Exercise 4.3 Class 10 Maths | Quadratic equations Class 10 Maths
Exercise 4.3 class 10 maths
1. Find the nature of the roots of the following quadratic equations. If the real roots exist ,find them.
(i) 2x² – 3x + 5 =0
Here a = 2 , b = -3 , c = 5
D = b² – 4ac
D = (-3)² – 4 × 2 × 5
D = 9 – 40
D = – 31
Since D < 0 so there is no real roots exist.
(ii) 3x² – 4√3 x + 4 = 0
Here a = 3 , b = -4√3 , c = 4
D = b² – 4ac
D = (-4√3)² – 4 × 3 × 4
D = 48 – 48
D = 0
Since D = 0 so it has 2 real and equal roots exist.
Using quadratic formula
(iii) 2x² – 6x + 3 = 0
Here a= 2 , b = -6 , c = 3
D = b²- 4ac
D = (-6)² – 4 × 2 × 3
D = 36 – 24
D = 12
Since D > 0 so it has 2 real and distinct roots exits.
Using Quadratic formula
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
Here a = 2 , b = k , c = 3
Since it has 2 equal roots
So D = 0
b² – 4ac = 0
k² – 4× 2 × 3 = 0
k² -24= 0
k² = 24
k = √24
k = 2√6 or k = – 2√6
(ii) kx (x – 2) + 6 = 0
kx² – 2kx +6 = 0
Here a = k , b = -2k. , c = 6
Since it has 2 equal roots so
D = 0
b² – 4ac = 0
(-2k)² – 4× k× 6 = 0
4k² -24k = 0
4k ( k – 6 ) = 0
Either 4k = 0 so k =0 but it doesn’t satisfy the equation so rejected.
Or k-6= 0 then k= 6 ans.
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.
Solutions: Let length be x m so breadth will be 2x m.
Area of rectangular grove = 800 m²
x (2x) =800
2x² -800 = 0
Here a = 2, b = 0 and c = 800
Discriminant (D) = b² – 4ac
D = 0² – 4 × 2 × 800
D = – 6400
Since D < 0 so this quadratic equation has no real roots exist. Hence this is not possible to design such rectangular 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 4 years ago the product of their ages in years was 48.
Solution: let present age of first friend be x years
and present age of second friend be (20 – x) years
four years age of first friend = (x- 4)yrs
and age of second friend =(20 – x-4)yrs
= (16-x)yrs
ATQ (x-4) (16-x) =48
16x -x²-64 +4x =48
x² -20 x +112 = 0
Here a = 1 , b = -20 , c = 112
D = b²- 4ac
D = (-20)²-4×1×112
D = 400 – 448
D = -48
Since D <0 so no real roots exist. Therefore the given situation is not possible.
5. Is it possible to design a rectangular park of perimeter 80 meter and area 400 m square if so find its length and breadth.
Solution: Let length be x m and breadth be y m
perimeter of rectangular park = 80m
2( x + y) = 80
x+y = 40
y = 40 – x ……….(i)
Area of rectangular park= 400m²
x × y = 400
Using equation (i)
x ( 40 – x) =400
40x – x² = 400
x² – 40x + 400 =0
Here a = 1 , b = -40 , c = 400
Discriminant (D) = b²-4ac
D = (-40)² – 4 × 1 ×400
D = 1600 – 1600
D = 0. Since D = 0 so this quadratic equation has 2 real and equal roots exist.do this is possible to design such rectangular park.
Using quadratic formula
x = -b/2a, -b/2a
x = – (-40)/2 , -(-40)/2
x = 20, 20. Put x= 20 in (i)
y = 20
So length and breadth are 20m , 20m
Ncert Solutions Class 10 Maths Exercise 1.1 https://10thmathsguide.com/exercise-11-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 1.2
Ncert Solutions Class 10 Maths Exercise 2.1 https://10thmathsguide.com/exercise-21-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 2.2
Ncert Solutions Class 10 Maths Exercise 3.1 https://10thmathsguide.com/exercise-31-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 3.2 https://10thmathsguide.com/exercise-32-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 3.3 https://10thmathsguide.com/exercise-33-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 4.1 https://10thmathsguide.com/exercise-41-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 4.2 https://10thmathsguide.com/exercise-42-class-10-maths/
Ncert Solutions Class 10 Maths Exercise 4.3 https://10thmathsguide.com/exercise-43-class-10-maths/
MCQ VIDEOS ALL CHAPTERS CLASS 10 MATHShttps://www.youtube.com/watch?v=-eBlHyBxjLg&list=PL2uPMjJCHErQZZNipbsnagBqPrCU_WRN8
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