Equations of the Form acosθ+bsinθ = c

Q.

$\tan \left[\left(\frac{\pi }{4}\right)+\frac{1}{2}{\cos }^{-1}\left(x\right)\right]+\tan \left[\left(\frac{\pi }{4}\right)-\frac{1}{2}{\cos }^{-1}\left(x\right)\right]=1$ has how many solutions?

1. One solution

2. Two solutions

3. Three solutions

4. No solution

Q.

Evaluate : $\left(\frac{\sin 3\theta +\sin 5\theta +\sin 7\theta +\sin 9\theta }{\cos 3\theta +\cos 5\theta +\cos 7\theta +\cos 9\theta }\right)$

1. $\tan \left(3\theta \right)$

2. $\cot \left(3\theta \right)$

3. $\tan \left(6\theta \right)$

4. $\cot \left(6\theta \right)$

Q.

If $\tan (x+y)=33$ and $x={\tan }^{-1}3$ , then y will be

1. $0.3$

2. ${\tan }^{-1}1.3$

3. ${\tan }^{-1}0.3$

4. ${\tan }^{-1}\frac{1}{18}$

Q. The number of integral values of $k$ for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in R$ is

Q.

$\text{The general value of x satisfying the equation}\sqrt{3}sinx+cosx=\sqrt{3}isgivenby$

1. $x=n\pi +(-1{)}^n\frac{\pi }{4}+\frac{\pi }{3},n\in Z$

2. $x=n\pi +(-1{)}^n\frac{\pi }{3}+\frac{\pi }{6},n\in Z$

3. $x=n\pi \pm \frac{\pi }{6},n\in Z$

4. $x=n\pi \pm \frac{\pi }{3},n\in Z$

Q. Find the particular solution of differential equation :

$\frac{dy}{dx}=-\frac{x+y\cos x}{1+\sin x}$, given that $y=1$ when $x=0$.

Q.

The order of the differential equation whose solution is $y=a\cos \left(x\right)+b\sin \left(x\right)+c{e}^{-x}$ is

1. $3$

2. $2$

3. $1$

4. None of these

Q. If $x$ $(|x|<\pi )$ and $y$ are the solutions of the equation $12\sin x+5\cos x=2y^2-8y+21,$ then the value of $12\cot \left(\frac{xy}{2}\right)$ is

Q.

Evaluate the definite integrals.
${\int }_0^1(x{e}^x+sin\frac{\pi .x}{4})dx$

Q. The general solution for the equation $\sqrt{3}\sin x-\cos x=2$ is

1. $(6n+1)\frac{\pi }{6}+(-1{)}^n\frac{\pi }{2},n\in Z$
2. $(6n-1)\frac{\pi }{4}+(-1{)}^n\frac{\pi }{2},n\in Z$
3. $(6n+1)\frac{\pi }{4}+(-1{)}^n\frac{\pi }{2},n\in Z$
4. $(6n-1)\frac{\pi }{6}+(-1{)}^n\frac{\pi }{2},n\in Z$

Q. The value of $A=(1+\tan 1^{\circ })(1+\tan 2^{\circ })(1+\tan 3^{\circ })\cdots (1+\tan {45}^{\circ })\text{is}$

1. $2^{22}$
2. $2^{23}$
3. $2^{44}$
4. $-2^{23}$

Q. General solution of the equation $4\cos x-3\sec x=\tan x,(\cos x\ne 0)$ can be

1. $n\pi +(-1{)}^n\alpha ,\sin \alpha =\frac{-1+\sqrt{17}}{8},n\in Z$
2. $n\pi +(-1{)}^n\beta ,\sin \beta =\frac{-1-\sqrt{17}}{8},n\in Z$
3. $n\pi +(-1{)}^n\alpha ,\sin \alpha =\frac{1+\sqrt{17}}{8},n\in Z$
4. $n\pi +(-1{)}^n\beta ,\sin \beta =\frac{1-\sqrt{17}}{8},n\in Z$

Q. If the equation $2{\sin }^{2}x+\frac{\sin 2x}{2}=k$ has atleast one real solution, then the sum of all integral value of $k$ is

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

Q.

$Avalueof\theta satisfyingcos\theta +\sqrt{3}sin\theta =2is$

1. $\frac{5\pi }{3}$

2. $\frac{4\pi }{3}$

3. $\frac{2\pi }{3}$

4. $\frac{\pi }{3}$

Q. If the straight line through the point P(3, 4) makes an angle $\frac{\pi }{6}$ with the x–axis and meets the line 12x + 5y + 10 = 0 at Q then the length PQ is

1. 
2. 
3. 
4. 

Q.

Discuss the continuity of the following functions :

(b) f(x) = sin x + cos x

Q.

$\frac{\sqrt{sinA}-\sqrt{sinB}}{\sqrt{sinA}+\sqrt{sinB}}=\frac{a+b-2\sqrt{ab}}{a-b}$

Q. The area enclosed by the curves $y=|sinx+cosx|$ and $y=|cosx-sinx|$ in $[0,\frac{\pi }{2}]$ is (in sq. units)

1. $4-2\sqrt{2}$
2. $4-\sqrt{2}$
3. $8-\sqrt{2}$
4. $8-2\sqrt{2}$

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

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

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 CORRECT combination?

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

Q. Let $\alpha ,\beta$ be two distinct roots of $a\cos \theta +b\sin \theta =c$, where $a,b$ and $c$ are three real constants and $\theta \in [0,2\pi ]$. Then $\alpha +\beta$ is also a root of the same equation if

1. $c+a=b$
2. $c=a$
3. $a+b=c$
4. $b+c=a$

Q.

Find $\frac{dy}{dx}$, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x=\frac{si{n}^{3}t}{\sqrt{cost2t}},y=\frac{co{s}^{3}t}{\sqrt{cos2t}}$

Q. If $\alpha ,\beta$ are the solution of a $cos2\theta +b$ $sin2\theta =c$, then $tan\alpha tan\beta$

1. 
2. 
3. 
4. 

Q.

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

1. 

2. 

3. 

4. 

Q.

If $acos2\theta +bsin2\theta =c$ has $\alpha and\beta$ as its roots, that prove that

(i) $tan\alpha +tan\beta =\frac{2b}{a+c}$

(ii) $tan\alpha tan\beta =\frac{c-a}{c+a}$

(iii) $tan(\alpha +\beta )=\frac{b}{a}$

Q.

$If\sqrt{3}cos\theta +sin\theta =\sqrt{2},thengeneralvalueof\theta is$

1. $n\pi +(-1{)}^n\frac{\pi }{4},n\in Z$

2. $(-1{)}^n\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

3. $n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

4. $n\pi +\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

Q.

$\text{Write the set of values of a for which the equation}\sqrt{3}sinx-xcosx=ahasnosolution.$

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