Which of the following functions are periodic? (Where · denotes the greatest integer function.)

p x = x + x + 1 3 + x + 2 3 - 3 x + 1 0 , where [.] denotes the greatest integer function.

Column – I Column – II A. In an acute angled triangle A B C , the least values of ∑ sec ⁡ A and ∑ tan 2 ⁡ A are λ and μ respectively, then p. λ + μ = 2 B. In a triangle A B C , the least values of ∑ cosec A 2 and ∑ sec 2 ⁡ A 2 are λ and μ respectively, then q. μ - λ = 3 C. In a triangle A B C , the least values of ∏ cosec A 2 and ∑ cosec 2 A are λ and μ respectively, then r. λ - μ = 4 s. 3 λ - 2 μ = 0 t. 2 λ - 3 μ = 2

A → q , s ; B → p , t ; C → r

A → t , s ; B → p , r ; C → q

A → q , p ; B → t , r ; C → p

A → r , s ; B → p , q ; C → t

EASY

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The period of the function $f (x) = \tan (4 x - 1)$

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Important Questions on Trigonometric Ratios and Identities

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

a, b

be two fixed positive integers such that

f (a + x) = b + {[b^{3} + 1 - 3 b^{2} f (x) + 3 b {\{f (x)\}}^{2} - {\{f (x)\}}^{3}]}^{1 / 3}

for all real

x

then

f (x)

is a periodic function with period

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The period of

\sin^{2} θ

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

Consider the function

f (x) = \cos x^{2}

. Then

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The function $f (x) = |sin 4 x| + |cos 2 x|,$ is a periodic function with a fundamental period

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

Which one is not periodic?

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The period of the function

f (x) = [\sin 3 x] + |\cos 6 x|

is (

[.]

denotes the greatest integer less than or equal to

x

)

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

f (x) = {(- 1)}^{[\frac{2 x}{π}]}, g(x) = |sin x| - |cos x|

and

ϕ (x) = f (x) \cdot g (x)

(where,

[.]

denote the greatest integer function), then the respective fundamental periods of

f (x), g (x)

and

ϕ (x)

are

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

f (3 x + 2) + f (3 x + 29) = 0 \forall x \in R

, then the period of

f (x)

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

If f be a real value function such that for any real $x, f (15 + x) = f (15 - x) & f (30 + x) = - f (30 - x)$ , then which of the following statement is true ?

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The period of the function

f (x) = c o s e c^{2} 3 x + \cot 4 x

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The period of the function

f (x) = 4 \sin^{4} (\frac{4 x - 3 π}{6 π^{2}}) + 2 \cos (\frac{4 x - 3 π}{3 π^{2}})

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

Which of the following functions are periodic? (Where

[\cdot]

denotes the greatest integer function.)

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

The period of the function

f (θ) = \sin \frac{θ}{3} + \cos \frac{θ}{2}

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Periodic Functions

	Column – I		Column – II
A.	In an acute angled triangle $A B C$ , the least values of $\sum \sec A$ and $\sum \tan^{2} A$ are $λ$ and $μ$ respectively, then	p.	$λ + μ = 2$
B.	In a triangle $A B C$ , the least values of $\sum cosec (\frac{A}{2})$ and $\sum \sec^{2} (\frac{A}{2})$ are $λ$ and $μ$ respectively, then	q.	$μ - λ = 3$
C.	In a triangle $A B C$ , the least values of $\prod cosec (\frac{A}{2})$ and $\sum {cosec}^{2} A$ are $λ$ and $μ$ respectively, then	r.	$λ - μ = 4$
		s.	$3 λ - 2 μ = 0$
		t.	$2 λ - 3 μ = 2$