In which way have microbes played a major role in controlling diseases caused by harmful bacteria?
Question: In which way have microbes played a major role in controlling diseases caused by harmful bacteria? Solution: Several micro-organisms are used for preparing medicines. Antibiotics are medicines produced by certain micro-organisms to kill other disease-causing micro-organisms. These medicines are commonly obtained from bacteria and fungi. They either kill or stop the growth of disease-causing micro-organisms. Streptomycin, tetracycline, and penicillin are common antibiotics.Penicillium n...
Read More →Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
Question: Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively. Solution: TO FIND: The smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively. L.C.M of 28 and 32. $28=2^{2} \times 7$ $32=2^{5}$ L.C.M of 28 , and $32=2^{5} \times 7$ $=224$ Hence 224 is the least number which exactly divides 28 and 32 i.e. we will get a remainder of 0 in this case. But we need the smallest number which leaves remainders 8 and 12 when divi...
Read More →Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
Question: Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively. Solution: TO FIND: The smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively. L.C.M of 28 and 32. $28=2^{2} \times 7$ $32=2^{5}$ L.C.M of 28 , and $32=2^{5} \times 7$ $=224$ Hence 224 is the least number which exactly divides 28 and 32 i.e. we will get a remainder of 0 in this case. But we need the smallest number which leaves remainders 8 and 12 when divi...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $e^{x}+e^{x^{2}}+\ldots+e^{x^{5}}$ Solution: $\frac{d}{d x}\left(e^{x}+e^{x^{2}}+\ldots+e^{x^{5}}\right)$ $=\frac{d}{d x}\left(e^{x}\right)+\frac{d}{d x}\left(e^{x^{2}}\right)+\frac{d}{d x}\left(e^{x^{3}}\right)+\frac{d}{d x}\left(e^{x^{4}}\right)+\frac{d}{d x}\left(e^{x^{3}}\right)$ $=e^{x}+\left[e^{x^{2}} \times \frac{d}{d x}\left(x^{2}\right)\right]+\left[e^{x^{3}} \cdot \frac{d}{d x}\left(x^{3}\right)\right]+\left[e^{x^{4}} \cdot \frac{d}{d x}\l...
Read More →If the lines y = 3x + 1 and 2y = x + 3 are equally
Question: If the linesy= 3x+ 1 and 2y=x+ 3 are equally inclined to the liney=mx+ 4, find the value ofm. Solution: The equations of the given lines are y= 3x+ 1 (1) 2y=x+ 3 (2) y=mx+ 4 (3) Slope of line $(1), m_{1}=3$ Slope of line $(2), m_{2}=\frac{1}{2}$ Slope of line $(3), m_{3}=m$ It is given that lines (1) and (2) are equally inclined to line (3). This means that the angle between lines (1) and (3) equals the angle between lines (2) and (3). $\therefore\left|\frac{m_{1}-m_{3}}{1+m_{1} m_{3}}...
Read More →Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8,
Question: Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21. Solution: TO FIND: The number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21. L.C.M Of 8, 15 and 21. $8=2^{3}$ $15=3 \times 5$ $21=3 \times 7$ L.C.M of 8,15 and $21=2^{3} \times 3 \times 5 \times 7=840$ When 110000 is divided by 840, the remainder is obtained as 800. Now, 110000 800 = 109200 is divisible by each of 8, 15 and 21...
Read More →Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8,
Question: Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21. Solution: TO FIND: The number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21. L.C.M Of 8, 15 and 21. $8=2^{3}$ $15=3 \times 5$ $21=3 \times 7$ L.C.M of 8,15 and $21=2^{3} \times 3 \times 5 \times 7=840$ When 110000 is divided by 840, the remainder is obtained as 800. Now, 110000 800 = 109200 is divisible by each of 8, 15 and 21...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $\log \left(\cos e^{x}\right)$ Solution: Let $y=\log \left(\cos e^{x}\right)$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}\left[\log \left(\cos e^{x}\right)\right]$ $=\frac{1}{\cos e^{x}} \cdot \frac{d}{d x}\left(\cos e^{x}\right)$ $=\frac{1}{\cos e^{x}} \cdot\left(-\sin e^{x}\right) \cdot \frac{d}{d x}\left(e^{x}\right)$ $=\frac{-\sin e^{x}}{\cos e^{x}} \cdot e^{x}$ $=-e^{x} \tan e^{x}, e^{x} \neq(2 n+1) \frac{\pi}{2}, n \in \m...
Read More →Name some traditional Indian foods made of wheat,
Question: Name some traditional Indian foods made of wheat, rice and Bengal gram (or their products) which involve use of microbes. Solution: (a)Wheat: Product: Bread, cake, etc. (2) Rice: Product:Idli, dosa (2) Bengal gram: Product:Dhokla, Khandvi...
Read More →In a morning walk three persons step off together, their steps measure 80 cm,
Question: In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps? Solution: GIVEN: In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm. TO FIND: minimum distance each should walk so that all can cover the same distance in complete steps. The distance covered by each of them is required to be same as well ...
Read More →What is the smallest number that, when divided by 35,
Question: What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case? Solution: TO FIND: Smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case L.C.M OF 35, 56 and 91 $35=5 \times 7$ $56=2^{3} \times 7$ $91=13 \times 7$ L.C.M of 35,56 and $91=2^{3} \times 5 \times 7 \times 13$ $=3640$ Hence 84 is the least number which exactly divides 28, 42 and 84 i.e. we will get a remainder of 0 in this case. But we need the smallest num...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $\sin \left(\tan ^{-1} e^{-x}\right)$ Solution: Let $y=\sin \left(\tan ^{-1} e^{-x}\right)$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}\left[\sin \left(\tan ^{-1} e^{-x}\right)\right]$ $=\cos \left(\tan ^{-1} e^{-x}\right) \cdot \frac{d}{d x}\left(\tan ^{-1} e^{-x}\right)$ $=\cos \left(\tan ^{-1} e^{-x}\right) \cdot \frac{1}{1+\left(e^{-x}\right)^{2}} \cdot \frac{d}{d x}\left(e^{-x}\right)$ $=\frac{\cos \left(\tan ^{-1} e^{-x}\...
Read More →What is the smallest number that, when divided by 35,
Question: What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case? Solution: TO FIND: Smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case L.C.M OF 35, 56 and 91 $35=5 \times 7$ $56=2^{3} \times 7$ $91=13 \times 7$ L.C.M of 35,56 and $91=2^{3} \times 5 \times 7 \times 13$ $=3640$ Hence 84 is the least number which exactly divides 28, 42 and 84 i.e. we will get a remainder of 0 in this case. But we need the smallest num...
Read More →In which food would you find lactic acid bacteria? Mention some of their useful applications.
Question: In which food would you find lactic acid bacteria? Mention some of their useful applications. Solution: Lactic acid bacteria can be found in curd. It is this bacterium that promotes the formation of milk into curd. The bacterium multiplies and increases its number, which converts the milk into curd. They also increase the content of vitamin B12in curd. Lactic acid bacteria are also found in our stomach where it keeps a check on the disease-causing micro-organisms....
Read More →Find the image of the point (3, 8)
Question: Find the image of the point (3, 8) with respect to the linex+ 3y= 7 assuming the line to be a plane mirror. Solution: The equation of the given line is x+ 3y= 7 (1) Let point B (a,b) be the image of point A (3, 8). Accordingly, line (1) is the perpendicular bisector of AB. Slope of $\mathrm{AB}=\frac{b-8}{a-3}$, while the slope of line $(1)=-\frac{1}{3}$ Since line (1) is perpendicular to AB, $\left(\frac{b-8}{a-3}\right) \times\left(-\frac{1}{3}\right)=-1$ $\Rightarrow \frac{b-8}{3 a-...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $e^{x^{3}}$ Solution: Let $y=e^{x^{3}}$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}\left(e^{x^{3}}\right)=e^{x^{3}} \cdot \frac{d}{d x}\left(x^{3}\right)=e^{x^{3}} \cdot 3 x^{2}=3 x^{2} e^{x^{3}}$...
Read More →Give examples to prove that microbes release gases during metabolism.
Question: Give examples to prove that microbes release gases during metabolism. Solution: The examples of bacteria that release gases during metabolism are: (a)Bacteria and fungi carry out the process of fermentation and during this process, they release carbon dioxide. Fermentation is the process of converting a complex organic substance into a simpler substance with the action of bacteria or yeast. Fermentation of sugar produces alcohol with the release of carbon dioxide and very little energy...
Read More →Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).
Question: Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive). Solution: ANSWER: TO FIND: Least number that is divisible by all the numbers between 1 and 10 (both inclusive) Let us first find the L.C.M of all the numbers between 1 and 10 (both inclusive) 1 = 1 2 = 2 3 = 3 4 = 22 5 = 5 6 = 2 3 7 = 7 8 = 23 9 = 32 10 = 2 5 L.C.M $=2520$ Hence 2520 is the least number that is divisible by all the numbers between 1 and 10 (both inclusive)...
Read More →Bacteria cannot be seen with the naked eyes, but these can be seen with the help of a microscope.
Question: Bacteria cannot be seen with the naked eyes, but these can be seen with the help of a microscope. If you have to carry a sample from your home to your biology laboratory to demonstrate the presence of microbes under a microscope, which sample would you carry and why? Solution: Curdcan be used as a sample for the study of microbes. Curd contains numerous lactic acid bacteria (LAB) orLactobacillus. These bacteria produce acids that coagulate and digest milk proteins. A small drop of curd...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $e^{x^{3}}$ Solution: Let $y=e^{x^{3}}$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\sin ^{-1} x}\right)$ $\Rightarrow \frac{d y}{d x}=e^{\sin ^{-1} x} \cdot \frac{d}{d x}\left(\sin ^{-1} x\right)$ $\quad=e^{\sin ^{-1} x} \cdot \frac{1}{\sqrt{1-x^{2}}}$ $\quad=\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}$ $\therefore \frac{d y}{d x}=\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}, x \in(-1,1)$...
Read More →A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. i
Question: A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. it is to be paved with square tiles of the same size. Find the least possible number of such tiles. Solution: GIVEN: A rectangular yard is 18 m 72 cm long and 13 m 20 cm broad .It is to be paved with square tiles of the same size. TO FIND: Least possible number of such tiles. Length of the yard = 18 m 72 cm = 1800 cm + 72 cm = 1872 cm (∵ 1 m = 100 cm) Breadth of the yard =13 m 20 cm = 1300 cm + 20 cm = 1320 cmThe size of ...
Read More →A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. i
Question: A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. it is to be paved with square tiles of the same size. Find the least possible number of such tiles. Solution: GIVEN: A rectangular yard is 18 m 72 cm long and 13 m 20 cm broad .It is to be paved with square tiles of the same size. TO FIND: Least possible number of such tiles. Length of the yard = 18 m 72 cm = 1800 cm + 72 cm = 1872 cm (∵ 1 m = 100 cm) Breadth of the yard =13 m 20 cm = 1300 cm + 20 cm = 1320 cmThe size of ...
Read More →Find the direction in which a straight line must be drawn through the point
Question: Find the direction in which a straight line must be drawn through the point$(-1,2)$ so that its point of intersection with the line $x+y=4$ may be at a distance of 3 units from thispoint. Solution: Lety=mx+cbe the line through point (1, 2). Accordingly, 2 =m(1) +c. $\Rightarrow 2=-m+c$ $\Rightarrow c=m+2$ $\therefore y=m x+m+2 \ldots$ (1) The given line is $x+y=4 \ldots(2)$ On solving equations (1) and (2), we obtain $x=\frac{2-m}{m+1}$ and $y=\frac{5 m+2}{m+1}$ $\therefore\left(\frac{...
Read More →Differentiate the following w.r.t. x:
Question: Differentiate the following w.r.t.x: $e^{\sin ^{-1} x}$ Solution: Let $y=e^{\sin ^{-1} x}$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\sin ^{-1} x}\right)$ $\Rightarrow \frac{d y}{d x}=e^{\sin ^{-1} x} \cdot \frac{d}{d x}\left(\sin ^{-1} x\right)$]' $=e^{\sin ^{-1} x} \cdot \frac{1}{\sqrt{1-x^{2}}}$ $=\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}$ $\therefore \frac{d y}{d x}=\frac{e^{\sin ^{-1} x}}{\sqrt{1-x^{2}}}, x \in(-1,1)$...
Read More →Find the greatest number of 6 digits exactly divisible by 24,
Question: Find the greatest number of 6 digits exactly divisible by 24, 15 and 36. Solution: TO FIND: Greatest number of 6 digits exactly divisible by 24, 15 and 36 The greatest 6 digit number be 999999 24, 15 and 36 $24=2^{3} \times 3$ $15=3 \times 5$ $36=2^{2} \times 3^{2}$ L.C.M of 24,15 and $36=360$ Since $\frac{999999}{360}=2777 \times 360+279$ Therefore, the remainder is 279. Hence the desired number is equal to $=999999-279$ $=999720$ Hence $=999720$ is the greatest number of 6 digits exa...
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