Find :
Question: Find $\frac{d y}{d x}$ : $y=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right),-\frac{1}{\sqrt{3}}x\frac{1}{\sqrt{3}}$ Solution: The given relationship is $y=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ $y=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ $\Rightarrow \tan y=\frac{3 x-x^{3}}{1-3 x^{2}}$ ...(1) It is known that, $\tan y=\frac{3 \tan \frac{y}{3}-\tan ^{3} \frac{y}{3}}{1-3 \tan ^{2} \frac{y}{3}}$ ...(2) Comparing equations (1) and (2), we obtain $x=\tan \frac{y}{...
Read More →Differentiate the following and give examples of each:
Question: Differentiate the following and give examples of each: (a)Innate and acquired immunity (b)Active and passive immunity Solution: (a)Innate and acquired immunity (b)Active and passive immunity...
Read More →105 goats, 140 donkeys and 175 cows have to be taken across a river.
Question: 105 goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals went in each trip? Solution: We are given that, 105 goats, 140 donkeys an...
Read More →The following are some well-known abbreviations, which have been used in this chapter. Expand each one to its full form:
Question: The following are some well-known abbreviations, which have been used in this chapter. Expand each one to its full form: (a)MALT (b)CMI (c)AIDS (d)NACO (e)HIV Solution: (a)MALT- Mucosa-Associated Lymphoid Tissue (b)CMI- Cell-Mediated Immunity (c)AIDS- Acquired Immuno Deficiency Syndrome (d)NACO- National AIDS Control Organization (e)HIV- Human Immuno Deficiency virus...
Read More →The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively.
Question: The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly. Solution: We are given the length, breadth and height of a room as 8m 25cm, 6m 75cm and 4m 50cm, respectively. We need to determine the largest room which can measure the three dimensions of the room exactly. We first convert each dimension in cm Length of room = 8m 25cm = 825cm Breadth of room = 6m 75cm = 6...
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Question: Find $\frac{d y}{d x}$ : $y=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$ Solution: We have, $y=\sin ^{-1}\left[\frac{2 x}{1+x^{2}}\right]$ put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$ Now, $y=\sin ^{-1}\left[\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right]$ $\Rightarrow y=\sin ^{-1}(\sin 2 \theta), \quad\left(\right.$ as $\left.\sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$ $\Rightarrow y=2 \theta, \quad\left(\right.$ as $\left.\sin ^{-1}(\sin x)=x\right)$ $\Righta...
Read More →Name the primary and secondary lymphoid organs.
Question: Name the primary and secondary lymphoid organs. Solution: (a)Primary lymphoid organs include the bone marrow and the thymus. (b)Secondary lymphoid organs are the spleen, lymph nodes, tonsils, Peyers patches of small intestine, and appendix....
Read More →Find the equation of a line drawn perpendicular to the line
Question: Find the equation of a line drawn perpendicular to the line $\frac{x}{4}+\frac{y}{6}=1$ through the point, where it meets the $y$-axis. Solution: The equation of the given line is $\frac{x}{4}+\frac{y}{6}=1$. This equation can also be written as $3 x+2 y-12=0$ $y=\frac{-3}{2} x+6$, which is of the form $y=m x+c$ $\therefore$ Slope of the given line $=-\frac{3}{2}$ Let the given line intersect the $y$-axis at $(0, y)$. On substituting $x$ with 0 in the equation of the given line, we obt...
Read More →The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively.
Question: The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly. Solution: We are given the length, breadth and height of a room as 8m 25cm, 6m 75cm and 4m 50cm, respectively. We need to determine the largest room which can measure the three dimensions of the room exactly. We first convert each dimension in cm Length of room = 8m 25cm = 825cm Breadth of room = 6m 75cm = 6...
Read More →Discuss with your teacher what does ‘a suitable gene’ means, in the context of DNA vaccines.
Question: Discuss with your teacher what does a suitable gene means, in the context of DNA vaccines. Solution: A suitable gene refers to a specific DNA segment which can be injected into the cells of the host body to produce specific proteins. This protein kills the specific disease-causing organism in the host body and provides immunity....
Read More →What measure would you take to prevent water-borne diseases?
Question: What measure would you take to prevent water-borne diseases? Solution: Water-borne diseases such as cholera, typhoid, hepatitis B, etc. spread by drinking contaminated water. These water-borne diseases can be prevented by ensuring proper disposal of sewage, excreta, periodic cleaning. Also, measures such as disinfecting community water reservoirs, boiling drinking water, etc. should be observed...
Read More →Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively
Question: Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively Solution: Find the greatest number which divides 2011 and 2623 leaving remainder 9 and 5 respectively. The required number when divides 2011 and 2623 leaves remainders 9 and 5 this means $2011-9=2002$ and $2623-5=2618$ are completely divisible by the number. Therefore, the required number = H.C.F. of 2002 and 2618 By applying Euclids division lemma $2618=2002 \times 1+616$ $2002=616 \times 3+15...
Read More →How does the transmission of each of the following diseases take place?
Question: How does the transmission of each of the following diseases take place? (a)Amoebiasis (b)Malaria (c)Ascariasis (d)Pneumonia Solution:...
Read More →Find the equation of the line parallel to y-axis and drawn through
Question: Find the equation of the line parallel toy-axis and drawn throughthe point of intersection of the lines $x-7 y+5=0$ and $3 x+y=0$. Solution: The equation of any line parallel to they-axis is of the form x=a (1) The two given lines are x 7y+ 5 = 0 (2) 3x+y= 0 (3) On solving equations $(2)$ and $(3)$, we obtain $x=-\frac{5}{22}$ and $y=\frac{15}{22}$. Therefore, $\left(-\frac{5}{22}, \frac{15}{22}\right)$ is the point of intersection of lines (2) and (3). Since line $x=$ a passes through...
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Question: Find $\frac{d y}{d x}$ : $\sin ^{2} x+\cos ^{2} y=1$ Solution: The given relationship is $\sin ^{2} x+\cos ^{2} y=1$ Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}\left(\sin ^{2} x+\cos ^{2} y\right)=\frac{d}{d x}(1)$ $\Rightarrow \frac{d}{d x}\left(\sin ^{2} x\right)+\frac{d}{d x}\left(\cos ^{2} y\right)=0$ $\Rightarrow 2 \sin x \cdot \frac{d}{d x}(\sin x)+2 \cos y \cdot \frac{d}{d x}(\cos y)=0$ $\Rightarrow 2 \sin x \cos x+2 \cos y(-\sin y) \cdot \frac{d...
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Question: Find $\frac{d y}{d x}$ : $\sin ^{2} x+\cos ^{2} y=1$ Solution: The given relationship is $\sin ^{2} x+\cos ^{2} y=1$ Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}\left(\sin ^{2} x+\cos ^{2} y\right)=\frac{d}{d x}(1)$ $\Rightarrow \frac{d}{d x}\left(\sin ^{2} x\right)+\frac{d}{d x}\left(\cos ^{2} y\right)=0$ $\Rightarrow 2 \sin x \cdot \frac{d}{d x}(\sin x)+2 \cos y \cdot \frac{d}{d x}(\cos y)=0$ $\Rightarrow 2 \sin x \cos x+2 \cos y(-\sin y) \cdot \frac{d...
Read More →Find the greatest number that will divide 445,
Question: Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively. Solution: Find the greatest number that divides 445, 572 and 699 and leaves remainders of 4, 5 and 6 respectively. The required number when divides 445,572 and 699 leaves remainders 4,5 and 6 this means $445-4=441,572-5=567$ and $699-6=693$ are completely divisible by the number. Therefore, the required number = H.C.F. of 441, 567 and 693. First consider 441 and 567. By applying Eucli...
Read More →In which way has the study of biology helped us to control infectious diseases?
Question: In which way has the study of biology helped us to control infectious diseases? Solution: Various advancements that have occurred in the field of biology have helped us gain a better understanding to fight against various infectious diseases. Biology has helped us study the life cycle of various parasites, pathogens, and vectors along with the modes of transmission of various diseases and the measures for controlling them. Vaccination programmes against several infectious diseases such...
Read More →What are the various public health measures, which you would suggest as safeguard against infectious diseases?
Question: What are the various public health measures, which you would suggest as safeguard against infectious diseases? Solution: Public health measures are preventive measures which are taken to check the spread of various infectious diseases. These measures should be taken to reduce the contact with infectious agents. Some of these methods are: (1)Maintenance of personal and public hygiene:It is one of the most important methods of preventing infectious diseases. This measure includes maintai...
Read More →What is the largest number that divides 626,
Question: What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively. Solution: We need to find the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively. The required number when divides 626,3127 and 15628 leaves remainders 1,2 and 3 this means $626-1=625,3127-2=3125$ and $15628-3=15625$ are completely divisible by the number. Therefore, the required number = H.C.F. of 625, 3125 and 15625. First we cons...
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Question: Find $\frac{d y}{d x}$ : $\sin ^{2} y+\cos x y=\pi$ Solution: The given relationship is $\sin ^{2} y+\cos x y=\pi$ Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}\left(\sin ^{2} y+\cos x y\right)=\frac{d}{d x}(\pi)$ $\Rightarrow \frac{d}{d x}\left(\sin ^{2} y\right)+\frac{d}{d x}(\cos x y)=0$ ...(1) Using chain rule, we obtain $\frac{d}{d x}\left(\sin ^{2} y\right)=2 \sin y \frac{d}{d x}(\sin y)=2 \sin y \cos y \frac{d y}{d x}$ ...(2) $\frac{d}{d x}(\cos x ...
Read More →Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3,
Question: Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively. Solution: We need to find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively. The required number when divides 280 and 1245 , leaves remainder 4 and 3 , this means $280-4=276$ and $1245-3=1242$ are completely divisible by the number. Therefore, the required number = H.C.F. of 276 and 1242. By applying Euclids division lemma $1242=276 \times 4+...
Read More →Find the perpendicular distance from the origin to the line joining the points
Question: Find the perpendicular distance from the origin to the line joining the points$(\cos \theta, \sin \theta)$ and $(\cos \phi, \sin \phi)$ Solution: The equation of the line joining the points $(\cos \theta, \sin \theta)$ and $(\cos \phi, \sin \phi)$ is given by $y-\sin \theta=\frac{\sin \phi-\sin \theta}{\cos \phi-\cos \theta}(x-\cos \theta)$ $y(\cos \phi-\cos \theta)-\sin \theta(\cos \phi-\cos \theta)=x(\sin \phi-\sin \theta)-\cos \theta(\sin \phi-\sin \theta)$ $x(\sin \theta-\sin \phi)...
Read More →Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.
Question: Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively. Solution: We need to find the greatest number which divides 285 and 1249 leaving remainder 9 and 7 respectively. The required number when divides 285 and 1249 , leaves remainder 9 and 7 , this means $285-9=276$ and $1249-7=1242$ are completely divisible by the number. Therefore, the required number = H.C.F. of 276 and 1242. By applying Euclids division lemma $1242=276 \times 4+138$ $276=138 \ti...
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Question: Find $\frac{d y}{d x}$ : $x^{3}+x^{2} y+x y^{2}+y^{3}=81$ Solution: The given relationship is $x^{3}+x^{2} y+x y^{2}+y^{3}=81$ Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}\left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)=\frac{d}{d x}(81)$ $\Rightarrow \frac{d}{d x}\left(x^{3}\right)+\frac{d}{d x}\left(x^{2} y\right)+\frac{d}{d x}\left(x y^{2}\right)+\frac{d}{d x}\left(y^{3}\right)=0$ $\Rightarrow 3 x^{2}+\left[y \frac{d}{d x}\left(x^{2}\right)+x^{2} \frac{d y}{d...
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