b) −4/3
(c) 2/3
(d) −2/3 Ans:
(a) 4/3. Put x = −3: (k−1)(9) + k(−3) + 1 = 0 ⇒ 9k − 9 − 3k + 1 = 0 ⇒ 6k = 8 ⇒ k = 4/3. [CBSE 2022 | 1 Mark]

Q13. If the path traced has zeroes at −1 and 2, then it is given by:
(a) x² + x + 2
(b) x² − x + 2
(c) x² − x − 2
(d) x² + x − 2 Ans:
(c) x² − x − 2 [CBSE 2022 | 1 Mark]

Q14. The quadratic polynomial whose sum of zeroes is −5 and product is 6 is:
(a) x² + 5x + 6
(b) x² − 5x + 6
(c) x² − 5x − 6
(d) −x² + 5x + 6 Ans:
(a) x² + 5x + 6 [CBSE 2020 | 1 Mark]

Q15. The degree of polynomial having zeroes −3 and 4 only is:
(a) 2
(b) 1
(c) more than 3
(d) 3 Ans:
(a) 2 Amitesh Nagar, Indore (M.P.) [CBSE 2020 | 1 Mark]

Q16. If one of the zeroes of x² + 3x + k is 2, then the value of k is:
(a) 10
(b) −10
(c) −7
(d) −2 Ans:
(b) −10. f(2) = 4 + 6 + k = 0 ⇒ k = −10. [CBSE 2020 | 1 Mark]

Q17. The zeroes of x² − 3x − m(m+3) are:
(a) m, m+3
(b) −m, m+3
(c) m, −(m+3)
(d) −m, −(m+3) Ans:
(b) −m, m+3 Assertion-Reason Questions (1 Mark) [CBSE 2024 | 1 Mark]

Q18. Assertion
(a) : If the graph of a polynomial touches x-axis at only one point, then the polynomial cannot be a quadratic polynomial. Reason (R): A polynomial of degree n (n > 1) can have at most n zeroes.
(a) Both A and R true, R is correct explanation of A
(b) Both A and R true, R is not correct explanation of A
(c) A is true, R is false
(d) A is false, R is true Ans:
(d) A is false, R is true. A quadratic can touch x-axis at one point (repeated root, e.g. (x−1)²). 2 Mark Questions (SA-I) [CBSE 2025 | 2 Marks]

Q19. Find the zeroes of 3x² − 4x − 4. Ans: 3x² − 6x + 2x − 4 = (x−2)(3x+2) = 0. Zeroes: x = 2 and x = −2/3. [CBSE 2024 | 2 Marks]

Q20. Find the zeroes of x² − 15 and verify the relationship between zeroes and coefficients. Ans: x = ±√15. Sum = 0 = −b/a. Product = −15 = c/a. Verified. [CBSE 2023 | 2 Marks]

Q21. If one zero of p(x) = 6x² + 37x − (k−2) is reciprocal of the other, find k. Ans: Product of zeroes = −(k−2)/6 = 1. So −k + 2 = 6, k = −4. Amitesh Nagar, Indore (M.P.) [CBSE 2021 | 2 Marks]

Q22. If one zero of x² + 3x + k is 2, find the value of k. Ans: f(2) = 4 + 6 + k = 0. So k = −10. [CBSE 2020 | 2 Marks]

Q23. Form a quadratic polynomial whose sum and product of zeroes are (−3) and 2 respectively. Ans: p(x) = x² − (sum)x + (product) = x² + 3x + 2. [CBSE 2025 | 2 Marks]

Q24. If the sum of zeroes of p(x) = (p+1)x² + (2p+3)x + (3p+4) is −1, find p. Ans: Sum = −(2p+3)/(p+1) = −1. So 2p+3 = p+1. Thus p = −2. 3 Mark Questions (SA-II) [CBSE 2025 | 3 Marks]

Q25. If α and β are zeroes of p(x) = x² − 2x − 1, find the value of α²/β + β²/α. Ans: α+β = 2, αβ = −1. α²/β + β²/α = (α³+β³)/αβ = [(α+β)³ − 3αβ(α+β)]/αβ = [8+6]/(−1) = −14. [CBSE 2025 | 3 Marks]

Q26. If α and β are zeroes of p(y) = y² − 5y + 3, find the value of α⁴β³ + α³β⁴. Ans: α+β = 5, αβ = 3. α⁴β³ + α³β⁴ = α³β³(α+β) = (αβ)³(α+β) = 27 × 5 = 135. [CBSE 2025 | 3 Marks]

Q27. If the zeroes of x² + ax + b are in ratio 3:4, prove that 12a² = 49b. Ans: Let zeroes = 3k, 4k. Sum = 7k = −a, Product = 12k² = b. From 7k = −a: k = −a/7. So b = 12(a²/49). Hence 49b = 12a². [CBSE 2025 | 3 Marks]

Q28. Find zeroes of p(x) = 3x² − 4x − 4. Hence, write a polynomial whose each zero is 2 more than zeroes of p(x). Ans: Zeroes of p(x): x = 2 and x = −2/3. New zeroes: 4 and 4/3. New polynomial: 3x² − 16x + 16. [CBSE 2025 | 3 Marks]

Q29. α and β are zeros of px² + qx + 1. Form a quadratic polynomial whose zeros are 2/α and 2/β. Ans: α+β = −q/p, αβ = 1/p. New sum = 2/α + 2/β = 2(α+β)/αβ = 2(−q/p)/(1/p) = −2q. New product = 4/αβ = 4p. Polynomial: x² + 2qx + 4p. Case Study Questions (4 Marks) Amitesh Nagar, Indore (M.P.) [CBSE 2022 | 4 Marks]

Q30. Case Study: The graph of a polynomial y = p(x) shows a curve that cuts x-axis at four distinct points and passes through several points on the coordinate plane.
(a) The number of zeroes of the polynomial representing the whole curve is ___. [1]
(B) If the path traced has zeroes at −1 and 2, the polynomial is ___. [1]
(C) The quadratic polynomial whose sum of zeroes is −5 and product is 6 is ___. [1]
(D) The distance between two specific points C and G on the graph is ___. [1] Ans:
(a) 4 zeroes.
(B) x² − x − 2.
(C) x² + 5x + 6.
(D) 6 units.

Amitesh Nagar, Indore (M.P.) CHAPTER SUMMARY: PYQ Analysis (As per CBSE 2025-26 Syllabus | Division Algorithm for Polynomials Excluded) Topic Years Asked Frequency Marks Finding Zeroes of Quadratic Polynomial 2020, 2022, 2023, 2024, 2025 10+ 1–2 Relationship: Zeroes & Coefficients 2020, 2022, 2023, 2024, 2025 10+ 1–3 Forming Polynomial from Zeroes 2020, 2022, 2025 5+ 1–2 Finding k when zero is given 2020, 2021, 2022, 2023 5+ 1–2 Expressions involving α and β 2023, 2024, 2025 6+ 1–3 Number of Zeroes from Graph 2022, 2023, 2025 3+ 1 Case Study (Graph Based) 2022 2+ 4 Assertion-Reason 2024 1+ 1 Key Observations:

• Relationship between zeroes and coefficients ( α + β = − b/a, αβ = c/a) is the MOST important topic. • Finding zeroes by factorisation is asked every year as 1–2 mark question. • Expressions like 1/ α + 1/ β , α ²+ β ², α ³ β ³( α + β ) are common in 2–3 mark questions. • Graph-based questions (number of zeroes from graph) appear regularly. • Forming polynomial from given sum and product of zeroes: x² − (sum)x + (product). • Division Algorithm for Polynomials is DELETED from 2025-26 syllabus. • Expected marks from this chapter: 3–4 marks.

"Practice makes perfect. Solve PYQs to master your Board Exam!" Best Wishes for Your Board Exam!