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NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations : NATURE OF ROOTS

This content is part of NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

NATURE OF ROOTS :

Consider the quadratic equation, ax2 + bx + c = 0 having as its roots and b2 - 4ac is called discriminate of roots of quadratic equation. It is denoted by D or D.

               Roots of the given quadratic equation may be

               (i) Real and unequal         (ii) Real and equal             (iii) Imaginary and unequal.

               Let the roots of the quadratic equation ax2 + bx + c = 0 (where ) be and then

                                        .....(i)

               and                  ....(ii)

The nature of roots depends upon the value of expression ‘b2 - 4ac’ with in the square root sign. This is known as discriminate of the given quadratic equation.

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NCERT Solutions for Class 10 Maths Chapter 2 GRAPH OF POLYNOMIALS

This content is part of NCERT Solutions for Class 10 Maths Chapter 2 Polynomials

GRAPH OF POLYNOMIALS :

In algebraic or in set theoretic language the graph of a polynomial f(x) is the collection (or set) of all points (x, y), where y = f(x). In geometrical or in graphical language the graph of a polynomial f(x) is a smooth free hand curve passing through points x1, y1), (x2, y2), (x3, y3), ..... etc. where y1, y2, y3,.... are the values of the polynomial f(x) at x1, x2, x3,.... respectively.

               In order to draw the graph of a polynomial f(x), follow the following algorithm.

                           ALGORITHM :

Step (i) Find the values y1, y2, ...... ynof polynomial f(x) on different points x1, x2, ....... xn………… and prepare a table that gives values of y or f(x) for various values of x.

               x:

x1            x2     ……               xn            xn+1         ….

y = f(x)

y1=f(x1)  y2=f(x2)  ….          Yn=f(xn) yn+1 = f(xn+1)         …..

Step (ii) Plot that points (x1, y1), (x2, y2), (x3, y3),.....(xn, yn).... on rectangular co-ordinate system. In plotting these points use different scales on the X and Y axes.

Step (iii) Draw a free hand smooth curve passing through points plotted in step 2 to get the graph of the polynomial f(x).

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No. 232

A new geometric design every day

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Management of Natural Resources

This content is part of NCERT Solutions for Class 10 Science Chapter 16 Management of Natural Resources

Question 1. What changes can you make in your habits to become more environment-friendly? Answer: We should use the formula of three R’s in our home in order to be environment friendly.

Reduce, i.e. to use less, and save the water and the electricity by stopping their wastage.

Recycle, i.e. to segregate the waste so that materials which can be recycled should be dumped in a place for recycling.

Reuse, i.e. to reuse certain materials like bottles Of jams and pickles for storing other kitchen items, etc.

Question 2. What would be the advantages of exploiting resources with short-term aims? Answer: There would be no advantage of exploiting resources with short-term aims. Their exploitation may appear to be advantageous in the short-term but it is highly disadvantageous in the long- term. By doing so, we may be able to enjoy the comforts Of life but we would damage our environment gradually.