Domain and Range of Trigonometric Ratios

Q. The maximum value of $cos{\alpha }_1.cos{\alpha }_2......cos{\alpha }_n,$
under the restrictions $0\le {\alpha }_1{\alpha }_2,.....,{\alpha }_n\le \frac{\pi }{2}andcot{\alpha }_1.cot{\alpha }_2......cot{\alpha }_n=1$ is

1. $\frac{1}{2^{\frac{n}{2}}}$
2. $\frac{1}{2}$
3. $\frac{1}{2n}$
4. $1$

Q. If $\theta$ is an acute angle and $sin\theta =\frac{p-6}{8-p},$ then p must satisfy

1. $6\le p$ < 8
2. $6\le p$ < 7
3. $3\le p\le 4$
4. $4\le p$ < 7

Q. If $cose{c}^2\theta =\frac{4xy}{(x+y{)}^2}$, then

1. $x=y$
2. $x=-y$
3. $x=\frac{1}{x}$
4. $x=\frac{1}{y}$

Q. If $A=si{n}^8\theta +co{s}^{14}\theta ,$ then for all real values of $\theta$

1. $A\ge 1$
2. 0 < $A\le 1$
3. 1 < $2A\le 3$
4. $Noneofthese$

Q. The number of ordered solution pairs $(x,y)$ in $[0,2\pi ]$ for the system of the equations $x+y=\frac{2\pi }{3}$ and $\cos x+\cos y=\frac{3}{2}$ is

Q. The maximum value of $cos{\alpha }_1.cos{\alpha }_2......cos{\alpha }_n,$
under the restrictions $0\le {\alpha }_1{\alpha }_2,.....,{\alpha }_n\le \frac{\pi }{2}andcot{\alpha }_1.cot{\alpha }_2......cot{\alpha }_n=1$ is

1. $\frac{1}{2^{\frac{n}{2}}}$
2. $\frac{1}{2}$
3. $\frac{1}{2n}$
4. $1$

Q. If $sinx+cosecx=2,thensi{n}^{n}x+cose{c}^{n}x$ is equal to

1. $2$
2. $2^n$
3. $2^{n-1}$
4. $2^{n-2}$

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If the quadratic equation}{x}^2-2\alpha x+2\alpha =0\text{has two real}& \text{(P)}& 0\\ & \text{and distinct roots}{x}_1\text{and}{x}_2\text{such that}|x_1–x_2|\le 2\sqrt{3},\text{then}& & \\ & \alpha \text{can be}& & \\ \text{(B)}& \text{If}y=\frac{x^2+4x}{x^2+4x+6},\text{then the integers in range of}y\text{is/are}& \text{(Q)}& 1\\ \text{(C)}& \text{If}(2-n)x^2-n<8x+4\forall x\in R,\text{then}\frac{n}{4}\text{can be}& \text{(R)}& 2\\ \text{(D)}& \text{If the equation}\cos 2x+2a\cos x+6a=17\text{has a solution,}& \text{(S)}& 3\\ & \text{then}a\text{can be}& & \\ & & \text{(T)}& -1\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(P)},\text{(Q)}$
2. $\text{(C)}\to \text{(Q)},\text{(R)}$
3. $\text{(D)}\to \text{(Q)},\text{(T)}$
4. $\text{(D)}\to \text{(R)},\text{(S)}$

Q.

If $|cos\theta +\{sin\theta +\sqrt{si{n}^2\theta +si{n}^2\alpha }\}|\le k$, then the value of k is

1. $\sqrt{1+co{s}^2\alpha }$

2. $\sqrt{1+si{n}^2\alpha }$

3. $\sqrt{2+si{n}^2\alpha }$

4. $\sqrt{2+co{s}^2\alpha }$

Q. If $\theta$ is an acute angle and $sin\theta =\frac{p-6}{8-p},$ then p must satisfy

1. $6\le p$ < 8
2. $6\le p$ < 7
3. $3\le p\le 4$
4. $4\le p$ < 7

Q. The quadratic equation $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
(where $x\in R$) has real roots if $p$ lies in

1. $(0,2\pi )$
2. $(-\pi ,0)$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. $(0,\pi ]$

Q.

If $e^{sinx}-e^{-sinx}=a$ has alteast one real solution, then

1. $|a|\in (\frac{e^2-1}{e},\infty )$
2. $|a|\in [0,\frac{e^2-1}{e}]$
3. $|a|\in (e,\infty )$
4. $|a|\in [0,e]$

Q. The number of solutions of the equations $y=\frac{1}{3}[\sin x+[\sin x+\left[\sin x\right]\left]\right]$ and $[y+[y\left]\right]=2\cos x$, where $[.]$ denotes the greatest integer function is

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

Q. If $A=si{n}^8\theta +co{s}^{14}\theta ,$ then for all real values of $\theta$

1. $A\ge 1$
2. 0 < $A\le 1$
3. 1 < $2A\le 3$
4. $Noneofthese$

Q. Let $f\left(x\right)=\log ({\log }_{1/3}\left({\log }_7\right(\sin x+a\left)\right))$ be defined for every value of $x$, then the possible value of $a$ is

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

Q. The quadratic equation $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
(where $x\in R$) has real roots if $p$ lies in

1. $(0,2\pi )$
2. $(-\pi ,0)$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. $(0,\pi ]$

Q.

For 0< $\varphi$ < $\frac{\pi }{2},ifx={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,y={\sum }_{n=0}^{\infty }si{n}^{2n}\varphi ,z={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,then$

1. xyz = xz + y

2. xyz = xy + z

3. xyz = x + y + z

4. xyz = yz + x

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If the quadratic equation}{x}^2-2\alpha x+2\alpha =0\text{has two real}& \text{(P)}& 0\\ & \text{and distinct roots}{x}_1\text{and}{x}_2\text{such that}|x_1–x_2|\le 2\sqrt{3},\text{then}& & \\ & \alpha \text{can be}& & \\ \text{(B)}& \text{If}y=\frac{x^2+4x}{x^2+4x+6},\text{then the integers in range of}y\text{is/are}& \text{(Q)}& 1\\ \text{(C)}& \text{If}(2-n)x^2-n<8x+4\forall x\in R,\text{then}\frac{n}{4}\text{can be}& \text{(R)}& 2\\ \text{(D)}& \text{If the equation}\cos 2x+2a\cos x+6a=17\text{has a solution,}& \text{(S)}& 3\\ & \text{then}a\text{can be}& & \\ & & \text{(T)}& -1\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(R)},\text{(S)}$
2. $\text{(A)}\to \text{(S)},\text{(T)}$
3. $\text{(B)}\to \text{(P)},\text{(Q)}$
4. $\text{(B)}\to \text{(Q)},\text{(T)}$

Q.

$Iff\left(x\right)=co{s}^{2}x+se{c}^{2}x,then$

1. f(x) <1

2. f(x) = 1

3. 1 < f(x) < 2

4. f(x) $\ge$ 2

Q. Let $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ be defined for every real values of $x$, then the range of $a$

1. $(2,6)$
2. $[2,6]$
3. $(-\infty ,2)\cup [6,\infty )$
4. $(-\infty ,2]\cup (6,\infty )$