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NCERT SOLUTION

CLASS XTH: REAL NUMBERS

Q1:   Use Euclid’s division algorithm to find the HCF of

(I)               135 and 225  (ii) 196 and 38220  (iii) 867, 255

Solution

To start the solution, first, we write the Euclid division lemma

Let a, b, q, and r be integers such that a = bq + r.

Where 0 < a, a is the dividend, b is the quotient, and r is the remainder. if the remainder r = 0, then HCF = b.

For the HCF of 135 and 225

225 = 135x1 + 90

135 = 90x1 + 45

90 = 45x2 + 0. Here, the remainder is 0; hence, we can consider 45 as an HCF

HCF = 45

(ii) 38220 is greater than 196, so divide 38220 by 196 and note down the quotient and remainder, a = 38220, b = 196, r = 0

We can write by the Euclid division lemma

38220 = 196x195 + 0

Hence HCF = 196

(iii) 867 and 255

In this question, 867 is greater than 255, so we divide 855 by 255 and note the remainder and the quotient

867 =  255x3+102

Here, remainder is not zero

Again 255 = 102x2 + 51.

102 = 2x51 + 0

Here, HCF is 251. 

Chapter 2 Polynomials 

Polynomials are expressions containing more than one term.

Types

Monomial: The expression containing only one term,  ex. P(x) = 2x

Binomial: The expression containing two terms

 Ex: 2x+3y has two terms

Trinomial: An expression  containing three terms 

P(x) = x+2y+3z 

Zeros of the polynomial

 The value of the variable that makes the polynomial zero is said to be za ero of the polynomial

ex: (x+2)

If we put x+2 = 0 or x = -2, it will be zero

 to get such a type of answer, we write P(x) =0 

and solve for x, or try  to find the value of x

In Figure 1, the curve does not touch the x-axis, so it has no real root. 

In figure (ii), we can see the curve crosses the x-axis at time, so it has one zero

In 3, the curve touches the x-axis only once, so it has one zero

In fig. V, the graph touches the axis 3 times, so the number of solutions = =3

In Figure6, the curve touches or crosses the x-axis 6 times, so the number of solutions = 6

Exercise 2.1 

Q1. To find the zeros first, see the graph 

(i)  The first graph does not touch the X axis; it is parallel to the X axis, so this function has no solution.

(ii) Graph cuts the x-axis time, so it has only one zero

(iii) The third graph cuts the X axis three times, so the number of zeros is =3 

(iv) This graph cuts the x-axis twice, so the number of zeros = 2

(v) This cuts the x-axis three times, so the number of zeros = 3