Class 10 Zeroes and Coefficients — practice questions with answers

Algebra — Polynomials · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.

1. For the quadratic polynomial x² - 5x + 6, what is the sum of its zeroes?

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Answer: A) 5

Sum of zeroes =

- \frac{b}{a}

- \frac{- 5}{1}

= 5.

2. For the cubic 2x³ - 5x² - 14x + 8, the product of the zeroes is:

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Answer: A) -4

Product =

- \frac{d}{a}

- \frac{8}{2}

= -4.

3. If alpha and beta are the zeroes of x² - 6x + k and alpha² + beta² = 40, find the value of k.

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Answer: C) -2

alpha² + beta² = (sum)² - 2(product). 40 = 36 - 2k, so 2k = 36 - 40 = -4, k = -2.

4. If α, β and γ are the zeroes of 2x³ + x² − 5x + 2, the value of 1/α + 1/β + 1/γ is:

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Answer: A)

\frac{5}{2}

Put the fractions over a common denominator:

\frac{1}{α} + \frac{1}{β} + \frac{1}{γ} = \frac{α β + β γ + γ α}{α β γ}

. Here

α β + β γ + γ α = \frac{c}{a} = - \frac{5}{2}

and

α β γ = - \frac{d}{a} = - \frac{2}{2} = - 1

, so the value is

\frac{- 5/2}{- 1} = \frac{5}{2}

- \frac{5}{2}

misses the minus sign in

- \frac{d}{a}

;

\frac{1}{2}

puts the sum

α + β + γ = - \frac{1}{2}

on top instead of the pairwise sum;

\frac{2}{5}

turns the fraction upside down.

5. The zeroes α and β of x² − 20x + 64 are both positive. What is the value of √α + √β?

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Answer: D) 6

Square the expression first:

(α + β)^{2} = α + β + 2 α β = 20 + 264 = 20 + 16 = 36

. Since both zeroes are positive,

α + β = 36 = 6

. (Check with the zeroes 16 and 4:

4 + 2 = 6

.) 36 forgets the final square root; 20 uses only

α + β

;

20

omits the

2 α β

cross term.

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