Equations of the Form acosθ+bsinθ = c

Q.

If $y={\cos }^{-1}\left[\frac{3\cos x-4\sin x}{5}\right]$ then $\frac{dy}{dx}$ equals to

1. $0$

2. $1$

3. $-1$

4. $\frac{1}{2}$

Q.

If ${\sin }^{-1}x+{\sin }^{-1}2x=\frac{\pi }{3}$ then $x=$

1. $\frac{\sqrt{3}}{2\sqrt{7}}$

2. $\sqrt{\frac{3}{7}}$

3. $0$

4. $1$

Q. Let $S=\left\{xϵ\right(-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\}$. The sum of all distinct solutions of the equation $\sqrt{3}secx+cosecx+2(tanx-cotx)=0$ in the set S is equal to

1. $-\frac{7\pi }{9}$
2. $-\frac{2\pi }{9}$
3. $0$
4. $\frac{5\pi }{9}$

Q. The equation $\sqrt{3}\sin x+\cos x=4$ has

1. only one solution
2. two solutions
3. infinitely many solutions
4. no solution

Q. The general solution of $\sin x+\cos x=1$ is

1. $x=2n\pi ,n\in Z$
2. $x=2n\pi +\frac{\pi }{2},n\in Z$
3. $x=\left\{2n\pi \right\}\cup \{2n\pi +\frac{\pi }{2}\},n\in Z$
4. $x=\varphi$

Q.

If sin x + cos x = t, then sin 3x - cos 3x is equal to

1. 3t (2$t^2$ + 1)

2. t (2$t^2$ - 3)

3. t (2$t^2$ + 3)

4. -2t $(3t^2$ + 1)

Q. The number of solutions of $5{\cos }^2\theta -3{\sin }^2\theta +6\sin \theta \cos \theta =7$ in the interval $[0,2\pi ]$ is

1. $2$
2. $4$
3. $0$
4. $1$

Q. $\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{Number of solutions of the equation}& \text{(P)}& 0\\ & e^x+e^{-x}=\tan x\forall x\in [0,\frac{\pi }{2})& & \\ \text{(II)}& \text{Number of solutions of the equations}& \text{(Q)}& 1\\ & x+y=\frac{2\pi }{3}\text{and}\cos x+\cos y=\frac{3}{2}\text{is}& & \\ \text{(III)}& \text{Number of solutions of the equation}& \text{(R)}& 2\\ & \cos x+2\sin x=1,x\in [0,2\pi )\text{is}& & \\ \text{(IV)}& \text{Number of solutions of the equation}& \text{(S)}& \text{Infinite}\\ & (\sqrt{3}\sin x+\cos x{)}^{\sqrt{\sqrt{3}\sin 2x-\cos 2x+2}}=4\text{is}& & \end{array}$

Which of the following is only INCORRECT combination?

1. $\left(I\right)\to \left(Q\right)$
2. $\left(II\right)\to \left(P\right)$
3. $\left(III\right)\to \left(S\right)$
4. $\left(IV\right)\to \left(S\right)$

Q.

Find the value of $xϵ(0,\pi )$ which satisfies the equation sin x +$\sqrt{3}$ cos x = $\sqrt{2}$

1. $\frac{\pi }{12}$

2. $\frac{5\pi }{12}$

3. $\frac{11\pi }{12}$

4. $\frac{7\pi }{12}$

Q. The general solution of $\sin x-\cos x=\sqrt{2},$ for any integer n is

1. $n\pi$
2. $2n\pi +\frac{3\pi }{4}$
3. $2n\pi$
4. $(2n+1)\pi$

Q.

If sin 3x + cos 2x = -2, then find the value of x

1. (2m+1) $\frac{\pi }{2}$

2. (4m-1) $\frac{\pi }{6}$

3. (4m+1) $\frac{\pi }{2}$

4. (2m+1) $\frac{\pi }{6}$