Range of Trigonometric Expressions

Q.

The value of $\cos 1°·\cos 2°...\cos 100°$ is

1. $-1$

2. $0$

3. $1$

4. None of these

Q. Let $x,y,z$ be elements from the interval $(0,2\pi )$ satisfying the inequality $(4+\sin 4x)(2+{\cot }^{2}y)(1+{\sin }^{4}z)\le 12{\sin }^{2}z.$ Which of the following statements is FALSE?

1. The number of ordered pairs $(x,y)$ is $8.$
2. The number of ordered pairs $(y,z)$ is $4.$
3. The number of ordered pairs $(z,x)$ is $4.$
4. The number of ordered triplets $(x,y,z)$ is $16.$

Q.

Minimum value of $5si{n}^2\theta +4co{s}^2\theta$ is

1. 1

2. 2

3. 3

4. 4

Q. If $\cos \alpha =x+\frac{1}{x},x\in R-\left\{0\right\}$, then $\alpha \in$

1. $(0,\frac{\pi }{2})$
2. $(\frac{\pi }{2},\pi )$
3. $\left\{\frac{\pi }{2}\right\}$
4. $\varphi$

Q. The range of the function $|\sec x|+5$ is

1. $[5,\infty )$
2. $[4,\infty )$
3. $[6,\infty )$
4. $(6,\infty )$

Q. If $y=\cot \theta ({\sin }^2\theta +\sin \theta \cos \theta )$, then
(where $\theta \ne n\pi ,n\in Z$)

1. $y_{min}=\frac{1-\sqrt{2}}{4}$
2. $y_{min}=\frac{1-\sqrt{2}}{2}$
3. $y_{max}=\frac{1+\sqrt{2}}{4}$
4. $y_{max}=\frac{1+\sqrt{2}}{2}$

Q.

Let n be a positive integer such that $sin\frac{\pi }{2^n}+cos\frac{\pi }{2^n}=\frac{\sqrt{n}}{2}$. Then

1. $6\le n\le 8$

2. 4 < $n\le 8$

3. $4\le$ n < 8

4. 4 < n < 8

Q. Consider the quadratic equation $x^2+2x\sin a+\cos a+1=0,a\in R.$ If
$D$ denotes the discriminant of the given quadratic equation, then which of the following is/are correct?

1. Maximum value of $D$ is $8$
2. Minimum value of $D$ is $-8$
3. Maximum value of $D$ occurs when $\cos a=0$
4. Maximum value of $D$ occurs when $\cos a=-\frac{1}{2}$

Q. The maximum value of $(\sin \theta \cos \theta {)}^{42}$ is

1. $\frac{1}{2^{41}}$
2. $\frac{1}{2^{42}}$
3. $\frac{1}{2^{43}}$
4. $\frac{1}{2^{40}}$

Q. Let $x,y,z$ be real numbers with $x\ge y\ge z\ge \frac{\pi }{12}$ such that $x+y+z=\frac{\pi }{2}$. If $p=\cos x\sin y\cos z,$ then

1. the maximum value of $p$ is $\frac{2+\sqrt{3}}{8}$
2. the minimum value of $p$ is $\frac{1}{8}$
3. the maximum value of $p$ is attained when $x=y=\frac{5\pi }{24}$ and $z=\frac{\pi }{12}$
4. the minimum value of $p$ is attained when $x=y=z=\frac{\pi }{6}$

Q.

If x and y are the maximum and minimum values of sin $\theta$ + cos $\theta$, find the value of $x^2$ + $y^2$

__

Q. The number of integral value(s) in the range of the expression $\frac{6\tan x+4-4{\tan }^{2}x}{1+{\tan }^{2}x}$ is

Q. Which among the following option(s) is/are NOT possible?

1. $\sin \theta =2020$
2. $\tan \theta =2020$
3. $\sec \theta =\frac{2020}{2021}$
4. $\cos \theta =\frac{2020}{2021}$

Q. The number of integral values of $p$, for which the equation $5\cos x+4\sin x=2p+3$ has real solutions is

Q. The range of the function $2\sin x+7$ is

1. $[0,7]$
2. $[-7,7]$
3. $[5,7]$
4. $[5,9]$

Q. The least value of $3{\sin }^2\theta +4{\cos }^2\theta$ is

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

Q. The minimum value of $({\sin }^2\theta +{\cos }^2\theta +{\sec }^2\theta +{\text{cosec}}^2\theta +{\tan }^2\theta +{\cot }^2\theta )$ is

Q.

Let n be a positive integer such that $sin\frac{\pi }{2^n}+cos\frac{\pi }{2^n}=\frac{\sqrt{n}}{2}$. Then

1. $6\le n\le 8$

2. 4 < $n\le 8$

3. $4\le$ n < 8

4. 4 < n < 8

Q. If $a=3{\cos }^{2}A+{\sin }^{4}A$, then which of the following is/are correct?

1. The minimum value of $a$ is $1$
2. The minimum value of $a$ is $0$
3. The maximum value of $a$ is $3$
4. The maximum value of $a$ is $4$

Q. If $p=4\cos (x-\frac{\pi }{3})+3\sqrt{3}\sin x,$ then the maximum value of $\left[p\right]$ is
(where [.] denotes greatest integer function)

Q. If $A,B,C$ are acute positive angles such that $A+B+C=\pi$ and $\cot A\cot B\cot C=k$, then

1. $k<\frac{1}{9}$
2. $k>\frac{1}{3}$
3. $k\le \frac{1}{3\sqrt{3}}$
4. $k\ge \frac{1}{3\sqrt{3}}$

Q. If ${\log }_{3x}\left(45\right)={\log }_{4x}\left(40\sqrt{3}\right),$ then the integral part of ${\log }_3{x}^3$ is equivalent to

1. Maximum value of $({\sin }^{82}x+{\cos }^{164}x+2)$
2. Number of solutions of $4\sin x=x$ in $x\in [0,2\pi ]$
3. Number of real roots of the equation $2x^{98}+5x^{97}+5x^{96}+...+5x^2+5x+3=0$
4. Value of $\left(\frac{a_{100}-3a_{98}}{24a_{99}}\right)$ if $a_n={\alpha }^n-{\beta }^n,$ where $n\ge 1$ and $\alpha ,\beta$ are the roots of $x^2-72x-3=0$

Q. The value of $a$ for which the equation $2\sin 2\theta -\sqrt{a}\cos 2\theta =\sqrt{2}+\sqrt{2-a}$ has solution in $\theta$ is

1. $(3,\infty )$
2. $(0,\infty )$
3. $[1,2]$
4. $[\sqrt{5}-1,2]$

Q. The expression $\frac{36}{4\sin (x+\frac{\pi }{3})-4\sqrt{3}\cos x+8}$ lies in the interval

1. $[4,12]$
2. $[-4,4]$
3. $(4,12)$
4. $[3,9]$

Q. Which of the following quadratic equations does not have both of its roots lying in the range of $y=3\sin x$?

1. $x^2-4=0$
2. $x^2+x-2=0$
3. $3x^2-x=0$
4. $x^2-2x-8=0$

Q.

The maximum value of 3cos $\theta$ - 4 sin$\theta$ is

1. 3

2. 4

3. 5

4. None of these

Q. $\mathrm{The}\mathrm{minimum}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{function}f\left(x\right)={\cos }^{2}x+{\sin }^{4}x\mathrm{is}$.

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

Q. What is the least value of $\frac{25{\sec }^{4}x-50{\sec }^{2}x+74}{{\tan }^{2}x}$ ?

1. $50$
2. $70$
3. $75$
4. $90$

Q.

$\alpha ,\beta ,\gamma$ are real numbers satisfying $\alpha +\beta +\gamma =\pi$.

The value of the given expression

$sin\alpha +sin\beta +sin\gamma$ is

1. Zero

2. -3

3. Positive

4. Negative

Q. If $P=4\sin x+{\cos }^{2}x$, then which of the following is/are true ?

1. Maximum value of $P$ is $5$
2. Minimum value of $P$ is $-4$
3. Maximum value of $P$ occurs when $\sin x=0$
4. Minimum value of $P$ occurs when $\sin x=-1$