6 Views

Question : If $\alpha$ is an acute angle and $2\sin \alpha+15\cos^2\alpha=7$, then the value of $\cot \alpha$ is:

Option 1: $\frac{4}{3}$

Option 2: $\frac{4}{5}$

Option 3: $\frac{5}{4}$

Option 4: $\frac{3}{4}$


Team Careers360 3rd Jan, 2024
Answer (1)
Team Careers360 22nd Jan, 2024

Correct Answer: $\frac{3}{4}$


Solution : Given: If $\alpha$ is an acute angle and $2\sin \alpha+15\cos^2 \alpha=7$.
$2\sin \alpha+15\cos^2\alpha=7$
⇒ $2\sin \alpha+15(1–\sin^2 \alpha)=7$
Let $\sin \alpha$ be $x$.
⇒ $2x+15(1–x^2)=7$
⇒ $2x+15–15x^2=7$
⇒ $15x^2–2x–8=0$
⇒ $15x^2–12x+10x–8=0$
⇒ $3x(5x–4)+2(5x–4)=0$
⇒ $(5x–4)(3x+2)=0$
⇒ $x=\frac{4}{5},\frac{–2}{3}$
$\sin \alpha=\frac{4}{5},\frac{–2}{3}$
Since $\alpha$ is an acute angle, the value of $\sin\alpha=\frac{4}{5}$
So, the $\text{Perpendicular}$ is 4 and the $\text{Hypotenuse}$ is 5.
Using Pythagoras theorem, $(\text{Hypotenuse})^2=(\text{Base})^2+(\text{Perpendicular})^2$.
⇒ $(5)^2=(\text{Base})^2+4^2$
⇒ $25=(\text{Base})^2+16$
⇒ $(\text{Base})^2=9$
⇒ $\text{Base}=3$
The value of $\cot\alpha$ is given as $\frac{\text{Base}}{\text{Perpendicular}}$ i.e., $\cot \alpha=\frac{3}{4}$
Hence, the correct answer is $\frac{3}{4}$.

Know More About

Related Questions

TOEFL ® Registrations 2024
Apply
Accepted by more than 11,000 universities in over 150 countries worldwide
Manipal Online M.Com Admissions
Apply
Apply for Online M.Com from Manipal University
View All Application Forms

Download the Careers360 App on your Android phone

Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile

150M+ Students
30,000+ Colleges
500+ Exams
1500+ E-books