On a square cardboard sheet of area 784 cm2,
Question: On a square cardboard sheet of area 784 cm2, four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular plates. Find the area of the square sheet not covered by the circular plates. Solution: $\because \quad$ Area of square $=784$ $\therefore \quad(\text { Side })^{2}=(28)^{2}$ $\Rightarrow \quad$ Side $=28 \mathrm{~cm}$ Since, all four are congruent circular plates. $...
Read More →Four circular cardboard pieces of radii 7 cm
Question: Four circular cardboard pieces of radii 7 cm are placed on a paperin such a way that each piece touches other two pieces. Find the area of the portion enclosed between these pieces. Solution: Given that, four circular cardboard pieces arc placed on a paper in such a way that each piece touches other two pieces. Now, we join centre of all four circles to each other by a line segment. Since, radius of each circle is 7 cm. So, AB = 2 x Radius of circle = 27 = 14cm ⇒ AB = BC = CD = AD = 14...
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Question: Differentiate $(\log x)^{x}$ with respect to $\log x$ Solution: Let $u=(\log x)^{x}$ Taking log on both sides, $\log u=\log (\log x)^{x}$ $\Rightarrow \log u=x \log (\log x)$ $\Rightarrow \frac{1}{u} \frac{d u}{d x}=x \frac{d}{d x}\{\log (\log x)\}+\log (\log x) \frac{d}{d x}(x)$ $\Rightarrow \frac{1}{u} \frac{d u}{d x}=x\left(\frac{1}{\log x}\right) \frac{d}{d x}(\log x)+\log \log x(1)$ $\Rightarrow \frac{d u}{d x}=u\left[\frac{x}{\log x}\left(\frac{1}{x}\right)+\log \log x\right]$ $\...
Read More →Find the area of the sector of a circle
Question: Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm. Solution: Let the central angle of the sector be . Given that, radius of the sector of a circle (r) = 5 cm. and arc length $(l)=3.5 \mathrm{~cm}$ $\therefore$ Central angle of the sector, $\theta=\frac{\operatorname{arc} \text { length }(l)}{\text { radius }}$ $\left[\because \theta=\frac{l}{r}\right]$$\Rightarrow$ $\theta=\frac{3.5}{5}=0.7 R$ $\Rightarrow$$\theta=\left(0.7 \times \frac{1...
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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 $v=\tan ^{-1} x$ $\Rightarrow \frac{d u}{d x}=\frac{1}{\left(1+x^{2}\right)} \frac{d}{d x}\left(1+x^{2}\right)=\frac{2 x}{\left(1+x^{2}\right)}$ and $\frac{d v}{d x}=\frac{1}{1+x^{2}}$ $\therefore \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 x}{1+x^{2}} \times \frac{1+x^{2}}{1}=2 x$...
Read More →Give five examples of numbers,
Question: Give five examples of numbers, each one of which is divisible by 4 but not divisible by 8. Solution: For a number to be divisible by 4, the number formed by its last two digits should be divisible by 4. And for the number to be divisible by 8, the number formed by its last three digits should be divisible by 8. So, the numbers divisible by 4 and not by 8 will be 28, 36, 44, 52, 60....
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Question: Differentiate $x^{2}$ with respect to $x^{3}$ Solution: Let $u=x^{2}$ and $v=x^{3}$ $\Rightarrow \frac{d u}{d x}=2 x$ and $\frac{d v}{d x}=3 x^{2}$ $\therefore \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 x}{3 x^{2}}=\frac{2}{3 x}$...
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Question: Differentiate $x^{2}$ with respect to $x^{3}$ Solution: Let $u=x^{2}$ and $v=x^{3}$ $\Rightarrow \frac{d u}{d x}=2 x$ and $\frac{d v}{d x}=3 x^{2}$ $\therefore \frac{d u}{d v}=\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 x}{3 x^{2}}=\frac{2}{3 x}$...
Read More →Give five examples of numbers,
Question: Give five examples of numbers, each one of which is divisible by 3 but not divisible by 9. Solution: For a number to be divisible by 3, the sum of the digits should be divisible by 3. And for the number to be divisible by 9, the sum of the digits should be divisible by 9. Let us take the number 21. Sum of the digits is 2 + 1 = 3, which is divisible by 3 but not by 9. Hence, 21 is divisible by 3 not by 9. Similarly, lets check the number 24. Here, 2 + 4 = 6. This is divisible by 3 not b...
Read More →Three circles each of radius 3.5 cm
Question: Three circles each of radius 3.5 cm are drawm in such a way that each of them touches the other two. Find the area enclosed between these circles. Solution: Given that, three circles are in such a way that each of them touches the other two. Now, we join centre of all three circles to each other by a line segment. Since, radius of each circle is 3.5 cm. So; AB = 2 x Radius of circle = 2 x 3.5 = 7 cm. ⇒ AC = BC = AB = 7cm which shows that, ΔABC is an equilateral triangle with side 7 cm....
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Question: If $x=a(2 \theta-\sin 2 \theta)$ and $y=a(1-\cos 2 \theta)$, find $\frac{d y}{d x}$ when $\theta=\frac{\pi}{3}$. Solution: Given values are: $x=a(2 \theta-\sin 2 \theta)$ and $y=a(1-\cos 2 \theta)$ Applying parametric differentiation $\frac{d x}{d \theta}=2 a-2 a \cos 2 \theta$ $\frac{d y}{d \theta}=0+2 \operatorname{asin} 2 \theta$ $\frac{d y}{d x}=\frac{d y}{d \theta} \times \frac{d \theta}{d x}=\frac{\sin 2 \theta}{1-\cos 2 \theta}$ Now putting the value of $\theta=\frac{\pi}{3}$ $\...
Read More →Find the value of z for which the number 471z8 is divisible by 9.
Question: Find the value ofzfor which the number 471z8 is divisible by 9. Also, find the number. Solution: If a number is divisible by 9 , then the sum of the digits is also divisible by 9 . Sum of the digits of the given number $=4+7+1+z+8=20+z$ $20+z=27$, for $z=7$ 27 is divisible by 9 . Therefore, $471 z 8$ is divisible by 9 if $z$ is equal to 7 . The number is 47178 ....
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Question: If $x=a(2 \theta-\sin 2 \theta)$ and $y=a(1-\cos 2 \theta)$, find $\frac{d y}{d x}$ when $\theta=\frac{\pi}{3}$. Solution: Given values are: $x=a(2 \theta-\sin 2 \theta)$ and $y=a(1-\cos 2 \theta)$ Applying parametric differentiation $\frac{d x}{d \theta}=2 a-2 a \cos 2 \theta$ $\frac{d y}{d \theta}=0+2 \operatorname{asin} 2 \theta$ $\frac{d y}{d x}=\frac{d y}{d \theta} \times \frac{d \theta}{d x}=\frac{\sin 2 \theta}{1-\cos 2 \theta}$ Now putting the value of $\theta=\frac{\pi}{3}$ $\...
Read More →Find the value of x for which the number x806 is divisible by 9.
Question: Find the value ofxfor which the numberx806 is divisible by 9. Also, find the number. Solution: For a number to be divisible by 9 , the sum of the digits must be divisible by 9 . Sum of the digits in the given number $=x+8+0+6=x+14$ The sum of the digits is divisible by 9 , only in the following case: $x=4$ or $x+14=18$ Thus, the number x806 is divisible by 9 ifxxis equal to 4. The number is 4806....
Read More →Write the derivative of sinx with respect to cosx
Question: Write the derivative of sinxwith respect to cosx Solution: Let $u=\sin x$ and $v=\cos x$ $\Rightarrow \frac{d u}{d x}=\cos x$ and $\frac{d v}{d x}=-\sin x$ $\therefore \frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{\cos x}{-\sin x}$ $\Rightarrow \frac{d u}{d v}=-\cot x$...
Read More →Find all possible values of y for which the number 53y1 is divisible by 3.
Question: Find all possible values ofyfor which the number 53y1 is divisible by 3. Also, find each such number. Solution: If a number is divisible by 3, then the sum of the digits is also divisible by 3 . Sum of the digits $=5+3+y+1=9+y$ The sum of the digits is divisible by 3 in the following cases: $9+y=9$, or $y=0$ Then the number is 5301 . $9+y=12$, or $y=3$ Then the number is 5331 . $9+y=15$, or $y=6$ Then the number is 5361 . $9+y=18$, or $y=9$ Then the number is 5391 . y = 0, 3, 6 or 9 Th...
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Question: $\sin x=\frac{2 t}{1+t^{2}}, \tan y=\frac{2 t}{1-t^{2}}$, find $\frac{d y}{d x}$ Solution: $\sin x=\frac{2 t}{1+t^{2}}$ and $\tan y=\frac{2 t}{1-t^{2}}$ $\Rightarrow x=\sin ^{-1} \frac{2 t}{1+t^{2}}$ and $y=\tan ^{-1} \frac{2 t}{1-t^{2}}$ $\Rightarrow x=2 \tan ^{-1} t$ and $y=2 \tan ^{-1} t$ $\Rightarrow \frac{d x}{d t}=\frac{2 t}{1+t^{2}}$ and $\frac{d y}{d t}=\frac{2 t}{1+t^{2}}$ $\therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{\frac{2 t}{1+t^{2}}}{\frac{2 t}...
Read More →ind all possible values of x for which the number 7x3 is divisible by 3.
Question: ind all possible values ofxfor which the number 7x3 is divisible by 3. Also, find each such number. Solution: For a number to be divisible by 3, the sum of the digits must be divisible by 3. Sum of the digits $=7+x+3$ $=10+x$ $10+x$ will be divisible by 3 in the following cases: $10+x=12$, or $x=2$ Thus, the number will be 723 . $10+x=15$, or $x=5$ Thus, the number will be 753 . $10+x=18$, or $x=8$ Thus, the number will be 783 . So, the numbers can be 723, 753 or 783....
Read More →Test the divisibility of each of the following numbers by 7:
Question: Test the divisibility of each of the following numbers by 7: (i) 693 (ii) 7896 (iii) 3467 (iv) 12873 (v) 65436 (vi) 54636 (vii) 98175 (viii) 88777 Solution: (i) 693 69 (2 3) = 69 6 = 63, which is divisible by 7. Hence, 693 is divisible by 7. (ii) 7896 789 (2 6) = 789 12 = 777, which is divisible by 7. Hence, 7896 is divisible by 7. (iii) 3467 346 (2 7) = 346 14 = 332, which is not divisible by 7. Hence, 3467 is not divisible by 7. (iv) 12873 1287 (2 3) = 1287 6 = 1281, which is divisib...
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Question: If $x=\frac{1+\log t}{t^{2}}, y=\frac{3+2 \log t}{t}$, find $\frac{d y}{d x}$ Solution: $x=\frac{1+\log t}{t^{2}}$ and $y=\frac{3+2 \log t}{t}$ $\Rightarrow \frac{d x}{d t}=\frac{t-2 t-2 t \log t}{t^{4}}$ and $\frac{d y}{d t}=\frac{2-3-2 \log t}{t^{2}}$ $\Rightarrow \frac{d x}{d t}=\frac{-1-2 \log t}{t^{3}}$ and $\frac{d y}{d t}=\frac{-1-2 \log t}{t^{2}}$ $\therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{\frac{-1-2 \log t}{t^{2}}}{\frac{-1-2 \log t}{t^{3}}}=t$...
Read More →Test the divisibility of each of the following numbers by 11:
Question: Test the divisibility of each of the following numbers by 11: (i) 22222 (ii) 444444 (iii) 379654 (iv) 1057982 (v) 6543207 (vi) 818532 (vii) 900163 (viii) 7531622 Solution: A given number is divisible by 11, if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11. (i) 22222 For the given number, sum of the digits at odd places = 2 + 2 + 2 = 6 sum of digits at even places = 2 + 2 = 4 Difference of the ...
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Question: If $x=\cos t\left(3-2 \cos ^{2} t\right), y=\sin t\left(3-2 \sin ^{2} t\right)$ find the value of $\frac{d y}{d x}$ at $t=\frac{\pi}{4}$ Solution: $x=\cos t\left(3-2 \cos ^{2} t\right)$ and $y=\sin t\left(3-2 \sin ^{2} t\right)$ $\Rightarrow \frac{d x}{d t}=-\sin t\left(3-2 \cos ^{2} t\right)+\cos t(4 \cos t \sin t)$ and $\frac{d y}{d t}=\cos t\left(3-2 \sin ^{2} t\right)+\sin t(-4 \sin t \cos t)$ $\Rightarrow \frac{d x}{d t}=-3 \sin t+6 \sin t \cos ^{2} t$ and $\frac{d y}{d t}=3 \cos ...
Read More →Text the divisibility of each of the following numbers by 5:
Question: Text the divisibility of each of the following numbers by 5: (i) 95 (ii) 470 (iii) 1056 (iv) 2735 (v) 55053 (vi) 35790 (vii) 98765 (viii) 42658 (ix) 77990 Solution: A number is divisible by 8 only when the number formed by its last three digits is divisible by 8.(i) 6132The last three digits of the given number are 132 which is not divisible by 8. So, 6132 is not divisible by 8.(ii) 7304The last three digits of the given number are 304 which is divisible by 8. So, 7304 is divisible by ...
Read More →Test the divisibility of each of the following numbers by 8:
Question: Test the divisibility of each of the following numbers by 8: (i) 134 (ii) 618 (iii) 3928 (iv) 50176 (v) 39392 (vi) 56794 (vii) 86102 (viii) 66666 (ix) 99918 (x) 77736 Solution: A given number is divisible by 4 only when the number formed by its last two digits is divisible by 4.(i) 134The last two digits of 134 are '34' which is not divisible by 4. Hence, 134 is not divisible by 4.(ii) 618The last two digits of 618 are '18' which is not divisible by 4. Hence, 618 is not divisible by 4....
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Question: If $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$, show that at $t=\frac{\pi}{4}, \frac{d y}{d x}=\frac{b}{a} t=\frac{\pi}{4}, \frac{d y}{d x}=\frac{b}{a}$ Solution: $x=a \sin 2 t(1+\cos 2 t)$ and $y=b \cos 2 t(1-\cos 2 t)$ $\Rightarrow \frac{d x}{d t}=2 a \cos 2 t(1+\cos 2 t)+2 a \sin 2 t(1-\cos 2 t)$ and $\frac{d y}{d t}=-2 b \sin 2 t(1-\cos 2 t)+2 b \cos 2 t(1+\cos 2 t)$ $\Rightarrow \frac{d x}{d t}=2 a\left(\cos 2 t+\cos ^{2} 2 t+\sin 2 t-\sin 2 t \cos 2 t\right)$ and $\...
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