Elementary Properties and Identities MCQ Quiz in मल्याळम - Objective Question with Answer for Elementary Properties and Identities - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept - Use trigonometric identities.

Solution -   Let 

Now 

so use the formula - 

Calculation:

Let sin^-1 x = y. Then x = sin y

Since,  

therefore   and so 

We have,

cos y = 

  for        ....(i)

Now,       [From (i)]       

Concept:

Calculation:

Given: tan-1 a + tan-1 b = π/4

As we know that, 

⇒ tan^-1 [a + b / 1 - ab] = π/4

⇒ [a + b / 1 - ab] = tan (π/4) = 1

⇒ a + b = 1 - ab

⇒ a + b + ab = 1

Concept:

2 tan-1 x = tan-1 (2x / 1- x2), - 1 ≤ x ≤ 1

Calculation:

Given: 2 tan-1 (cos x) = tan-1 (2 cosec x)

As we know that, 2 tan-1 x = tan-1 (2x / 1- x2), - 1 ≤ x ≤ 1

⇒ 2 tan-1 (cos x) = tan^-1 (2 cos x / 1 - cos² x)

As we know that, sin² x + cos² x = 1

⇒ 2 tan-1 (cos x) = tan-1 (2 cos x / sin² x)

⇒ tan-1 (2 cos x / sin2 x) = tan-1 (2 cosec x) = tan^-1 (2 / sin x)

⇒ 2 cos x / sin2 x = 2 / sin x

⇒ cos x sin x - sin² x = 0

⇒ sin x (cos x - sin x) = 0

⇒ sin x = 0 or cos x - sin x = 0

⇒ x = 0 or π/4

As we can see that for x = 0 the given equation does not exist

Hence, x = π/4 is the only solution for the given equation.

Concept -

Use trigonometric identities

Solution -

The given question is also written in the form

Now use the identity 

............(i) 

Let 

Now  then 

put the value of A we get 

put this value in equation (i) we get 

Now taking cos of both side we get the value of x

So the final answer is  hence option 2 is correct.

Concept:

Calculation:

⇒ 

⇒  

Since 

⇒ x = 

∴ The value of x is 

Concept:

tan-1 (1/x) = cot x, x > 0

tan-1 x + tan-1 y = tan-1 (x + y) / (1 - xy), xy

Calculation

Given: tan-1 (2x / 1 - x2) + cot-1 (1 - x2 / 2x) = π/3 such that -1

As we know that, tan-1 (1/x) = cot x, x > 0

⇒ cot-1 (1 - x2 / 2x) = tan-1 (2x / 1 - x2)

⇒ 2 tan-1 (2x / 1 - x2) = π/3

⇒ tan-1 (2x / 1 - x2) = π/6

⇒ 2x / 1 - x² = 1/√3

⇒ x² + 2√3 x - 1 = 0

By comparing the above equation with ax² + bx + c = 0, we get

⇒ a = 1, b = 2√3 and c = -1

By using the formula x = [-b ± √(b² - 4ac)] / 2a, we get

⇒ x = 2 - √3 or - (2 + √3)

But as it is given that -1

Hence, x = 2 - √3

Calculation: 

Given

We know that  and 

 = 

= 

= 

= 

= 

= 

=  = sin(π/2) = 1

∴  = 1

The correct answer is option (4).

Concept:

Calculation:

Let S = 

S =            [∵ ]

S = 

S = 

S = 

S = π 

Concept:

If sin θ = x then θ = sin^-1 x

sin-1 (sin θ) = θ, ∀ θ ∈ [-π/2, π/2]

sin-1 x + cos^-1 x = π / 2, where x ∈ [-1, 1]

Calculation:

Given: 

As we know that, if sin θ = x then θ = sin-1 x

⇒ sin^-1 (1/5) + cos^-1 x = sin^-1 (1)

As we know that, sin π/2 = 1

⇒ sin-1 (1/5) + cos-1 x = sin-1 (sin π/2)

As we know that, sin-1 (sin θ) = θ, ∀ θ ∈ [-π/2, π/2]

⇒ sin-1 (1/5) + cos-1 x = π/2

As we know that, sin-1 x + cos-1 x = π / 2, where x ∈ [-1, 1]

Hence, x = 1/5