Trigonometric elements MCQ Quiz - Objective Question with Answer for Trigonometric elements - Download Free PDF

Concept:

When A² + B² + C² = 0, it implies A = B = C = 0 (since the squares of real numbers are non-negative).

Substitute the values of A, B, and C for determinant calculation into the matrix

Calculation:

Since, Cos0 =1

Thus Matrix becomes 

Now determinant = 1[(1×1 - 1×1)] - 1[(1×1 - 1×1)] + 1[(1×1 - 1×1)]

= = 1(0) - 1(0) + 1(0) = 0

∴ The value of the determinant is 0.

Hence, the correct answer is Option 2.

Given:

Integral: ∫Sin(mx) × Cos(nx) dx

Concept Used:

The trigonometric product-to-sum formula is used here:

Sin(A) × Cos(B) = 1/2 × [Sin(A+B) + Sin(A-B)]

Calculation:

Using the formula:

Sin(mx) × Cos(nx) = 1/2 × [Sin(mx + nx) + Sin(mx - nx)]

Substituting values:

⇒ Sin(mx) × Cos(nx) = 1/2 × [Sin((m+n)x) + Sin((m-n)x)]

Comparing this result with the given options, the correct answer is:

Option 4: 1/2[Sin(m+n)x + Sin(m-n)x]

Conclusion:

∴ The solution satisfies the formula, and the correct answer is Option 4.

Calculation

∵ P^T P = I

B = PAP^T 

Pre multiply by P^T (Given)

P^TB = P^TP AP^T = AP^T

Now post multiply by P

P^TBP = AP^T P = A 

So A² = 

A² = P^TB² P

Similarly A¹⁰ = P^TB¹⁰ P = C 

⇒ 

Similarly check A³ and so on since C = A¹⁰

⇒ Sum of diagonal elements of C is 

= 

g cd(m, n) = 1 (Given) 

⇒ m + n = 65

Hence option 1 is correct

Calculation

Expanding through R₃ we get

Independent of

Also,

Therefore,

at

Hence option 2 and 4 are correct

The correct option is: 90 

Explanation: We start with the matrix ( A ) and the given equation ( A + A^T - 2I = 0 ).

Given:

• First, determine the transpose of ( A ):
• Next, add ( A ) and ( A^T ): [ A + A^T =
• Now, subtract ( 2I ) from ( A + A^T ):
• Set ( A + A^T - 2I ) equal to the zero matrix:
• This results in the following equations:
• Solving these gives:
• The only angle that satisfies both equations is

Thus, the value of (θ) is 90.

Concept:

cos² θ - sin² θ = cos 2θ

Calculation:

Given: A =  

|A| = cos θ × cos θ - sin θ × sin θ

= cos2 θ - sin2 θ

= cos 2θ

Calculation: 

m sinθ + n cosθ = sin θ  + cos θ 

= +

= 

Now,

|m sinθ + n cosθ| = 

=  + 

= 2()

= 2

Hence, option (4) is correct. 

Formula used:

det(A) = 

= a₁(b₂ c₃ - b₃ c₂) - a₂(b₁c₃ - b₃c₁) + a₃ (b₁c₂ - b2c1)

Calculation:

= cos C (0 + tan A sin B) - tan A (sin B cos C) 

= tan A sin B cos C - tan A sin B cos C

= 0

Explanation:

We have, Δ =  

Applying R₁ →  R₁ - R₂ we get,

Δ = 

Expanding Δ along row R₁

⇒ Δ = 0 - (-sinθ)(1 - 1 - cosθ) + 0

⇒ Δ =  - (-sinθ)(- cosθ)

⇒ Δ = -sinθcosθ

⇒ Δ =  

⇒ Δ =    (∵ 2 sin θ cos θ = sin 2θ )

Now, Δ will be maximum when sin2θ will be minimum.

Minimum value of sin2θ = - 1

∴ Maximum value of Δ = ( -1/2) ×  ( -1) = 1/2

Calculation:

A = 

A = cos² x - sin² x

A = cos 2x

Concept: For x,y ∈ R:

sin(x + y) = sin x cos y + cos x sin y

cos(x + y) = cos x cos y - sin x sin y

Explanation:

We have Δ = 

⇒ Δ = 

Applying R₃ → R₃ - cosyR₁ - sinyR₂ 

⇒ Δ = 

Expanding along row R₃,

⇒ Δ = 0 - 0 + (sin y - cos y)(sin²x + cos²x)

⇒ Δ = sin y - cos y  (∵ sin2x + cos2x = 1)

⇒ Δ = √2[sin y - cos y]

⇒ Δ = √2[ ]

⇒ Δ = √2[sin(y - )]

Now -1 ≤ sin(y - ) ≤ 1

⇒ -√2 ≤ √2 sin(y - ) ≤ √2

∴ Δ ∈ [-√2, √2]

CONCEPT:

• sin A cos B + sin B cos A = sin (A + B)
• If is a square matrix of order 2, then determinant of A is given by |A| = (a­11 × a22) – (a12 – a21)

CALCULATION:

Given: 

As we know that, if  is a square matrix of order 2, then determinant of A is given by |A| = (a­11 × a22) – (a12 – a21)

⇒ 

As we know that, sin A cos B + sin B cos A = sin (A + B)

⇒ k = sin 90° = 1

Hence, the correct option is 4.

Calculation:

Let Δ = 

= sec² x - tan² x

= 1 + tan² x -  tan2 x       [∵ 1 + tan2 x = sec2 x]

= 1 

Calculation:

Let Δ = 

= sec² x - tan² x

= 1 + tan² x -  tan2 x       [∵ 1 + tan2 x = sec2 x]

= 1 

Concept :

A=  

Then determinant of matrix A , | A | = = a₁₁ × {(a₂₂ × a₃₃) – (a₂₃ × a₃₂)} - a₁₂ × {(a₂₁ × a₃₃) – (a₂₃ × a₃₁)} + a₁₃ × {(a₂₁ × a₃₂) – (a₂₂ × a₃₁)} . 

Calculation:

⇒ Δ =  

⇒ Δ =  

⇒ Δ =  

⇒ Δ =                

⇒ Δ = - x³                                        [∵ sin² θ + cos² θ = 1]

The correct option is 4.