JEE Main 2021 — Mathematics Algebra

$$|\begin{matrix}1+{\mathrm{cos}}^{2}\theta & {\mathrm{sin}}^{2}\theta & 4\mathrm{sin}3\theta \\ {\mathrm{cos}}^{2}\theta & 1+{\mathrm{sin}}^{2}\theta & 4\mathrm{sin}3\theta \\ {\mathrm{cos}}^{2}\theta & {\mathrm{sin}}^{2}\theta & 1+4\mathrm{sin}3\theta \end{matrix}|=0$$∣1+cos2θcos2θcos2θ​sin2θ1+sin2θsin2θ​4sin3θ4sin3θ1+4sin3θ​∣=0

$${R}_{3}\rightarrow {R}_{3}-{R}_{2}$$R3​→R3​−R2​

$$|\begin{matrix}1+{\mathrm{cos}}^{2}\theta & {\mathrm{sin}}^{2}\theta & 4\mathrm{sin}3\theta \\ {\mathrm{cos}}^{2}\theta & 1+{\mathrm{sin}}^{2}\theta & 4\mathrm{sin}3\theta \\ 0 & -1 & 1\end{matrix}|$$∣1+cos2θcos2θ0​sin2θ1+sin2θ−1​4sin3θ4sin3θ1​∣

$${C}_{2}\rightarrow {C}_{2}+{C}_{1}$$C2​→C2​+C1​

$$|\begin{matrix}1+{\mathrm{cos}}^{2}\theta & 2 & 4\mathrm{sin}3\theta \\ {\mathrm{cos}}^{2}\theta & 2 & 4\mathrm{sin}3\theta \\ 0 & -1 & 1\end{matrix}|$$∣1+cos2θcos2θ0​22−1​4sin3θ4sin3θ1​∣

$${C}_{2}\rightarrow {C}_{2}+{C}_{3}$$C2​→C2​+C3​

$$|\begin{matrix}1+{\mathrm{cos}}^{2}\theta & 2+4\mathrm{sin}3\theta & 4\mathrm{sin}3\theta \\ {\mathrm{cos}}^{2}\theta & 2+4\mathrm{sin}3\theta & 4\mathrm{sin}3\theta \\ 0 & 0 & 1\end{matrix}|=0$$∣1+cos2θcos2θ0​2+4sin3θ2+4sin3θ0​4sin3θ4sin3θ1​∣=0

Expanding along ${R}_{3}$R3​

$$\Rightarrow (1+{\mathrm{cos}}^{2}\theta )(2+4\mathrm{sin}3\theta )-(2+4\mathrm{sin}3\theta )({\mathrm{cos}}^{2}\theta )=0$$⇒(1+cos2θ)(2+4sin3θ)−(2+4sin3θ)(cos2θ)=0

$$\Rightarrow (2+4\mathrm{sin}3\theta )(1+{\mathrm{cos}}^{2}\theta -{\mathrm{cos}}^{2}\theta )=0$$⇒(2+4sin3θ)(1+cos2θ−cos2θ)=0

$$\Rightarrow \mathrm{sin}3\theta =\frac{-1}{2}$$⇒sin3θ=2−1​

$$\Rightarrow 3\theta =\frac{7\pi }{6}$$⇒3θ=67π​

$$\Rightarrow \theta =\frac{7\pi }{18}$$⇒θ=187π​