2 Marks

Question 12 Marks

Find the approximate value of (1.999)⁵.

Answer

Let x = 2
And $\triangle\text{x}=-0.001$ $[\because2-0.001=1.999]$
Let y = x⁵
On differentiating both sides w.r.t. x, we get
$\frac{\text{dy}}{\text{dx}}=5\text{x}^4$
Now, $\triangle\text{y}=\frac{\text{dy}}{\text{dx}}\triangle\text{x}=5\text{x}^4\times\triangle\text{x}=5\times2^4\times[-0.001]=-80\times0.001=-0.080$
$\therefore\ (1.999)^5-\text{y}+\triangle\text{y}$
$=2^5+(-0.080)$
$=32-0.080=31.920$

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Question 22 Marks

Find an angle $\theta,0<\theta<\frac{\pi}{2},$ which increases twice as fast as its sine.

Answer

Let $\theta$ increases twice as fast as its sine
$\Rightarrow\ \theta=2\sin\theta$
$\Rightarrow\ \frac{\text{d}\theta}{\text{dt}}=2\cdot\cos\theta\cdot\frac{\text{d}\theta}{\text{dt}}$
$\Rightarrow\ 1=2\cos\theta$
$\Rightarrow\ \frac{1}{2}=\cos\theta$
$\Rightarrow\ \cos\theta=\cos\frac{\pi}{3}$
$\therefore\ \theta=\frac{\pi}{3}$
So, the required angle is $\frac{\pi}{3}$

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Maths 2 Marks

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