Domain or Range MCQ Quiz in मल्याळम - Objective Question with Answer for Domain or Range - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Use simply trigonometric identities.

Solutions: 

We know that  and this implies 

Using the rule  then 

Now taking tan of both side 

we again know that the value of  then 

The final answer to this question is  , and option 2 is correct.

Concept:

Range of sin θ is [-1, 1]

Calculation:

Given that

f(x) = π + sin2 x.

The given function is a combination of a constant term (π) and the

square of the sine function sin2 x.

We know that 

-1 ≤ sin x ≤ 1 

⇒ 0 ≤ sin²x ≤ 1 

⇒ π + 0  ≤ π + sin2x ≤ π + 1 

Therefore, the range of π + sin2 x is [π, π + 1].

Concept:

We know that, 

Calculations:

We know that, 

Adding - 2 on both side, we get

⇒

⇒

If we take mod inequality will be changed.

⇒

∴  

Hence, the maximum value of |-2 + sin θ| is 3.

Concept:

The minimum and maximum value of a.sin x ± b.cos x is given by:

Calculation:

Given: sin x - √8cos x - 2 

Compare sin x - √8cox x is a.sinx - b.cosx 

Here a = 1 and b = √8

⇒  

⇒ 

Subtract 2 in the above equation

⇒ -3 - 2 ≤ sin x - √8.cos x - 2 ≤ 3 - 2

⇒ -5 ≤ sin x - √8.cos x - 2 ≤ 1

Range [- 5, 1]

Concept: 

The period of the function f(x) = π + sin2 x is determined by the

periodicity of the sine function, as the sin2 x.

Calculation:

Given that 

f(x) = π + sin2 x    ----(1)

The cos x function has a period of 2π.

The cos 2x function has a period of π.

We know that 

sin²x = (1 = cos 2x)/2

As, adding or subtracting a constant number does not affect the period

of the function.

Hence, The period of the function π + sin2 x is π.

Concept:

• If the given equation is in quadratic form ax² + bx + c=0, then its solution is found as .
• -1 ≤ sin x ≤ 1
• cos 2x = 1 - 2sin2x

Calculation:

k sin x + cos 2x = 2k - 7

⇒ k sin x + 1 - 2sin²x = 2k - 7

⇒ -2sin2x + k sin x + 1 - 2k + 7 = 0 

⇒ -2sin²x + k sin x - 2k + 8 = 0

⇒ 2sin2x - k sinx + 2k - 8 = 0  ...(Multiply by -)

After comparing with Quadratic equation, a = 2, b = -k, c = (2k - 8)

⇒ 

⇒ sin x =  and sin x = 2,

Since, sin x cannot be greater than 1,

Thus, sin x ≤ 1

k - 4 ≤ 2

k ≤ 6

Thus, the maximum value of k is 6.

Concept:

• The domain of a function is the set of values that we are allowed to plug into our function.
•  The range is the set of all possible values that the function will give when we give in the domain as input.

Calculation:

The above graph is of the function cosecx

Since , the domain of the cosec function is the set {x : x ∈ R and x ≠ nπ, n ∈ Z} or R - {x : x = nπ, n ∈ Z} and 

the range set is the set {y : y ∈ R and y ≥ 1 or y ≤ -1} 

i.e., the set R - (-1,1).

It means that y = cosec x assumes all real values except -1

Hence, the correct answer is option 2).

Concept: 

Where the function is defined is called the range of the inverse trigonometric function and the values we get are called the domain of the inverse trigonometric function.

Solutions  -

So the final answer is  [-1,1] hence option 2 is correct.

Alternate Method

Let  

and we all know that the Sin function is defined on a whole real line so the range of the given function is . and the values which we get of sin function for the whole real line is in between [-1,1] because sin function is periodic and oscillates between 1 and -1 on the whole real line.

So the final answer is  [-1,1] hence option 2 is correct.

Concept:

The range of a function is the set of all its outputs.

Example:

Let us consider the function f: A→ B,

where f(x) = 2x and A and B = {set of natural numbers}.

Here we say A is the domain and B is the co-domain.

The set of ƒ images of all the elements of A is known as the range of ƒ.  

Calculation:

f(x) = 1 - sinx 

Here the domain of f(x) is (-∞, ∞)

At this domain, the range of sinx is [-1, 1]

Range of f(x) = 1 - sin2x = 1 - [-1, 1] 

⇒ f(x) = [0 , 2]

∴ Range of f(x) is [0 , 2].

Concept Used:

The range of cos(x) is [-1, 1]

Calculation

Given:

f(x) = 7 cos(10x + 4π)

The range of cos(10x + 4π) is [-1, 1]

Multiply by 7:

⇒ 7 × [-1, 1] = [-7, 7]

∴ The range of f(x) = 7 cos(10x + 4π) is [-7, 7]

Hence option 4 is correct