Question

\[ \text{If } A \text{ lies in second quadrant and } 3\tan A+4=0, \]

\[ \text{then the value of } 2\cot A-5\cos A+\sin A \]

is equal to

(a) \(-\frac{53}{10}\)
(b) \(\frac{23}{10}\)
(c) \(\frac{37}{10}\)
(d) \(\frac{7}{10}\)

Solution

\[ 3\tan A+4=0 \]

\[ \tan A=-\frac43 \]

Since \(A\) lies in second quadrant,

\[ \sin A>0,\quad \cos A<0 \]

Take

\[ \text{Perpendicular}=4, \quad \text{Base}=-3 \]

\[ \text{Hypotenuse} = \sqrt{4^2+(-3)^2} = 5 \]

\[ \sin A=\frac45, \quad \cos A=-\frac35, \quad \cot A=-\frac34 \]

Now,

\[ 2\cot A-5\cos A+\sin A \]

\[ = 2\left(-\frac34\right) -5\left(-\frac35\right) +\frac45 \]

\[ = -\frac32+3+\frac45 \]

\[ = \frac{-15+30+8}{10} \]

\[ =\frac{23}{10} \]

Answer

\[ \boxed{\frac{23}{10}} \]

Correct Option: (b)