Question Bank — Polynomials

NCERT Class 10 Mathematics · Chapter 2 · CBSE Board Pattern

Class 10 Mathematics Chapter 2 Question Bank CBSE Board Pattern

QUESTION BANK

Class 10 Mathematics – Chapter: Polynomials (NCERT)

Section A — MCQs (10 questions, 1 mark each)

  1. The degree of the polynomial \(7u^6 - 4u^3 + 2u^2 + 8u\) is
    (a) 3 (b) 4 (c) 6 (d) 8

  2. Which of the following is a quadratic polynomial?
    (a) \(2x + 5 - x^2\) (b) \(3x^3 - 2x^2 + x - 1\) (c) \(2y + 3\) (d) \(x^3 + 1\)

  3. If \(p(x) = x^2 - 3x - 4\), then the value of \(p(-1)\) is
    (a) 0 (b) –6 (c) 4 (d) –4

  4. The graph of \(y = ax^2 + bx + c\) (where \(a \neq 0\)) is a parabola. The number of zeroes of the polynomial is equal to the number of points where the parabola
    (a) intersects the y-axis (b) intersects the x-axis (c) touches the vertex (d) intersects itself

  5. A linear polynomial \(ax + b\) (\(a \neq 0\)) has
    (a) no zero (b) exactly one zero (c) exactly two zeroes (d) at most two zeroes

  6. For the quadratic polynomial \(x^2 - 3x - 4\), the zeroes are the x-coordinates of the points where its graph
    (a) intersects the y-axis (b) intersects the x-axis (c) attains maximum value (d) attains minimum value

  7. Assertion (A): A polynomial of degree \(n\) can have at most \(n\) zeroes.
    Reason (R): The graph of \(y = p(x)\) intersects the x-axis at most at \(n\) points.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

  8. Assertion (A): The quadratic polynomial \(x^2 + 7x + 10\) has zeroes –2 and –5.
    Reason (R): Sum of zeroes = –(coefficient of \(x\))/coefficient of \(x^2\).
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

  9. If \(\alpha\) and \(\beta\) are zeroes of \(ax^2 + bx + c\) (\(a \neq 0\)), then \(\alpha\beta =\)
    (a) \(b/a\) (b) \(-b/a\) (c) \(c/a\) (d) \(-c/a\)

  10. The polynomial \(x^3\) has how many zeroes?
    (a) 0 (b) 1 (c) 2 (d) 3

Section B — Very Short Answer (6 questions, 2 marks each)

  1. Find the zeroes of the linear polynomial \(2x + 3\).

  2. State the geometrical meaning of the zeroes of a quadratic polynomial.

  3. If \(\alpha\) and \(\beta\) are zeroes of \(x^2 - 2x - 8\), find the value of \(\alpha + \beta\) and \(\alpha\beta\).

  4. Write the general form of a cubic polynomial.

  5. How many zeroes can a cubic polynomial have? Give one example from the chapter.

  6. Verify that the zero of the linear polynomial \(ax + b\) (\(a \neq 0\)) is \(-b/a\).

Section C — Short Answer (5 questions, 3 marks each)

  1. Find the zeroes of \(x^2 + 7x + 10\) and verify the relationship between the zeroes and the coefficients.

  2. Find the zeroes of \(x^2 - 3\) and verify the relationship between the zeroes and the coefficients.

  3. Find a quadratic polynomial whose sum and product of zeroes are –3 and 2 respectively.

  4. Find the zeroes of \(2x^2 - 8x + 6\) by factorisation and verify the sum and product of zeroes.

  5. From the graphs given in the chapter (Fig. 2.9), state the number of zeroes of each polynomial and justify.

Section D — Long Answer (3 questions, 5 marks each)

  1. Draw the graph of \(y = x^2 - 3x - 4\) (using values from the chapter) and explain the geometrical meaning of its zeroes. Also state the three possible cases for the number of zeroes of any quadratic polynomial \(ax^2 + bx + c\) (\(a \neq 0\)) with suitable diagrams described in words.

  2. Verify that 3, –1 and \(-\frac{1}{3}\) are the zeroes of the cubic polynomial \(3x^3 - 5x^2 - 11x - 3\). Hence verify the relationships between the zeroes and the coefficients.

  3. Find the zeroes of the quadratic polynomial \(3x^2 + 5x - 2\) by splitting the middle term. Verify the sum and product of zeroes. Also state how many zeroes a quadratic polynomial can have at most.

Section E — Case/Source-Based (2 questions, 4 marks each)

Case 1: In a playground, the path of a ball thrown upwards follows the curve \(y = x^2 - 3x - 4\) (where \(x\) is horizontal distance in metres and \(y\) is height in metres).
(i) At what points does the path intersect the ground (x-axis)?
(ii) How many times does the path touch the ground?
(iii) What is the relationship between the zeroes and the coefficients of this quadratic?
(iv) If the same path is represented by \(k(x^2 - 3x - 4)\), do the zeroes change? Justify.

Case 2: A rectangular garden has length and breadth related by the quadratic expression \(x^2 - 2x - 8\) (area in square metres when one side is \(x\) metres).
(i) Factorise the expression to find possible dimensions.
(ii) Find the zeroes of the expression.
(iii) Verify the sum and product of zeroes with coefficients.
(iv) What does the number of zeroes tell about the possible dimensions of the garden?

▶ Answer Key Attempt all questions first,
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Section A

  1. (c) 6
  2. (a) \(2x + 5 - x^2\)
  3. (a) 0
  4. (b) intersects the x-axis
  5. (b) exactly one zero
  6. (b) intersects the x-axis
  7. (a)
  8. (a)
  9. (c) \(c/a\)
  10. (b) 1

Section B

  1. \(2k + 3 = 0 \implies k = -3/2\) (1 mark for equation, 1 mark for value).
  2. The zeroes are the x-coordinates of the points where the graph of \(y = p(x)\) intersects the x-axis.
  3. \(\alpha + \beta = 2\), \(\alpha\beta = -8\) (1 mark each).
  4. \(ax^3 + bx^2 + cx + d\) (\(a \neq 0\)).
  5. At most three zeroes. Example: \(x^3 - 4x\) has three zeroes (–2, 0, 2).
  6. Set \(p(k) = 0 \implies ak + b = 0 \implies k = -b/a\) (correct substitution 1 mark, simplification 1 mark).

Section C

  1. \((x + 2)(x + 5) = 0 \implies x = -2, -5\). Sum = –7, product = 10 (matches –b/a and c/a). (Factorisation 1 mark, zeroes 1 mark, verification 1 mark).
  2. \(x = \sqrt{3}, -\sqrt{3}\). Sum = 0, product = –3 (matches coefficients).
  3. Let polynomial be \(x^2 + bx + c\). Then \(b = 3\), \(c = 2\). Required polynomial: \(x^2 + 3x + 2\).
  4. \(2(x - 1)(x - 3) = 0 \implies x = 1, 3\). Sum = 4, product = 3 (matches coefficients).
  5. (i) 1 (ii) 2 (iii) 3 (iv) 1 (v) 1 (vi) 4 (each correct count with reason from graph ½ mark).

Section D

  1. Table values and graph drawn; intersects x-axis at x = –1 and x = 4. Three cases: two distinct zeroes, one zero (repeated), no zero (parabola above/below x-axis). (Graph & explanation 2 marks, three cases with justification 3 marks).
  2. \(p(3) = 0\), \(p(-1) = 0\), \(p(-1/3) = 0\). Sum of zeroes = 5/3 = –b/a; sum of products two at a time = –11/3 = c/a; product = 1 = –d/a. (Verification of zeroes 2 marks, three relations 3 marks).
  3. \(3x^2 + 5x - 2 = (3x - 1)(x + 2)\). Zeroes: \(1/3, -2\). Sum = –5/3, product = –2/3 (matches). Quadratic has at most two zeroes.

Section E

Case 1

(i) x = –1 and x = 4 (roots of equation).
(ii) Twice.
(iii) Sum of zeroes = 3, product = –4.
(iv) No, zeroes remain the same (k cancels).

Case 2

(i) \((x - 4)(x + 2)\).
(ii) 4 and –2.
(iii) Sum = 2, product = –8 (matches).
(iv) Two possible pairs of dimensions.

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All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.