Maths-

General

Easy

Question

Statement-I The equation has exactly one solution in [0, 2].

Statement-II For equations of type to have real solutions in should hold true.

- Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
- Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I.
- Statement-I is true, Statement-II is false.
- Statement-I is false, Statement-II is true.

Hint:

### In this question, given two statement. It is like assertion and reason. Statement1 is assertion and statement 2 is reason , Find the statement 1 is correct or not and the statement 2 correct or not if correct then is its correct explanation.

## The correct answer is: Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I.

### Here , we have to find the which statement is correct and if its correct explanation or not.

Firstly ,

Statement-I: The equation √(3 ) cosx- sinx=2 has exactly one solution in [ 0 , 2 π ]

We have,

√(3 ) cosx- sinx=2

⇒√(3 )/2 cosx-1/2 sinx=1

⇒cos〖π/6〗 cosx-sin〖π/6〗 sinx=1 [ since we know that sinπ/3 = √(3 )/2 and cos〖π/3〗 = 1/2 ]

⇒cos〖(π/6+ x )=1〗 [ since ,cos a cosb -sin a sinb = cos ( a + b ) ]

⇒cos〖(x+ π/6)=1〗

⇒ x + π/6 = 2nπ

⇒ x = 2nπ - π/6

For n = 0 , we have x = - π/6 , and ∉ ( 0,2π)

For n = 1 , we have x = 11 π /6 , and ∈ ( 0, 2π)

For n = 2, we have x = 23 π /6 and ∉ ( 0, 2π)

it have only one solution in [ 0 , 2π]

x = 11 π /6

Therefore, the Statement-I is true.

Now,

Statement-II : For equations of type acos θ + bsin θ = c to have real solution in [ 0 , 2π], |c| ≤ √(a^{2}+ b^{2} ) should hold true.

We have ,

acos θ + bsin θ = c

⇒a /c cos θ + b/c sin θ = 1

Let a/c = sinα and b/c = cosα

We have,

⇒ sinα cos θ + cosα sin θ = 1

⇒ sin(θ + α ) = 1 [ since, sin a cosb + cosa sinb = sin( a+ b)]

Therefore, (θ + α ) is definitely lies in [ 0 , 2π]. Hence statement – II is true. But it is not correct explanation of statement – I .

The correct answer is , Statement-I is true, Statement-II is true ;Statement-II is NOT a correct explanation for Statement-I

In this question, we have to find the statements are the correct or not and statement 2 is correct explanation or not , is same like assertion and reason. Here, Start solving first Statement and try to prove it . Then solve the Statement-II . Remember cos a cosb -sin a sinb = cos ( a + b ) and sin a cosb + cosa sinb = sin( a+ b) .

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