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0 votes
15.0k views
in Mathematics by (33.5k points)
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if \(\alpha =\lim\limits_{x \to \pi/4 }\) \(\cfrac{tan^3x-tan\,x}{cos\left(x+\cfrac{\pi}{4}\right)}\) and \(\beta= \lim\limits_{x \to 0}\) (cos x)cot x are

if α = lim (x→π/4) tan3 x-tan x/cos(x+π/4) and β = lim (x→0) (cos x)cot x are

the roots of the equation, ax2 + bx – 4 = 0, then the ordered pair (a, b) is :

(1) (1, –3)

(2) (–1, 3)

(3) (–1, –3)

(4) (1, 3)

2 Answers

+2 votes
by (44.0k points)
selected by
 
Best answer

Correct answer is: (4) (1, 3)

ß = \(\lim\limits_{x\to 0}(cos x)^{cot x}\) 

 = Exp {\(\lim\limits_{x\to 0}\frac{cos x-1}{tan x}\)} (0/0 type)

= Exp{\(\lim\limits_{x\to 0}\frac{-sin x}{sec^2x}\)} (By using D.L.H. Rule)

 = Exp{\(\frac{-sin 0}{sec^20}\)}

 = Exp{0} = e0 = 1

\(\therefore\) ß = 1

α = –4 ; β β = 1

If ax2 + bx – 4 = 0 are the roots then

6a – 4b – 4 = 0 & a + b – 4 = 0 

⇒ a = 1 & b = 3

+1 vote
by (38.6k points)

Correct answer is: (4) (1, 3)

\(\alpha\) = –4 ; \(\beta\) = 1

If ax2 + bx – 4 = 0 are the roots then

16a – 4b – 4 = 0 & a + b – 4 = 0

\(\Rightarrow\) a = 1 & b = 3

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