This page focuses on the detailed Polynomials question answers for Class 10 Mathematics Polynomials, addressing the question: 'Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
'. The solution provides a thorough breakdown of the question, highlighting key concepts and approaches to arrive at the correct answer. This easy-to-understand explanation will help students develop better problem-solving skills, reinforcing their understanding of the chapter and aiding in exam preparation.

Question 2

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Answer

(i) Let α, β are the zeroes of the polynomial ax^{2}+ bx +c, therefore

Sum of zeroes (α + β) = 1/4 = -b/a

Product of zeroes (αβ) = -1 = c/a

On comparing,

a = 4, b = -1 and c = - 4

Hence, the required polynomial is 4x^{2} –x – 4

(ii)

(iii)

Let α, β are the zeroes of the polynomial ax^{2}+ bx +c, therefore

Sum of zeroes (α + β) = 0 = -b/a

Product of zeroes (αβ) = √5 = √5/1 = c/a

On comparing,

a = 1, b = 0 and c = √5

Hence, the required polynomial is x^{2} + √5

- Q:-
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

- Q:-
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x

^{2}– 2x – 8 (ii) 4s^{2}– 4s + 1 (iii) 6x^{2}– 3 – 7x (iv) 4u^{2}+ 8u (v) t^{2 }– 15 (vi) 3x^{2 }– x – 4 - Q:-
Find two consecutive positive integers, sum of whose squares is 365.

- Q:-
Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. - Q:-
Refer to Example 13. (i) Complete the following table:

(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability

Do you agree with this argument? Justify your answer.

- Q:-
Find two numbers whose sum is 27 and product is 182.

- Q:-
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

- Q:-
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x

^{2 }– 3x + 5 = 0 (iii) 2x^{2}– 6x + 3 = 0 - Q:-
Prove that 3 + 2√5 is irrational.

- Q:-
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines (ii) parallel lines (iii) coincident lines

- Q:-
A piggy bank contains hundred 50p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin (i) will be a 50 p coin ? (ii) will not be Rs 5 coin?

- Q:-
In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

- Q:-
Prove that the parallelogram circumscribing a circle is a rhombus.

- Q:-
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

- Q:-
Find the roots of the following quadratic equations by factorisation:

- Q:-
Find the roots of the following quadratic equations, if they exist, by the method of

completing the square:

(i) 2x^{2 }– 7x + 3 = 0 (ii) 2x^{2 }+ x – 4 = 0 (iv) 2x^{2}+ x + 4 = 0 - Q:-
Sum of the areas of two squares is 468 m

^{2}. If the difference of their perimeters is 24 m, ind the sides of the two squares. - Q:-
How many tangents can a circle have?

- Q:-
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

- Q:-
A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3.

Find the number of blue balls in the jar.

- NCERT Chapter