The complete solution of the equation $7 {cos}^{2} x + sin x cos x - 3 = 0$ is given by

n π + \frac{π}{2}, (n \in I)

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n π - \frac{π}{4}, (n \in I)

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n π - {tan}^{- 1} \frac{4}{3}, (n \in I)

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n π + {tan}^{- 1} \frac{4}{3}, (n \in I)

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Solution

The correct option is C $n π - \frac{π}{4}, (n \in I)$
$7 {cos}^{2} x + sin x cos x - 3 = 0 \Rightarrow 1 + 7 cos 2 x + sin 2 x = 0$
Substituting
$y = tan x \Rightarrow sin 2 x = \frac{2 y}{1 + y^{2}}, cos 2 x = \frac{1 - y^{2}}{1 + y^{2}}$
$\Rightarrow - \frac{2 (3 y^{2} - y - 4)}{y^{2} + 1} = 0 \Rightarrow 3 y^{2} - y - 4 = 0 \Rightarrow (y + 1) (3 y - 4) = 0$
$\Rightarrow y = - 1$ or $3 y - 4 = 0$
$\Rightarrow tan x = - 1$ or $tan x = \frac{4}{3}$
$\Rightarrow x = π n - \frac{π}{4}$ or $x = π m + {tan}^{- 1} \frac{4}{3}$

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The value of x $\in (- 2 π, 2 π)$ such that $\frac{s i n x + i c o s x}{1 + i}$ , where $i = \sqrt{- 1}$ , is purely imaginary are given by

Q. Match the equations with their general solutions. (

n \in I

)

$\begin{matrix} T r i g o n o m e t r i c E q u a t i o n s & G e n e r a l S o l u t i o n s 1. t a n x = 2 & P . 2 n π \pm \frac{2 π}{3}, n \in I 2. s i n x = \frac{\sqrt{3}}{2} & Q . n π - \frac{π}{4}, n \in I 3. c o s x = \frac{- 1}{2} & R . n π + (- 1)^{n} \frac{π}{3}, n \in I 4. c o t x = - 1 & S . n π + t a n^{- 1} (2), n \in I \end{matrix}$

\begin{matrix} T r i g o n o m e t r i c E q u a t i o n s & G e n e r a l S o l u t i o n s 1. 7 c o s^{2} x + s i n^{2} x = 4 & P . n π \pm \frac{π}{4}, w h e r e n \in I 2. s i n^{2} x = \frac{1}{2} & Q . n π \pm \frac{π}{3}, w h e r e n \in I 3. t a n^{2} x + 3 = 0 & R . n π \pm \frac{π}{6}, w h e r e n \in I 4. 3 t a n^{2} x - 1 = 0 & S . N o s o l u t i o n \end{matrix}

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The complete solution of the equation 7cos2x+sinxcosx−3=0 is given by

The correct option is C nπ−π4,(n∈I)7cos2x+sinxcosx−3=0⇒1+7cos2x+sin2x=0Substituting y=tanx⇒sin2x=2y1+y2,cos2x=1−y21+y2⇒−2(3y2−y−4)y2+1=0⇒3y2−y−4=0⇒(y+1)(3y−4)=0⇒y=−1 or 3y−4=0⇒tanx=−1 or tanx=43⇒x=πn−π4 or x=πm+tan−143

The complete solution of the equation $7 {cos}^{2} x + sin x cos x - 3 = 0$ is given by