Graph of Trigonometric Ratios

Q. If range of the function $f\left(x\right)={\sin }^{-1}x+2{\tan }^{-1}x+x^2+4x+1$ is $[p,q]$, then the value of $(p+q)$ is

Q. The number of points of intersection of the curve $y=\sin 3x$ with $x$-axis in the interval $(0,\pi )$ is

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

Q. If $f\left(x\right)=\sin \left(\sqrt{\left[\lambda \right]}x\right)$ is a periodic function with fundamental period $\pi ,$ (where $[.]$ denotes the greatest integer function), then $\lambda$ belongs to

1. $(2,3)$
2. $[4,5)$
3. $[2,3)$
4. $(4,5)$

Q. If $x=\sin 2^{\circ }$ and $y=\sin 2$ (where $2$ is in radian), then the correct relation between $x$ and $y$ is

1. $x=y$
2. $x>y$
3. $x<y$
4. $x=2y$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Which of the following option(s) is/are correct?

निम्नलिखित में से कौनसा/कौनसे विकल्प सही है/हैं?

1. sin1 < cos2
2. sin1 > cos2
3. sin1° > sin1
4. cos1° > cos1

Q. If $x=\sin 1+\cos 1,y=\sqrt{\sin 1}+\sqrt{\cos 1}$ and $z=\sin 1-\cos 1,$ then which among the following relation between $x,y,z$ is correct ?

1. $x>y>z$
2. $y>x>z$
3. $z>x>y$
4. $z>y>x$

Q. $\mathrm{The}\mathrm{function}f\left(x\right)=\cot x\mathrm{is}\mathrm{defined}\mathrm{for}\mathrm{all}x\in R~\mathrm{n\pi },n\in Z.$

1. False
2. True

Q. Let $f\left(x\right)=\cos (x+\frac{\pi }{3})$ and $g\left(x\right)=\sin (x-\frac{\pi }{6})$ be two functions. Which of the following is/are correct ?
( where $x\in [0,2\pi ]$ )

1. $f\left(x\right)=g\left(x\right)$ has two solutions
2. Period of $f\left(x\right)$ is $\frac{5\pi }{3}$
3. Period of $g\left(x\right)$ is $\frac{11\pi }{6}$
4. $g\left(x\right)>f\left(x\right)$ when $x\in (\frac{\pi }{6},\frac{7\pi }{6})$

Q.

How many of following statements are true?

(1)Period of sin $\theta$ is $\pi$, because sin 0 and sin $\pi$ = 0

(2) Period of cos $\theta$ is $\pi$

(3) Period of tan $\theta$ is $\pi$

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Q. The function $f\left(x\right)=\sin \frac{\pi x}{2}+2\cos \frac{\pi x}{3}-\tan \frac{\pi x}{4}$ is periodic with period

1. $3$
2. $4$
3. $12$
4. $6$

Q. $\mathrm{The}\mathrm{function}f\left(x\right)=\mathrm{cosec}x\mathrm{is}\mathrm{discontinuous}\mathrm{at}x=\mathrm{n\pi },n\in Z$.

1. False
2. True

Q. The value of $\sqrt{\frac{1+\sin A}{1-\sin A}}$ where $A\in [0,2\pi ]-\left\{\frac{\pi }{2}\right\}$ is

1. $\sec A-\tan A\text{if}A\in [0,\frac{\pi }{2})$
2. $-(\sec A+\tan A)\text{if}A\in (\frac{\pi }{2},\pi ]$
3. $-\sec A+\tan A\text{if}A\in [\pi ,\frac{3\pi }{2})$
4. $\sec A+\tan A\text{if}A\in (\frac{3\pi }{2},2\pi ]$

Q. Let $f\left(x\right)$ be a $\text{sine}$ function with fundamental period $3\pi$ and it passes through the origin. If the maximum value of $f\left(x\right)$ is $10$, then $f\left(x\right)$ is

1. $f\left(x\right)=\pm 10\sin \left(\frac{3x}{2}\right)$
2. $f\left(x\right)=10\sin \left(\frac{2x}{3}\right)$
3. $f\left(x\right)=\pm 10\sin \left(\frac{2x}{3}\right)$
4. $f\left(x\right)=10\sin \left(\frac{3x}{2}\right)$

Q. Fundamental period of $f\left(x\right)=|\sin 2x|$ is

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

Q.

$\begin{array}{ccc}\text{Function}& \text{Domain}& \text{Range}\\ \text{(i) sinx}& \text{(P) R}& \text{(L) [-1, 1]}\\ \text{(ii) cos x}& \text{(Q) R-n}\pi & \text{(M) R}\\ \text{(iii) tan x}& \text{(R) R- {(2n+1)}}& \text{(N)}(-\infty ,\text{-1]}\cup \text{[1,}\infty )\\ \text{(iv) cosec x}& & \\ \text{(v) sec x}& & \\ \text{(vi) cot x}& & \end{array}$

How many of the following are matched correct?

(i) -P-L, (ii) -P-L, (iii)-R-M, (iv) -Q-N, (v) -R-N, (vi) -Q-M

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Q. If $A=\tan 1,B=\tan 2$ and $C=\tan 3,$ then the descending order of $A,B$ and $C$ is

1. $A,B,C$
2. $C,B,A$
3. $A,C,B$
4. $B,C,A$

Q. The number of solution of the equation $\left|\cos (x-\frac{\pi }{6})\right|=|\sin x|$ in $[0,\pi ]$ is

Q. Which of the following statement(s) is/are true for the curve $f\left(x\right)=\cos 2x$

1. Graph of $f\left(x\right)$ is symmetrical with respect to $y-$axis
2. Range of $f\left(x\right)$ is $[-2,2]$
3. Range of $f\left(x\right)$ is $[-1,1]$
4. Period of $f\left(x\right)$ is $2\pi$