CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

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CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)
Exercise 4.4

CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

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2. Find the values of $ k $ for each of the following quadratic equations, so that they have two equal roots.

(i) $ 2 x^{2}+k x+3=0 $

Answer

We know that if an equation $ a x^{2}+b x+c=0 $ has two equal roots,
its discriminant
$ \left(b^{2}-4 a c\right) $ will be 0.
$ 2 x^{2}+k x+3=0 $
Comparing equation with $ a x^{2}+b x+\mathrm{c}=0 $, we obtain, $ a=2, b=k, c=3 $
Discriminate $ =b^{2}-4 a c=(k)^{2}-4(2)(3)=k^{2}-24 $
For equal roots,
Discriminant $ =0 $
$ k^{2}-24=0 $
$ k^{2}=24 $
$ =k= \pm \sqrt{24}= \pm 2 \sqrt{6} $

(ii) $ k x(x-2)+6=0 $

Answer

We know that if an equation $ a x^{2}+b x+c=0 $ has two equal roots,
its discriminant
$ \left(b^{2}-4 a c\right) $ will be 0.
$ k x(x-2)+6=0 $
or $ k x^{2}-2 k x+6=0 $
Comparing this equation with $ a x^{2}+b x+c=0 $, we obtain, $ a=k, b=-2 k, c=6 $
Discriminant $ =b^{2}-4 a c=(-2 k)^{2}-4(k)(6)=4 k^{2}-24 k $
For equal roots, $ b^{2}-4 a c=0 $
$ =4 k^{2}-24 k=0 $
$ =4 k(k-6)=0 $
Either $ 4 k=0 $ or $ k=6 $
$ =k=0 $ or $ k=6 $
However, if $ k=0 $, then the equation will not have the terms ' $ x^{2} $ ' and ' $ x $ '.
Therefore, if this equation has two equal roots, $ k $ should be 6 only.

CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

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3. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is $ 800 \mathrm{~m}^{2} $ ? If so, find its length and breadth.

Answer

Let the breadth of mango grove be $ l $.
Length of mango grove will be $ 2 l $.
Area of mango grove $ =(2 l)(l)=2 l^{2} $
$2l^{2}=800$
$\Rightarrow l^{2}-400=0$
$\Rightarrow l^{2}=400$
$l= \pm 20$
However, length cannot be negative.
Therefore, breadth of mango grove $ =20 \mathrm{~m} $
Length of mango grove $ =2 \times 20=40 \mathrm{~m} $

CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

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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.

Answer

Let the age of one friend be $ x $ years. Age of the other friend will be $ (20-x) $ years.
4 years ago,
age of $ 1^{\text {st }} $ friend $ =(x-4) $ years
And, age of $ 2^{\text {nd }} $ friend $ =(20-x-4)=(16-x) $ years
Given that,
$(x-4)(16-x)=48$
$16 x-64-x^{2}+4 x=48$
$x^{2}-20 x+112=0$
Comparing this equation with $ a x^{2}+b x+c=0 $, we obtain $a=1, b=-20, c=112$
$\text { Discriminant }=b^{2}-4 a c=(-20)^{2}-4(1)(112)=400-448=-48$
As $ b^{2}-4 a c < 0 $,
Therefore, no real root is possible for this equation and hence, this situation is not possible.

5. Is it possible to design a rectangular park of perimeter 80 m and area $ 400 \mathrm{~m}^{2} $ ? If so, find its length and breadth.

Answer

Let the length and breadth of the park be $ l $ and $ b $.
Perimeter $ =2(l+b)=80 $
$ l+b=40 $ Or, $ b=40-l $
Area $ =l \times b=l(40-l) $
$ =40 l-l^{2}=400 $ Given
$ l^{2}-40 l+400=0 $
Comparing this equation with $ a l^{2}+b l+\mathrm{c}=0 $, we obtain $ a=1, b=-40, c=400 $
Discriminant $ \mathrm{D}=b^{2}-4 a c=(-40)^{2}-4(1)(400)=1600-1600=0 $
As $ b^{2}-4 a c=0 $,
Therefore, this equation has equal real roots and hence, this situation is possible.
Root of this equation,
$ l=-\frac{b}{2 a} $
$ l=-\frac{(-40)}{2(1)}=\frac{40}{2} $
Therefore, length of park, $ 1=20 \mathrm{~m} $
And breadth of park, $ b=40-1=40-20=20 \mathrm{~m} $

CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

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CBSE Class 10 Maths Chapter 4 Quadratic Equations Ex 4.4 || NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations (English Medium)

Download the Math Ninja App Now