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Question: Differentiate $\tan ^{-1}\left(\frac{1+a x}{1-a x}\right)$ with respect to $\sqrt{1+a^{2} x^{2}}$. Solution: Let $u=\tan ^{-1}\left(\frac{1+a x}{1-a x}\right)$ and $v=\sqrt{1+a^{2} x^{2}}$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\tan ^{-1}\left(\frac{1+\mathrm{ax}}{1-\mathrm{ax}}\right)$ By substituting $a x=\tan \theta$, we have $\mathrm{u}=\tan ^{-1}\left(\frac{1+\tan \theta}{1-\tan \theta}\right)$...
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Question: Differentiate $\sin ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)$ with respect to $\cos ^{-1}\left(\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right)$, if $0\mathrm{x}1$. Solution: Let $u=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ and $v=\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ By substituting $x=\tan \theta$, we have $\mathrm{u}=\sin ...
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Question: Differentiate $(\cos x)^{\sin x}$ with respect to $(\sin x)^{\cos x}$. Solution: Let $u=(\cos x)^{\sin x}$ and $v=(\sin x)^{\cos x}$. We need to differentiate $u$ with respect to $v$ that is find $\frac{\mathrm{du}}{\mathrm{dv}}$. We have $u=(\cos x)^{\sin x}$ Taking log on both sides, we get $\log u=\log (\cos x)^{\sin x}$ $\Rightarrow \log u=(\sin x) \times \log (\cos x)\left[\because \log a^{m}=m \times \log a\right]$ On differentiating both the sides with respect to $x$, we get $\f...
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Question: Differentiate $\sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right)$ with respect to $\sec ^{-1}\left(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$, if $\mathrm{x} \in(1 / \sqrt{2}, 1)$ Solution: Let $\mathrm{u}=\sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right)$ and $\mathrm{v}=\sec ^{-1}\left(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ By su...
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Question: Differentiate $\sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right)$ with respect to $\sec ^{-1}\left(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$, if $x \in(0,1 / \sqrt{2})$ Solution: Let $u=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ and $v=\sec ^{-1}\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ We need to differentiate $u$ with respect to $v$ that is find $\frac{\mathrm{du}}{\mathrm{dv}}$. We have $u=\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$ By substituting $x=\sin \theta$, we have $u=...
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Question: Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$, if $-1x1, x \neq 0$. Solution: Let $\mathrm{u}=\tan ^{-1}\left(\frac{\sqrt{1+\mathrm{x}^{2}}-1}{\mathrm{x}}\right)$ and $\mathrm{v}=\sin ^{-1}\left(\frac{2 \mathrm{x}}{1+\mathrm{x}^{2}}\right)$ We need to differentiate $u$ with respect to $v$ that is find $\frac{\text { du }}{\text { dv }}$. We have $\mathrm{u}=\tan ^{-1}\left(\frac{\sqrt{1+\mathrm{x}^{2}}-1}{...
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Question: Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$, if $x \in\left(-\frac{1}{2}, \frac{1}{2 \sqrt{2}}\right)$ Solution: Let $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ and $v=\sqrt{1-4 x^{2}}$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ $\Rightarrow u=\sin ^{-1}\left(4 x \sqrt{1-(2 x)^{2}}\right)$ By substituting $2 x=\cos \theta$, we have $u=\sin...
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Question: Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$, if $x \in\left(\frac{1}{2 \sqrt{2}}, \frac{1}{2}\right)$ Solution: Let $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ and $v=\sqrt{1-4 x^{2}}$ We need to differentiate $u$ with respect to $v$ that is find $\frac{\text { du }}{\text { dv }}$. We have $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ $\Rightarrow u=\sin ^{-1}\left(4 x \sqrt{1-(2 x)^{2}}\right)$ By substituting $2 x=\cos \theta$...
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Question: Differentiate $\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ with respect to $\sqrt{1-4 x^{2}}$, if $x \in\left(-\frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}\right)$ Solution: Let $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ and $v=\sqrt{1-4 x^{2}}$. We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\sin ^{-1}\left(4 x \sqrt{1-4 x^{2}}\right)$ $\Rightarrow u=\sin ^{-1}\left(4 x \sqrt{1-(2 x)^{2}}\right)$ By substituting $2 x=\cos \theta$, we ha...
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Question: Differentiate $\sin ^{-1} \sqrt{1-x^{2}}$ with respect to $\cos ^{-1} x$, if $x \in(-1,0)$ Solution: Given $x \in(-1,0)$ However, $x=\cos \theta$ $\Rightarrow \cos \theta \in(-1,0)$ $\Rightarrow \theta \in\left(\frac{\pi}{2}, \pi\right)$ Hence, $u=\sin ^{-1}(\sin \theta)=\pi-\theta .$ $\Rightarrow u=\pi-\cos ^{-1} x$ On differentiating u with respect to $x$, we get $\frac{d u}{d x}=\frac{d}{d x}\left(\pi-\cos ^{-1} x\right)$ $\Rightarrow \frac{\mathrm{du}}{\mathrm{dx}}=\frac{\mathrm{d}...
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Question: Differentiate $\sin ^{-1} \sqrt{1-\mathrm{x}^{2}}$ with respect to $\cos ^{-1} \mathrm{x}$, if $x \in(0,1)$ Solution: Let $u=\sin ^{-1} \sqrt{1-x^{2}}$ and $v=\cos ^{-1} x$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. We have $u=\sin ^{-1} \sqrt{1-x^{2}}$ By substituting $x=\cos \theta$, we have $\mathrm{u}=\sin ^{-1} \sqrt{1-(\cos \theta)^{2}}$ $\Rightarrow \mathrm{u}=\sin ^{-1} \sqrt{1-\cos ^{2} \theta}$ $\Rightarrow \mathrm{u}=\sin ^{-1} \sqrt{\si...
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Question: Differentiate $(\log x)^{x}$ with respect to $\log x$. Solution: Let $u=(\log x)^{x}$ and $v=\log x$ We need to differentiate $u$ with respect to $v$ that is find $\frac{\text { du }}{\text { dv }}$. We have $u=(\log x)^{x}$ Taking log on both sides, we get $\log u=\log (\log x)^{x}$ $\Rightarrow \log u=x \times \log (\log x)\left[\because \log a^{m}=m \times \log a\right]$ On differentiating both the sides with respect to $x$, we get $\frac{\mathrm{d}}{\mathrm{du}}(\log \mathrm{u}) \t...
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Question: Differentiate $\log \left(1+x^{2}\right)$ with respect to $\tan ^{-1} x$. Solution: Let $u=\log \left(1+x^{2}\right)$ and $v=\tan ^{-1} x$ We need to differentiate $u$ with respect to $v$ that is find $\frac{\text { du }}{\text { dv }}$. On differentiating $u$ with respect to $x$, we get $\frac{\mathrm{du}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left(1+\mathrm{x}^{2}\right)\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarr...
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Question: Differentiate $x^{2}$ with respect to $x^{3}$. Solution: Let $u=x^{2}$ and $v=x^{3}$ We need to differentiate $u$ with respect to $v$ that is find $\frac{d u}{d v}$. On differentiating $u$ with respect to $x$, we get $\frac{d u}{d x}=\frac{d}{d x}\left(x^{2}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ $\Rightarrow \frac{d u}{d x}=2 x^{2-1}$ $\therefore \frac{\mathrm{du}}{\mathrm{dx}}=2 \mathrm{x}$ Now, on dif...
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Question: Find $\frac{d y}{d x}$, when If $\sin x=\frac{2 t}{1+t^{2}}, \tan y=\frac{2 t}{1-t^{2}}$, find $\frac{d y}{d x}$ Solution: $\sin x=\frac{2 t}{1+t^{2}}, \tan y=\frac{2 t}{1-t^{2}}$ $x=\sin ^{-1} \frac{2 t}{1+t^{2}}$ and $y=\tan ^{-1} \frac{2 t}{1-t^{2}}$ $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{1}{\sqrt{1-\left(\frac{2 \mathrm{t}}{1+\mathrm{t}^{2}}\right)^{2}}} \times \frac{2\left(1+\mathrm{t}^{2}\right)-(2 \mathrm{t})(2 \mathrm{t})}{\left(1+\mathrm{t}^{2}\right)^{2}}$ $\frac{d x}{d t}=\f...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when Solution: $x=3 \sin t-\sin 3 t, y=3 \cos t-\cos 3 t$ $\frac{d x}{d t}=3 \cos t-3 \cos 3 t$ $\frac{d y}{d t}=-3 \sin t+3 \sin 3 t$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{dx}}{\mathrm{dt}}}=\frac{-3 \sin \mathrm{t}+3 \sin 3 \mathrm{t}}{3 \cos \mathrm{t}-3 \cos 3 \mathrm{t}}$ When $t=\frac{\pi}{3}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-3 \sin \left(\frac{\pi}{3}\right)+3 \sin 3\left(\frac{\pi}{3}...
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Question: Find $\frac{d y}{d x}$, when If $x=\frac{1+\log t}{t^{2}}, y=\frac{3+2 \log t}{t}$, find $\frac{d y}{d x}$ Solution: $: x=\frac{1+\log t}{t^{2}}, y=\frac{3+2 \log t}{t}$ $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{t}^{2}\left(\frac{1}{\mathrm{t}}\right)-(1+\log \mathrm{t})(2 \mathrm{t})}{\mathrm{t}^{4}}=\frac{\mathrm{t}-2 \mathrm{t}-2 \mathrm{t} \log \mathrm{t}}{\mathrm{t}^{4}}=\frac{-2 \log \mathrm{t}-1}{\mathrm{t}^{3}}$ $\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{t}\left(\frac{...
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Question: Find $\frac{d y}{d x}$, when If $x=\cos t\left(3-2 \cos ^{2} t\right)$ and $y=\sin t\left(3-2 \sin ^{2} t\right)$ find the value of $\frac{d y}{d x}$ at $t=\frac{\pi}{4}$ Solution: considering the given functions, $x=\cos t\left(3-2 \cos ^{2} t\right)$ $x=3 \cos t-2 \cos ^{3} t$ $\frac{d x}{d t}=-3 \sin t+6 \cos ^{2} t \sin t$ ......(1) $\frac{d y}{d t}=3 \cos t+6 \sin ^{2} t \cos t$ .......(2) $\frac{d y}{d x}=\frac{\frac{d y}{d x}}{\frac{d x}{d t}}=\frac{3 \cos t+6 \sin ^{2} t \cos t...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when If $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$, show that $a t t=\frac{\pi}{4}, \frac{d y}{d x}=\frac{b}{a}$ Solution: considering the given functions, $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$ rewriting the above equations, $x=a \sin 2 t+\frac{a}{2} \sin 4 t$ differentiating bove function with respect to $t$, we have, $\frac{d x}{d t}=2 a \cos 2 t+2 a \cos 4 t$ .....(1) $y=b \cos 2 t-b \cos ^{2} 2 t$ differenti...
Read More →Given below is a frequency distribution table.
Question: Given below is a frequency distribution table. Read it and answer the question that follow. (a) What is the lower limit of the second class interval? (b) What is the upper limit of the last class interval? (c) What is the frequency of the third class? (d) Which interval has a frequency of 10? (e) Which interval has the lowest frequency? (f) What is the class size? Solution: (a) What is the lower limit of the second class interval? The lower limit of the second class interval 20 30 is 2...
Read More →Verify Euler's relation for each of the following:
Question: Verify Euler's relation for each of the following: (i) A square (ii) A tetrahedron (iii) A triangular prism (iv) A square pyramid Solution: Euler's relation is:(i) A square prism (There is an error in this question. It should have been a square prism rather than square.) (ii) A tetrahedron(iii) A triangular prism(iv) A square pyramid...
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Question: Verify Euler's relation for each of the following: (i) A square (ii) A tetrahedron (iii) A triangular prism (iv) A square pyramid Solution: Euler's relation is:(i) A square prism (There is an error in this question. It should have been a square prism rather than square.) (ii) A tetrahedron(iii) A triangular prism(iv) A square pyramid...
Read More →In a throw of a dice,
Question: In a throw of a dice, the probability of getting an even number is the same as that of getting an odd number. Solution: True. The even numbers in a dice are 2, 4 and 6. Probability = total number of even numbers/ Total numbers = 3/6 [divide numerator and denominator by 3] = The odd numbers in a dice are 1, 3 and 5. Probability = total number of odd numbers/ Total numbers = 3/6 [divide numerator and denominator by 3] =...
Read More →The probability of getting a prime number
Question: The probability of getting a prime number is the same as that of a composite number in a throw of a dice. Solution: False. The composite numbers in a dice are 4 and 6. Probability = total number of composite numbers/ Total numbers = 2/6 [divide numerator and denominator by 2] = 1/3 The prime numbers in a dice are 2, 3 and 5. Probability = total number of prime numbers/ Total numbers = 3/6 [divide numerator and denominator by 3] =...
Read More →The probability of getting number 6 in a throw
Question: The probability of getting number 6 in a throw of a dice is 1/6. Similarly the probability of getting a number 5 is 1/5. Solution: False. The probability of getting a number 5 is 1/6....
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