Solve the followings Questions.

1. Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

Answer:
We know that the quadratic equation ax²+bx+c = 0 has
(a) Two distinct real roots, if b²-4ac > 0,
(b) Two equal real roots, if b²-4ac = 0,
(c) No real roots, if b²-4ac < 0.

(i) 2x²-3x+5 = 0
Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax²+bx+c = 0 we get
a = 2, b = -3 and c = 5
Then, b²-4ac = (-3)²-4×2×5 = 9-40 = -31 < 0
Hence the given quadratic equation has no real roots.

(ii) 3x²-4√3x+4 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax²+bx+c = 0 we get
a = 3, b = -4√3 and c = 4
Then, b²-4ac = (-4√3)²-4×3×4 = 48-48 = 0
Therefore, real roots exist for the given equation and they are equal to each other.

And the roots will be

Therefore, the roots are 

(iii) 2x²-6x+3 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax2+bx+c = 0 we get
a = 2, b = -6 and c = 3
then, b²-4ac = (-6)²-4×2×3 = 36-24 = 12 > 0
Hence the given quadratic equation has two distinct real roots.

Applying quadratic formulato find roots,

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

(i) 2x²+kx+3 = 0
Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax²+bx+c = 0 we get
a = 2, b = k and c = 3
Now the given quadratic equation have two equal roots if
b²-4ac = 0

(k)2-4×2×3 = 0
k²-24 = 0
 k² = 24
k = ±√24
k = ± 2√6
Therefore, the required value of k is ± 2√6.

(ii)  kx(x-2)+6 = 0
Answer:
The given quadratic equation can be written as
kx²-2kx+6 = 0….(1)
Comparing the quadratic equation (1) with the general form of quadratic equation ax²+bx+c = 0 we get
a = k, b = -2k and c = 6
Now the given quadratic equation have two equal roots if
b²-4ac = 0
(-2k)²-4×k×6 = 0
4k²-24k = 0
4k(k-6) = 0
k(k-6) = 0
k = 0
Or, k-6 = 0
k = 6
If k = 0, then equation will not have x² and x, which is not possible because the given equation is quadratic equation.
Therefore, the required value of k is 6.

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.

Answer:
Let the breadth of the mango grove be x m and the length is 2x m.
Area = length × breadth
= x × 2x
= 2×2 m²
Then by the given condition,
2x² = 800
x² = 400
x = ±√400
x = ±20
Since, length cannot be negative, then x ≠ -20
Hence it is possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m² and its breadth is 20 m and length is 20×2 = 40 m

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 1st friend is x.
Then the age of 2nd friend is (20-x)
Four year ago their age was (x-4) and (20-x-4)
Then using the given condition we have
(x-4)(16-x) = 48
16x – 64 – x² +4x = 48
20x – x² – 64 – 48 = 0
x² – 20x + 112 = 0……(1)
Now comparing the above quadratic equation with the general form of quadratic equation ax²+bx+c = 0 we get
a = 1, b = -20 and c = 112
then, b²-4ac = (-20)²-4×1×112 = 400-448 = -48 > 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 m²? If so find its length and breadth.

Answer:
Let the length and breadth of the park be “l” and “b”
Area of rectangle = 2(l + b)
2(l + b) = 80
l + b = 40
b = 40 – l
Then, Area = l(40 – l)
Then by the given condition,
l(40 – l) = 400
40l  – l² -400 = 0
l² – 40l + 400 = 0….(1)
Now comparing the above quadratic equation with the general form of quadratic equation ax²+bx+c = 0 we get
a = 1, b = -40 and c = 400
Then, b²-4ac = (-40)²-4×1×400 = 1600-1600 = 0
As the quadratic equation has two equal roots then the given situation is possible.

Therefore, length of park, l = 20 m

And breadth of park, b = 40 – l = 40 – 20 = 20 m.

CHAPTER NAME	OLD NCERT	NEW NCERT
Real Numbers	EXERCISE 1.1
	EXERCISE 1.2	1.1	CLICK HERE
	EXERCISE 1.3	1.2	CLICK HERE
	EXERCISE 1.4
Polynomials	EXERCISE 2.1	2.1	CLICK HERE
	EXERCISE 2.2	2.2	CLICK HERE
	EXERCISE 2.3
	EXERCISE 2.4
Pair of Linear Equations in Two Variables	EXERCISE 3.1
	EXERCISE 3.2	3.1	CLICK HERE
	EXERCISE 3.3	3.2	CLICK HERE
	EXERCISE 3.4	3.3	CLICK HERE
	EXERCISE 3.5
	EXERCISE 3.6
	EXERCISE 3.7
Quadratic Equations	EXERCISE 4.1	4.1	CLICK HERE
	EXERCISE 4.2	4.2	CLICK HERE
	EXERCISE 4.3
	EXERCISE 4.4	4.3	CLICK HERE
Arithmetic Progressions	EXERCISE 5.1	5.1	CLICK HERE
	EXERCISE 5.2	5.2	CLICK HERE
	EXERCISE 5.3	5.3	CLICK HERE
	EXERCISE 5.4	5.4 (Optional)	CLICK HERE
Triangles	EXERCISE 6.1	6.1	CLICK HERE
	EXERCISE 6.2	6.2	CLICK HERE
	EXERCISE 6.3	6.3	CLICK HERE
	EXERCISE 6.4
	EXERCISE 6.5
	EXERCISE 6.6
Coordinate Geometry	EXERCISE 7.1	7.1	CLICK HERE
	EXERCISE 7.2	7.2	CLICK HERE
	EXERCISE 7.3
	EXERCISE 7.4
Introduction to Trigonometry	EXERCISE 8.1	8.1	CLICK HERE
	EXERCISE 8.2	8.2	CLICK HERE
	EXERCISE 8.3
	EXERCISE 8.4	8.3	CLICK HERE
Some Applications of Trigonometry	EXERCISE 9.1	9.1	CLICK HERE
Circles	EXERCISE 10.1	10.1	CLICK HERE
	EXERCISE 10.2	10.2	CLICK HERE
Construction
Areas Related to Circles	EXERCISE 12.1
	EXERCISE 12.2	11.1	CLICK HERE
	EXERCISE 12.3
Surface Areas and Volumes	EXERCISE 13.1	12.1	CLICK HERE
	EXERCISE 13.2	12.2	CLICK HERE
	EXERCISE 13.3
	EXERCISE 13.4
	EXERCISE 13.5
Statistics	EXERCISE 14.1	13.1	CLICK HERE
	EXERCISE 14.2	13.2	CLICK HERE
	EXERCISE 14.3	13.3	CLICK HERE
	EXERCISE 14.4
Probability	EXERCISE 15.1	14.1	CLICK HERE
	EXERCISE 15.2