Odd Extension of a Function

Q.

Solve for x

Cos^{​​​​-1}[(x^{​​​​​​2}​​​-1)/(x^​​​2+1)] + Tan^​-1[2x/(x^{​​​​​​2}​​​​​-1)] = 2π/3

^{​​​​​​}

Q. If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\mathrm{\pi }}{3}$ and ${\cos }^{-1}x-{\cos }^{-1}y=\frac{\mathrm{\pi }}{6}$, find the values of x and y.

Q. If ${\left(\cos x\right)}^y={\left(\cos y\right)}^x,\mathrm{find}\frac{dy}{dx}$

Q. Differentiate from first principle:
$\left(i\right){\tan }^{2}x$

Q.

Odd extension is obtained by replacing x by (-x) in the equation of f(x).

1. True

2. False

Q. I want the solution for sin9A=sinA.

Q.

If f(x) = $x^3+x^2$ for 0 $\le x\le 2$ then the odd extension of f(x) would be -

x +2 for 2 < x $\le$ 4

1. x³ + x² for -2 ≤ x ≤ 0

   f(x) =

   x +2 for -4 ≤ x < -2

2. -x³ + x² for -2 ≤ x ≤ 0

   f(x) =

   -x +2 for -4 ≤ x < -2

3. None of these

4. x³ - x² for -2 ≤ x ≤ 0

   f(x) =

   x -2 for -4 ≤ x < -2

Q. If $f\left(x\right)$ and $g_a\left(x\right)$ are real valued function defined as $g_a\left(x\right)=\frac{a^x}{a^x+\sqrt{a}},$ where $a>0$ and $f\left(x\right)=g_9\left(x\right)$, then the value of $\left[\sum _{r=1}^{1995}f\left(\frac{r}{1996}\right)\right]$ is
(where [.] denotes greatest integer function)

Q. If $g:[2:2]\to R$ where $g\left(x\right)=x^3+tanx+\left[\frac{x^2+1}{p}\right]$ is a odd function then the value of parametric P is

1. -5<P<5
2. None of these
3. P<5
4. P>5

Q. The number of values of x in the interval [0, 5 π] satisfying the equation is
(a) 0
(b) 5
(c) 6
(d) 10

Q.

Solve the following equations for x:
(i) tan⁻¹2x + tan⁻¹3x = nπ + $\frac{3\mathrm{\pi }}{4}$

(ii) tan⁻¹(x + 1) + tan⁻¹(x − 1) = tan⁻¹$\frac{8}{31}$

(iii) ${\tan }^{-1}\frac{1}{4}+2{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{6}+{\tan }^{-1}\frac{1}{x}=\frac{\mathrm{\pi }}{4}$

(iv) sin⁻¹x + sin⁻¹2x = $\frac{\mathrm{\pi }}{3}$

(v) $3{\sin }^{-1}\frac{2x}{1+x^2}-4{\cos }^{-1}\frac{1-x^2}{1+x^2}+2{\tan }^{-1}\frac{2x}{1-x^2}=\frac{\mathrm{\pi }}{3}$

(vi) ${\cos }^{-1}x+{\sin }^{-1}\frac{x}{2}=\frac{\mathrm{\pi }}{6}$

(vii) tan⁻¹(x −1) + tan⁻¹x tan⁻¹(x + 1) = tan⁻¹3x

(viii) tan (cos⁻¹x) = sin$\left({\cot }^{-1}\frac{1}{2}\right)$

(ix) tan⁻¹$\left(\frac{1-x}{1+x}\right)-\frac{1}{2}$tan⁻¹x = 0, where x > 0

(x) cot⁻¹x − cot⁻¹(x + 2) = $\frac{\mathrm{\pi }}{12}$, x > 0

(xi) ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\cot }^{-1}\left(\frac{1-x^2}{2x}\right)=\frac{2\mathrm{\pi }}{3},x>0$

(xii) tan⁻¹(x + 2) + tan⁻¹(x − 2) = tan⁻¹$\left(\frac{8}{79}\right)$, x > 0

(xiii) ${\tan }^{-1}\frac{x}{2}+{\tan }^{-1}\frac{x}{3}=\frac{\mathrm{\pi }}{4},0<x<\sqrt{6}$

Q. Consider the real-valued function f satisfying 2 f(sin x)+ f(cos x)=x. Then, the

1. domain of f(x) is R
2. domain of f(x) is[-1, 1]
3. range of f(x) is $\left(-\frac{2\pi }{3},\frac{\pi }{3}\right]$
4. range of f(x) is R

Q. If $g:[2:2]\to R$ where $g\left(x\right)=x^3+tanx+\left[\frac{x^2+1}{p}\right]$ is a odd function then the value of parametric P is where [.] denotes the Greatest integer function.

1. None of these
2. -5<P<5
3. P<5
4. P>5

Q. ${\int }_0^{\mathrm{\pi }}\cos x\left|\cos x\right|dx$

Q. Prove $ln(1+x)$ is larger than $\frac{ta{n}^{-1}x}{1+x}$, for x > 0.

Q.

solve: tan^-1 (under root x² +x) +sin^-1 (under root x²+x+1) = pie/2 ??

Q. If $A={\int }_1^{sin\theta }\frac{tdt}{1+t^2}dt,B={\int }_1^{cosec\theta }\frac{1}{t(1+t^2)}dt$ then
$\left|\begin{array}{ccc}A& A^2& B\\ e^{A+B}& B^2& -1\\ 1& A^2+B^2& -1\end{array}\right|=?$

1. $sin\theta$
2. $cosec\theta$
3. $0$
4. $1$

Q. If $g:[2:2]\to R$ where $g\left(x\right)=x^3+tanx+\left[\frac{x^2+1}{p}\right]$ is a odd function then the value of parametric P is where [.] denotes the Greatest integer function.

1. $-5<p<5$
2. $p<5$
3. $p>5$
4. None of these

Q. At how many points f(x)=[sinx]is discountinous if x belongs to [o, 2pie)

Q.

The odd extension of the function y = $x^2$ is y = $-x^2$

1. True

2. False

Q. Assertion :$\int e^x(\tan x+{\sec }^{2}x)dx=e^x\tan x+C$ Reason: $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+C$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Assertion :$\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}x+\frac{1}{3}{\tan }^{-1}{x}^3+C$ Reason: $\int \frac{\cos x+x\sin x}{x(x+\cos x)}dx=\log (x^2+x\cos x)+C$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Find $\frac{dy}{dx}$, when

$x=\frac{e^t+e^{-t}}{2}\mathrm{and}y=\frac{e^t-e^{-t}}{2}$

Q. $\int 2x^3{e}^{x^2}dx$

Q. (sin x + cos x) dy + (cos x − sin x) dx = 0

Q. Differential equations of the first order and first degree.
Q. ydx + $\sqrt{1-x^2}\sin {}^{-1}xdy=0$

Q. Prove that:
${\tan }^{-1}\frac{1}{4}+{\tan }^{-1}\frac{2}{9}=\frac{1}{2}{\tan }^{-1}\frac{4}{3}$

Q. If tan(π/4 + (1/2) cos^-1x) + tan(π/4 - (1/2) cos^-1x) = 1, then x =
a) 0 b) 2 c) 3
d) No solution e) -2/3

Q. What is the derivative of $\sqrt{\tan x}$?

Q. If sin (A + B) = 1 and cos (A − B) = 1, 0° < A + B ≤ 90°, A ≥ B find A and B.