Download App

Trigonometric Equations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Equations1 Mark
MCQ
The smallest positive angle which satisfies the equation ​$2\sin^2\text{x}+\sqrt{3}\cos\text{x}+1=0$ is:
  • ✓
    $\frac{5\pi}{6}$
  • B
    $\frac{2\pi}{3}$
  • C
    $\frac{\pi}{3}$
  • D
    $\frac{\pi}{6}$
✓

Answer

Correct option: A.
$\frac{5\pi}{6}$
Given:
$2\sin^2\text{x}+\sqrt{3}\cos\text{x}+1=0$
$\Rightarrow2(1-\cos^2\text{x})+\sqrt{3}\cos\text{x}+1=0$
$\Rightarrow2-2\cos^2\text{x}+\sqrt{3}\cos\text{x}+1=0$
$\Rightarrow2\cos^2\text{x}-\sqrt{3}\cos\text{x}-3=0$
$\Rightarrow2\cos^2\text{x}-2\sqrt{3}\cos\text{x}+\sqrt{3}\cos\text{x}-3=0$
$\Rightarrow2\cos\text{x}\Big(\cos\text{x}-\sqrt{3}\Big)+\sqrt{3}\Big(\cos\text{x}-\sqrt{3}\Big)=0$
$\Rightarrow\Big(2\cos\text{x}+\sqrt{3}\Big)\Big(\cos\text{x}-\sqrt{3}\Big)=0$
$\therefore\cos\text{x}+\sqrt{3}=0$ or, $\cos\text{x}=\sqrt{3}$ is not possible.
$\Rightarrow\cos\text{x}=\cos\Big(\frac{\pi}{6}\Big)$
$\Rightarrow\text{x}=2\text{n}\pi\pm\frac{5\pi}{6},\ \text{n}\in\text{Z}$
For n = 0, the value of x is $\pm\frac{5\pi}{6}.$
Hence, the smallest positive angle is $\frac{5\pi}{6}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Trigonometric Equations→Trigonometric Equations — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The probability of getting head and tail alternately in three throws of a coin (or a throw of three coins), is
→
Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ from any of its foci?
→
$\cos A + \cos (240^\circ + A) + \cos (240^\circ - A) = $
→
If $\cos \alpha + \cos \beta + \cos \gamma = \sin \alpha + \sin \beta + \sin \gamma = 0$ then $\cos 3\alpha + \cos 3\beta + \cos 3\gamma $ equals to
→
If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces, is through five times, then the probability that the product of the outcomes is positive, is :
→
The sum of the first $20$ terms of the series $5+11+$ $19+29+41+\ldots$ is $..........$.
→
Mean deviation is least if it is taken about :-
→
A bag contains $5$ brown and $4$ white socks. A man pulls out two socks. The probability that these are of the same colour is
→
If ${\log _{0.3}}(x - 1) < {\log _{0.09}}(x - 1),$ then $x$ lies in the interval
→
Area of the triangle with vertices $(a,b),\;({x_1},{y_1})$ and $({x_2},{y_2})$, where $a,\;{x_1}$ and ${x_2}$ are in $G.P.$ with common ratio r and b, ${y_1}$ and ${y_2}$ are in $G.P.$ with common ratio $s$, is given by
→