Chapter 3

Solve the following equations and tick the correct one. The number of solutions of the cquation \(2\left(\sin ^{4} 2 x+\cos ^{4} 2 x\right)+3 \sin 2 x \cos 2 x=0\) is (a) 0 (b) 1 (c) 2 (d) 3

Solve the following equations and tick the correct one. \(\cos 2 x+a \sin x=2 a-7\) possesses a solution for (a) all \(a\) (b) \(a>6\) (c) \(a \leq 2\) (d) \(a \in[2,6]\)

Solve the following equations and tick the correct one. If \(0

Solve the following equations and tick the correct one. If \(1+\sin \theta+\sin ^{2} \theta+\ldots \ldots\) to \(\infty=4+2 \sqrt{3}\), \(0<\theta<\pi, \theta \neq \frac{\pi}{2}\), then (a) \(\theta=\frac{\pi}{6}\) (b) \(\theta=\frac{\pi}{3}\) (c) \(\theta=\frac{\pi}{3}\) or \(\frac{\pi}{6}\) (d) \(\frac{\pi}{3}\) or \(\frac{2 \pi}{3}\)

Solve the following equations and tick the correct one. If \(\tan (\pi \cos \theta)=\cot (\pi \sin \theta)\), then the value of \(\cos \left(\theta-\frac{\pi}{4}\right)\) is (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{2 \sqrt{2}}\) (d) None

Solve the following equations and tick the correct one. The most general values of \(x\) for which \(\sin x+\cos x=\min _{a \in R}\left\\{1, a^{2}-4 a+6\right\\}\) are given by (a) \(2 n \pi, n \in N\) (b) \(2 n \pi+\frac{\pi}{2}, n \in N\) (c) \(n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{4}, n \in N\) (d) None

Solve the following equations and tick the correct one. The number of roots of the equation \(x+2 \tan x=\frac{\pi}{2}\) in the interval \([0,2 \pi]\) is (a) 1 (b) 2 (c) 3 (d) \(\infty\)

Solve the following equations and tick the correct one. The number of solutions of the equation \(\cos (\pi \sqrt{x-4})\) \(\cos (\pi \sqrt{x})=1\) is (a) None (b) 1 (c) 2 (d) \(>2\)

Solve the following equations and tick the correct one. The number of solutions of the equation \(\sin \left(\frac{\pi x}{2 \sqrt{3}}\right)=x^{2}-2 \sqrt{3} x+4\) (a) forms an empty set (b) only one (c) is only two (d) is greater then two

Solve the following equations and tick the correct one. Number of real roots of the equation \(\sec \theta+\operatorname{cosec} \theta=\sqrt{15}\) lying between 0 and \(2 \pi\) is (a) \(\underline{8}\) (b) 4 (c) 2 (d) 0

Solve the following equations and tick the correct one. The general solution of the equation \(\sin 100 x-\cos 100 x=1\), is (a) \(2 n \pi+\frac{\pi}{3}, n \in z\) (b) \(n \pi+\frac{\pi}{2}, n \in z\) (c) \(n \pi+\frac{\pi}{4} \pi, n \in z\) (d) \(2 n \pi \frac{\pi}{3}, n \in z\)

Solve the following equations and tick the correct one. The general solution of the equation \(2^{\cos ^{2} x}+1=3.2^{-\sin ^{2} x}\) is (a) \(n \pi \cdot n \in z\) (b) \((n+1) \pi, n \in z\) (c) \((n-1) \pi, n \in z\) (d) None

Solve the following equations and tick the correct one. If \(x \in(0,1)\), the greatest root of the equation \(\sin 2 \pi x=\sqrt{2} \cos \pi x\) is (a) \(\frac{1}{4}\) (b) \(\frac{1}{2}\) (c) \(\frac{3}{4}\) (d) None

Solve the following equations and tick the correct one. The number of solutions of tan \((5 \pi \cos \alpha)\) \(=\cot (5 \pi \sin \alpha)\) for \(\alpha \in(0,2 \pi)\) is (a) 7 (b) 14 (c) 21 (d) 3

Solve the following equations and tick the correct one. The number of solution of the equation \(1+\sin x \cdot \sin ^{2}\left(\frac{x}{2}\right)=0\) in \([-\pi, \pi]\) is (a) 0 (b) \(\underline{1}\) (c) 2 (d) 3

Solve the following equations and tick the correct one. The real roots of the equation \(\cos ^{7} x+\sin ^{4} x=1\) in \((-\pi, \pi)\) are (a) \(-\frac{\pi}{2}, 0\) (b) \(-\frac{\pi}{2}, 0, \frac{\pi}{2}\) (c) \(\frac{\pi}{2}, 0\) (d) \(0, \frac{\pi}{4}, \frac{\pi}{2}\)