MCQ
The solution of ${\sin ^{ - 1}}x - {\sin ^{ - 1}}2x = \pm \frac{\pi }{3}$ is
- A$ \pm \frac{1}{3}$
- B$ \pm \frac{1}{4}$
- C$ \pm \frac{{\sqrt 3 }}{2}$
- ✓$ \pm \frac{1}{2}$
✓
Answer
Correct option: D.
$ \pm \frac{1}{2}$
d
(d) ${\sin ^{ - 1}}2x = {\sin ^{ - 1}}x - {\sin ^{ - 1}}\frac{{\sqrt 3 }}{2}$
(d) ${\sin ^{ - 1}}2x = {\sin ^{ - 1}}x - {\sin ^{ - 1}}\frac{{\sqrt 3 }}{2}$
${\sin ^{ - 1}}2x = {\sin ^{ - 1}}\left( {x\sqrt {\left( {1 - \frac{3}{4}} \right)} - \frac{{\sqrt 3 }}{2}\sqrt {1 - {x^2}} } \right)$
$2x = \left( {\frac{x}{2} - \frac{{\sqrt 3 }}{2}\sqrt {1 - {x^2}} } \right)$
$\frac{{\sqrt 3 }}{2}\sqrt {1 - {x^2}} = \frac{x}{2} - 2x = \frac{{ - 3x}}{2}$
$\frac{{3(1 - {x^2})}}{4} = \frac{{9{x^2}}}{4}$
$ \Rightarrow 3 - 3{x^2} = 9{x^2}$
==> ${x^2} = \frac{1}{4} \Rightarrow x = \pm \frac{1}{2}$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Let $A$ be a $2 \times 2$ matrix with real entries such that $A ^{\prime}=\alpha A + I$, where $\alpha \in R -\{-1,1\}$. If det $\left(A^2-A\right)=4$, then the sum of all possible values of $\alpha$ is equal to
→Integers $1,2,3, \ldots \ldots, n,(n \geq 3)$ are written on a black board and an integer $k (1 < k < n )$ is erased. The average of the remaining numbers is $16$ . Then $n + k$ is
→Suppose the cubic ${x^3} - px + q$ has three distinct real roots where $ p >0 $ and $q>0$ . Then which one of the folllowing holds?
→The graph of $y = f(x)$ is shown then number of solutions of the equation $f(f(x)) =2$ is
→$\int_{\,0}^{\,3} {|2 - x|dx} $ equals
→If the foci and vertices of an ellipse be $( \pm 1,\;0)$ and $( \pm 2,\;0)$, then the minor axis of the ellipse is
→Let the set of all values of $p$, for which
→$f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does not have any critical point, be the interval $(a, b)$. Then $16 a b$ is equal to ..........
The system of equations
$
\begin{array}{l}
x+y+z=6 \\
x+2 y+5 z=9 \\
x+5 y+\lambda z=\mu
\end{array}
$
has no solution if
→$
\begin{array}{l}
x+y+z=6 \\
x+2 y+5 z=9 \\
x+5 y+\lambda z=\mu
\end{array}
$
has no solution if
If ${x^y}.{y^x} = 1$, then ${{dy} \over {dx}} =$
→Let $f\left( x \right) = \frac{{{x^2} - x}}{{{x^2} + 2x}}\,x \ne 0, - 2$. Then $\frac{d}{{dx}}\left[ {{f^{ - 1}}\left( x \right)} \right]$ (wherever it is defined) is equal to
→