Graph of Trigonometric Ratios

Q.

If cos a . cos 2a . cos 3a ............... cos999a = $\frac{1}{2^x}$.Where a = $\frac{2\pi }{1999}$.Find the value of x.

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Q.

Find the number of solutions of $e^x=x^4$.

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Q. 16.Prove that CotA +cosecA-1/cot A-cosec A+1=1+cosA/1+sinA

Q. 19. Draw the graph of the function Y = sin x + cos x for (-/2 ≤ x≤ /2)

Q. Sec 50 x Sin40+ Cos40 x Cosec5

Q. What is the minimum value of (sin theta plus cos theta) when theta lies between 0 and 90 degree?

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. if tan theta + 4 = 3 ( 4 cot theta +1) where 0° < theta < 90°, then the value of 15/sec theta cosec theta

Q.

If cos$\alpha$ is a root of $25x^2$ + 5x - 12 = 0, (-1 $<$ x $<$ 0). Then a possible value of sin (2α) is

1. 

2. 

3. -

4. -

Q. If sin θ : a = cos α : 1 and tan θ : b = cot α : 1, then the value of (a sin α) : b equals

Q.

In a triangle PQR, R = $\frac{\pi }{2}$ If tan $\frac{P}{2}$and tan$\frac{Q}{2}$are the roots of the equation \( ax^2\) +bx+c=0 (a$\ne$0), then:

1. a+b=c

2. b+c=a

3. b=c

4. a+c=b

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

The expression 2 cos2α – 1 can be simplified to

व्यंजक 2 cos2α – 1 का सरलीकरण रूप हो सकता है

1. 1 – 4 sin²α
2. 4 cos²α – 3
3. $\frac{\cos 3\alpha }{\cos \alpha }$
4. 3 – 4 sin²α

Q. $\mathrm{The}\mathrm{function}f\left(x\right)=\sec x\mathrm{is}\mathrm{continuous}\mathrm{for}\mathrm{all}x\in R.$

1. False
2. True

Q. At a given instant, the legs of a right angled triangle are $8\text{cm}$ and $6\text{cm}.$ If the first leg decreases at the rate of $1\text{cm}/\text{sec}$ and the second leg increases at the rate of $2\text{cm}/\text{sec}$ respectively, then the rate of change in the area after $2sec$ is equal to $\left({\text{in cm}}^2/\text{sec}\right)$

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 P = sin a .sin 2a . sin 3a . . . . . . sin 999a &

Q = sin 2a .sin 4a . sin 8a . . . . . . sin 1998a

Find the value of $\frac{Q}{P}$ Where a = $\frac{2\pi }{1999}$

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Q. In a triangle $ABC$, suppose that $\tan \frac{A}{2},\tan \frac{B}{2}$ and $\tan \frac{C}{2}$ are in harmonic progression.
What is the maximum possible value of angle $B$ ?

1. $\frac{\pi }{6}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. Match the following graphs with respective trigonometric ratio:

1. $\sin x$
2. $\cos x$
3. $\tan x$

Q.

If the lines represented by the equation $a{x}^2-bxy-y^2=0$ make angles $\alpha$ and $\beta$ with the x - axis, then $\tan (\alpha +\beta )$ =

1. 

2. 

3. 

4. None of these

Q.

Function Domain Range

(i)sinx (P) R (L) [-1, 1]

(ii) cosx (Q) R - {n $\pi$} (M) R

(iii) tanx (R) R- {(2n + 1})$\frac{\pi }{2}$} (N) $(-\infty$, -1] u [1, $\infty$)

(iv) cosecx

(v) secx

(vi) cotx

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. Find the derivative of the following function:
f(x)= $(a{x}^2+sinx)(p+qcosx)$

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

Given sinθ₁ + sinθ₂ + sinθ₃ = 3, which of the following is/are correct?

दिया गया है sinθ₁ + sinθ₂ + sinθ₃ = 3, निम्नलिखित में से कौनसा/कौनसे विकल्प सही है/हैं?

1. cosθ₁ + cosθ₂ + cosθ₃ = 0
2. sinθ₁ + cosθ₂ + cosθ₃ = 1
3. cosθ₁ + sinθ₂ + cosθ₃ = 2
4. cosθ₁ + cosθ₂ + sinθ₃ = 1

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

Which of the following(s) is/are correct for f(x) = 3 sin2x?

निम्नलिखित में से कौनसा/कौनसे विकल्प f(x) = 3 sin2x के लिए सही है/हैं?

1. Domain of f(x) is R
   
   f(x) का प्रांत R है
2. Range of f(x) is [–3, 3]
   
   f(x) का परिसर [–3, 3] है
3. Domain of f(x) is $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
   
   f(x) का प्रांत $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ है
4. Range of f(x) is [–1, 1]
   
   f(x) का परिसर [–1, 1] है

Q.

In any $\Delta ABC,$ prove that

$\frac{(b-c)}{b+c}=\frac{tan\frac{1}{2}(B-C)}{tan\frac{1}{2}(B+C)}$

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. Sketch the graph of the following pairs of function on the same axes:

(i) $y=\sin x,y=\sin \left(x+\frac{\mathrm{\pi }}{4}\right)$
(ii) y = sin x, y = sin 3x

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

Given $\cos \theta =\frac{1}{3},(\theta \in R)$, then sinθ can be

दिया है $\cos \theta =\frac{1}{3},(\theta \in R)$, तब sinθ हो सकता है

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

Q. The number of solution of $\cos (x+\frac{\pi }{4})=\frac{1}{5}$ in $[0,2\pi ]$ is

Q.

How many of following statements are true?
(1)Period of $sin\theta$is $pi$, because sin 0 = 0 and $sin\pi =0$,
(2) Period of $cos\theta$ is $\pi$
(3) Period of $tan\theta$ is $\pi$

Q. Write the least value of cos² θ + sec² θ.