Class ten maths Quadratic Equations NCERT solution
Chapter 4: Quadratic Equations
TEXTBOOK’S EXERCISE 4.4
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
Sol. (i) Given equation is
This is of the form
Where, a=2, b=-3,c=5
Discriminant = b2 – 4ac
= (–3)2 – 4 × 2 × 5
= 9 – 40 = –31 < 0
Hence, given the quadratic equation has no real root.
=48 −48 = 0
Thus, given equation have real and equal roots, which are given by
Hence, the roots of the given quadratic equation are
(iii) Given equation is :
This is of the form
Where, a= 2, b= -6, c=3
Thus, the given equation has real and distinct roots.
So, the roots are
Hence, the roots of the given quadratic equation are
2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2×2 + kx + 3 = 0
(ii) kx(x – 2) + 6 = 0
Sol. (i) Given equation is
2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2×2 + kx + 3 = 0
(ii) kx(x – 2) + 6 = 0
Sol. (i) Given equation is
This is of the form
Where, a = 2, b= k, c = 3
As, roots of the given quadratic equation are equal,
(ii) Given equation is
This is of the form
Where,
As roots of the given quadratic equation are equal.
3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2 ? If so, find its length and breadth.
Sol. Let the breadth of rectangular grove = x m
Then, the length of the rectangular grove = 2x m
Area of rectangular grove = length × breadth
As per condition :
= 400
∴ The breadth of rectangular grove = 20 m
Length of rectangular grove = 40 m
So, it is possible to design such a mango grove.
4. 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.
Sol. Let the age of first friend = x years. Then, the age of the second friend = (20 – x) years.
Four years ago :
Age of the first friend
Age of the second friend
As per condition :
This is of the form
So, roots are not real. Then no real value of x satisfies the quadratic equation (i).
Hence, a given situation is not possible.
5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Sol. Let length of rectangular park = x m
Breadth of rectangular park = y m
∴ Perimeter of rectangular park = 2(x + y) m
Area of rectangular park = xy m2
As per condition: 2(x + y) = 80
As per condition: xy = 400 …(ii)
Putting the value of y in (ii), we have
x(40 – x) = 400
When, x = 20 then from (i), y = 40 – 20 = 20
∴ Length and breadth of the rectangular park is each equal to 20 m. Therefore given rectangular park can exist if it is in the shape of a square.