Exercise - 2.1 (Mathematics NCERT Class 10th)
Q.1 The graphs of y = p(x) are given in figures below for some polynomials p(x). Find the number of zeroes of p(x) , in each case.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Sol.
(i) There are no zeroes as the graph does not intersect the x-axis.
(ii) The number of zeroes is one as the graph intersects the x-axis at one point only.
(iii) The number of zeroes is three as the graph intersects the x-axis at three points.
(iv) The number of zeroes is two as the graph intersects the x-axis at two points.
(v) The number of zeroes is four as the graph intersects the x-axis at four points.
(vi) The number of zeroes is three as the graph intersects the x-axis at three points.
Polynomials : Exercise - 2.2 (Mathematics NCERT Class 10th)
Q.1 Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients :
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Sol. (i) We have,
The value of
when x + 2 = 0 or x – 4 = 0 , i.e., when x = – 2 or x = 4.
So, The zeroes of
Therefore , sum of the zeroes = (– 2) + 4 = 2
and product of zeroes = (– 2) (4) = – 8
(ii) We have,
The value of
(2s – 1) (2s – 1) is zero, i.e., when 2s – 1 = 0 or 2s – 1 = 0,
i.e., when
So, The zeroes of
Therefore, sum of the zeroes
and product of zeroes
(iii) We have,
The value of
So, The zeroes of
Therefore, sum of the zeroes
and product of zeroes
(iv) We have,
The value of
So, The zeroes of
Therefore, sum of the zeroes = 0 + (– 2) = – 2
and , product of zeroes = (0) (–2) = 0
(v) We have
The value of
i.e., when
So, The zeroes of
Therefore , sum of the zeroes =
and, product of the zeroes =
(vi) We have,
The value of
So, The zeroes of
Therefore , sum of the zeroes
and, product of the zeroes
Q.2 Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i)
(ii)
(iii)
(iv) 1, 1
(v)
(vi) 4, 1
Sol. (i) Let the polynomial be
and,
If a = 4, then b = – 1 and c = – 4.
Therefore, one quadratic polynomial which fits the given conditions is
(ii) Let the polynomial be
and
If a = 3, then b
So, One quadratic polynomial which fits the given conditions is
(iii) Let the polynomial be
and
If a = 1, then b = 0 and c =
So, one quadratic polynomial which fits the given conditions is
(iv) Let the polynomial be
and
If a = 1, then b = – 1 and c = 1.
So, one quadratic polynomial which fits the given conditions is