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
MCQ Questions for Class 10 Maths all Chaptershttps://sharmatutorial.in/category/mcq-class-10-maths/