Trigonometry- MCQ Test 2

1. If x = a cos 0 and y = b sin 0, then b²x² + a²y²=?

A) ab
B) b² + a²
C) a²b²
D)None of these

ANSWER=> (C)
Explanation/Answer:-
given, x = a cos 0 and y = b sin 0
Putting sin 0=0,cos 0 =1 => x=a, y=0
b²x² + a²y² => b²a²

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2. If secθ+tanθ=x, then secθ is :

A) (x²+1)/2x
B) (x²-1)/2x
C) (x+1)/2x
D) (x-1)/2x

ANSWER=> (A)

Given, secθ+tanθ=x..... eqn (1)
As we know that
sec²θ−tan²θ=1
⟹(secθ−tanθ)(secθ+tanθ)=1
⟹(secθ−tanθ)=1/(secθ+tanθ)
⟹(secθ−tanθ)=1/x ..... eqn(2)
Adding eqn (1) and eqn(2)weget
2secθ=x+1/x=(x²+1)/2x
secθ=(x²+1)/2x

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3.If secθ+tanθ=x, then tanθ is :

A) (x²+1)/2x
B) (x²-1)/2x
C) (x+1)/2x
D) (x-1)/2x

ANSWER=> (B) Explanation/Answer:-
Given, secθ+tanθ=x..... eqn (1)
As we know that
sec²θ−tan²θ=1
⟹(secθ−tanθ)(secθ+tanθ)=1
⟹(secθ−tanθ)=1/(secθ+tanθ)
⟹(secθ−tanθ)=1/x ..... eqn(2)
Substracting eqn (2) from eqn(1)weget
2tanθ=x-1/x=(x²-1)/2x
tanθ=(x²-1)/2x

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4. If (1-cos x)/sinx = ?

A) sinx/(1-cos x)
B) sinx/(1+cos x)
C) cosx/(1-cos x)
D) None of these

ANSWER=> (B)
Explanation/Answer:-
(1-cos x)/sinx
Multiplaying (1+cos x) in numarator and denumarator weget
(1-cos x)(1+cos X)/sinA(1+cos x)
(1-cos² x)/sinx(1+cos x)
(sin² x)/sinx(1+cos x)

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5. What is the minimum value of sin A, 0 ≤ A ≤ 90°

A) 1
B) 0
C) -1
D) None of these

ANSWER=> (A)
Explanation/Answer: sin 0°=0

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6. What is the minimum value of cos θ, 0 ≤ θ ≤ 90°

A) 1/√2
B) 1
C) 0
D) -1

ANSWER=> (C)
Explanation/Answer:-
cos 90°=0

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7. Given that sin θ = x/y , then tan θ =

A) x/√y²+x²
B) y/2
C) x/√y²-x²
D) 1

ANSWER=> (C)
Explanation/Answer:- cos² θ=1-sin² θ=>cos θ= √1-(x/y)²
tan θ=sin θ/cos θ=x/√y²-x²

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8. If sin A – cos A = 0, then the value of sin^4 A + cos^4 A is

A) 2
B) 1
C) 3/4
D) 1/2

ANSWER= (D)
Explanation/Answer:- Given,sin A – cos A = 0
=>sin A = cos A
=> tanA = 1
=> A=45°
sin^4 A + cos^4 A={(1/√2)²}²+{(1/√2)²}²
=>{(1/4)+{(1/4)}=1/2

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9. If cos 9A = sin A and 9A < 90°, then the value of tan 5A is

A) 0
B) 1/√3
C) 1
D) √3

ANSWER= (C)
Explanation/Answer:-
⇒ cos 9A = sin A ⇒cos 9A = cos(90- A)
⇒ 9A = 90° - A
⇒ 10 A = 90°
⇒ A = = 9°
∴ 5A = 5 × 9° = 45°
⇒ tan45°=1

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10. Given that sin α = 12 and cos β = 12, then the value of (α + β) is

A) 0°
B) 30°
C) 45°
D) 90°

ANSWER= (D)
Explanation/Answer:- Given, sin α = 12 and cos β = 12
Therefore, sin α = cos β
=> sin α = sin (90 - β)
=> α = 90 - β
=> α + β = 90