1 markMCQSection A

The graph of $y = f(x)$ is given.

The number of zeroes of f(x) is :

(a) 0 (b) 1 (c) 2 (d) 4

Q: The graph of $y = f(x)$ is given. The number of zeroes of f(x) is : (a) 0 (b) 1 (c) 2 (d) 4

(c) 2 — the graph meets the X-axis at 2 points, so $f(x)$ has 2 zeroes (confirm against the printed figure).

Q: What is the geometrical meaning of a zero of a polynomial?

A zero of f(x) is a value of x for which $f(x) = 0$; geometrically, it is the x-coordinate of the point where the graph of $y = f(x)$ intersects the X-axis.

Wavy curve graph of y = f(x) crossing the X-axis, with labeled X, X', Y, Y' axes and origin O. VERIFIER NOTE: exact inte — Fig. for Q4

Polynomials

Zeroes of a Polynomial from its Graph

Options

(A)0

(B)1

(D)4

Official Answer

zeroes of polynomialgraph of y=f(x)X-axis intersectiongeometrical meaning of zeroesroots of polynomial

Marking Scheme

11 mark for correctly reading the number of X-axis intersection points from the graph (verify against the actual printed figure).

Hint

Count only the points where the curve actually touches or crosses the X-axis — those x-values are the zeroes of $f(x)$ .

Quick Oral Answer

The zeroes of $f(x)$ are the x-values where its graph meets the X-axis; here the wavy curve meets the X-axis at 2 points, so $f(x)$ has 2 zeroes.

Analysis & Explanation

Tests reading the number of zeroes of a polynomial from its graph.

Concept

The zeroes of f(x) are exactly the x-coordinates where the graph of $y = f(x)$ crosses or touches the X-axis, since $y = 0$ there.

Key points

Count only points where the curve meets the X-axis, not turning points (local maxima/minima).
OCR caveat: the exact count must be confirmed from the printed figure; option (c) 2 corresponds to the standard two-crossing version of this well-known CBSE graph.

Common mistakes

Counting turning points of the curve instead of X-axis intersections.
Miscounting tangential touches as full crossings or vice versa.

Real-world/exam link

Reading zeroes graphically connects directly to factorising and solving polynomial equations algebraically.

Common Mistakes

1Counting the turning points (peaks/troughs) of the curve instead of the X-axis crossing points.
2Miscounting when the curve touches the axis tangentially versus crossing through it.
3Confusing the degree of the polynomial with the number of real zeroes visible on the graph.

Interesting Facts

This graph-based zero-counting question format has appeared repeatedly in CBSE Class 10 sample papers since the NCERT graphical approach to polynomials was introduced.

A quadratic polynomial's graph is always a parabola, which can intersect the X-axis at 0, 1 (tangent), or 2 points, corresponding to 0, 1, or 2 real zeroes.

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Frequently Asked Questions

What is the geometrical meaning of a zero of a polynomial?

A zero of f(x) is a value of x for which $f(x) = 0$ ; geometrically, it is the x-coordinate of the point where the graph of $y = f(x)$ intersects the X-axis.

Can the number of zeroes exceed the degree of the polynomial?

No, a polynomial of degree n can have at most n real zeroes, though the graph may show fewer if some roots are complex or repeated.