Class 10 Maths Chapter 2: Polynomials worksheet with answer key
A 20-mark worksheet on Polynomials for about 40 minutes of class or homework: 5 multiple-choice, 3 very short answer and 3 short answer questions, written from the NCERT chapter and checked by a second reviewer. Print the PDF for the class; the answer key carries the marking scheme. Teachers may photocopy it freely.
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Answer: (a) −45
The zero of ax + b is −ba = −45. Check: 5(−45) + 4 = 0.
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Answer: (b) 4
The highest power of y that appears is y4, so the degree is 4. The terms need not be in order.
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Answer: (c) 21
p(−2) = 2(−2)2 − 5(−2) + 3 = 8 + 10 + 3 = 21. Taking (−2)2 as −4 or −5(−2) as −10 gives the distractors.
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Answer: (b) opens downwards
For y = ax2 + bx + c the parabola opens downwards when a < 0. Here a = −3.
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Answer: (a) a ≠ 0
Only the coefficient of the highest power must be non-zero; if a = 0 the degree drops below 3. The other coefficients may be zero (x3 − 8 is a cubic).
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p(−2) = 0: 12 − 2k − 10 = 0, so k = 1. The polynomial is 3x2 + x − 10; product of the zeroes = −103, so the other zero is −103 ÷ (−2) = 53. (3x2 + x − 10 = (x + 2)(3x − 5).)
- p(−2) = 0 gives k = 1 (1)
- other zero 53 (1)
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x2 − (sum)x + (product) = x2 + 32x − 1. Multiplying by 2 (any k ≠ 0 keeps the zeroes): 2x2 + 3x − 2. Check: −32 and −22 = −1.
- writes x2 + 32x − 1 (1)
- clears the fraction: 2x2 + 3x − 2 (or any k times it) (1)
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(a) The curve meets the x-axis at three points, so p(x) has 3 zeroes. (b) They are the x-coordinates of those points: −2, 1 and 3. (c) A polynomial of degree n has at most n zeroes, so with 3 zeroes the degree is at least 3.
- 3 zeroes, namely −2, 1 and 3 (1)
- least degree 3, since degree n allows at most n zeroes (1)
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α + β = 5, αβ = 2. New sum = 1α + 1β = α + βαβ = 52. New product = 1αβ = 12. Polynomial: x2 − 52x + 12, or, multiplying by 2, 2x2 − 5x + 1.
- α + β = 5 and αβ = 2 (1)
- new sum 52 and new product 12 (1)
- 2x2 − 5x + 1 (or any k times it) (1)
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4x2 + 20x + 25 = (2x + 5)2, so the zeroes are −52 and −52. Sum = −5 = −204; product = 254 = ca, both copies used. The graph touches the x-axis at one point, (−52, 0): two coincident points.
- (2x + 5)2, zeroes −52, −52 (1)
- sum −5 and product 254 verified with both copies (1)
- graph meets the axis at one point (1)
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α + β = 3, αβ = 32. (a) α2 + β2 = (α + β)2 − 2αβ = 9 − 3 = 6. (b) αβ + βα = α2 + β2αβ = 6 ÷ 32 = 4.
- α + β = 3 and αβ = 32 (1)
- α2 + β2 = 6 (1)
- αβ + βα = 4 (1)
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