sin ⁡ θ + 3 cos ⁡ θ = 6 x - x 2 - 11 , 0 ≤ θ ≤ 4 π , x ∈ R , holds for-

Two pairs of values of ( x , θ )

Two values of x and two values of θ

sin ⁡ θ + 3 cos ⁡ θ = 6 x - x 2 - 11 , 0 ≤ θ ≤ 4 π , x ∈ R , holds for-

Two pairs of values of ( x , θ )

Assertion ( A ) : If 4 sin 4 θ + sin 2 2 θ + 4 cos 2 π 4 - θ 2 = 2 , then θ lies in 3 rd quadrant or 4 th quadrant. Reason R : sin 2 θ = sin θ

Both ( A ) and ( R ) are true and ( R ) is the correct explanation of ( A )

Both ( A ) and ( R ) are true but ( R ) is not the correct explanation of ( A )

( A ) is false but ( R ) is true

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$\sin θ + \sqrt{3} \cos θ = 6 x - x^{2} - 11, 0 \leq θ \leq 4 π, x \in R,$ holds for-

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Important Questions on Trigonometric Ratios and Identities

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let

S

be the set of all

α \in R

such that the equation,

c o s 2 x + α s i n x = 2 α - 7

has a solution. Then

S

is equal to:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

For

x \in (0, π),

the equation

\sin x + 2 \sin 2 x - \sin 3 x = 3

has

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let

S = \{θ \in [- 2 π, 2 π] : 2 \cos^{2} θ + 3 \sin θ = 0\} .

Then the sum of the elements of

S

is:

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Number of solutions of the equation

\sin x - \sin 2 x + \sin 3 x = 2 \cos^{2} x - 2 \cos x

(0, π)

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let

X = {x \in R : \cos (\sin x) = \sin (\cos x)} .

The number of elements in

X

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The equation

\sin^{4} x - (k + 3) \sin^{2} x - k - 4 = 0

has a solution if

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

m

and

M

are the minimum and the maximum values of

4 + \frac{1}{2} \sin^{2} 2 x - 2 \cos^{4} x, x \in R,

then

M - m

is equal to:

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

\cos 2 x + 7 = a (2 - \sin x)

can have a real solution for

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

5 (\tan^{2} x - \cos^{2} x) = 2 \cos 2 x + 9,

then the value of

\cos 4 x

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions to the equation

\cos^{4} x + \frac{1}{\cos^{2} x} = \sin^{4} x + \frac{1}{\sin^{2} x}

in the interval

[0, 2 π]

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Assertion

(A) :

\sqrt{4 \sin^{4} θ + \sin^{2} 2 θ} + 4 \cos^{2} (\frac{π}{4} - \frac{θ}{2}) = 2,

then

θ

lies in

3^{rd}

quadrant or

4^{th}

quadrant.
Reason

(R) : \sqrt{\sin^{2} θ} = \sin θ

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let

a, b, c

be three non-zero real numbers such that the equation

\sqrt{3} a \cos x + 2 b \sin x = c, x \in [- \frac{π}{2}, \frac{π}{2}]

has two distinct real roots

α

and

β

with

α + β = \frac{π}{3}

. Then, the value of

\frac{a}{b}

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions of the equation

4 \cos 2 θ \cdot \cos 3 θ = \sec θ,

when

0 < θ < π,

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions of the equation

1 + \sin^{4} x = \cos^{2} 3 x, x \in [- \frac{5 π}{2}, \frac{5 π}{2}]

is:

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If $\sin 6 θ + \sin 4 θ + \sin 2 θ = 0,$ then general value of $θ$ is $(n$ is an integer $)$

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions

x

of the equation

\sin (x + x^{2}) - \sin (x^{2}) = \sin x

in the interval

[2, 3]

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of real solutions

x

of the equation

\cos^{2} (x \sin (2 x)) + \frac{1}{1 + x^{2}} = \cos^{2} x + \sec^{2} x

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The angles

α, β, γ

of a triangle satisfy the equations

2 \sin α + 3 \cos β = 3 \sqrt{2} a n d 3 \sin β + 2 \cos α = 1

. Then angle

γ

equals

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The equation

\sqrt{3} \sin x + \cos x = 4

has

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

0 \leq x < \frac{π}{2},

then the number of values of

x

for which

\sin x - \sin 2 x + \sin 3 x = 0,

is:

sin⁡θ+ 3cos⁡θ=6x-x2-11, 0 ≤θ ≤4π, x ∈R, holds for-

Important Questions on Trigonometric Ratios and Identities

Learn and Prepare for any exam you want !

$\sin θ + \sqrt{3} \cos θ = 6 x - x^{2} - 11, 0 \leq θ \leq 4 π, x \in R,$ holds for-