Equations of the Form acosθ+bsinθ = c

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. Number of solution of the equation ${\sin }^{-1}(4{\sin }^2\theta +\sin \theta )+{\cos }^{-1}(6\sin \theta -1)=\frac{\pi }{2}$ lying in the interval $[0,5\pi ],$ is

1. $7$
2. $9$
3. $6$
4. $5$

Q. The general solution for the equation $\cos x+\sin x=\sqrt{2}$ is ____.

Q.

The number of integral values of k for which the equation $7cosx+5sinx=2k+1$ has a solution is

1. 4
2. 8
3. 10
4. 12

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. $\left(IV\right)\to \left(S\right)$