SATHEE: Trigonometrical Equations 1 Question 2

Trigonometrical Equations 1 Question 2

2. 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

(a) 3

(b) 5

(d) 4

(2019 Main, 12 April I)

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Answer:

Correct Answer: 2. (b)

Solution:

Given equation is $1 + \sin^{4} x = \cos^{2} (3 x)$

Since, range of $(1 + \sin^{4} x) = [1, 2]$

and range of $\cos^{2} (3 x) = [0, 1]$

So, the given equation holds if

$\begin{array}{ll} 1 + \sin^{4} x = 1 = \cos^{2} (3 x) \\ \Rightarrow & \sin^{4} x = 0 and \cos^{2} 3 x = 1 \end{array}$

Since, $x \in - \frac{5 π}{2}, \frac{5 π}{2}$

$∴ x = - 2 π, - π, 0, π, 2 π .$

Thus, there are five different values of $x$ is possible.

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Trigonometrical Equations 1 Question 2

2. The number of solutions of the equation 1+sin4⁡x=cos2⁡3x,x∈−5π2,5π2 is

Answer:

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2. 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