Determine the prime factorisation of each of the following positive integer:
Question: Determine the prime factorisation of each of the following positive integer: (i) 20570 (ii) 58500 (iii) 45470971 Solution: TO EXPRESS: each of the following numbers as a product of their prime factors (i) 20570 $20570=2 \times 5 \times 11^{2} \times 17$ (ii) 58500 $58500=2^{2} \times 3^{2} \times 5^{3} \times 13$ (iii) 45470971 $45470971=7^{2} \times 13^{2} \times 17^{2} \times 19$...
Read More →In your view what motivates youngsters to take to alcohol or drugs and how can this be avoided?
Question: In your view what motivates youngsters to take to alcohol or drugs and how can this be avoided? Solution: Many factors are responsible for motivating youngsters towards alcohol or drugs. Curiosity, need for adventure and excitement, experimentation are the initial causes of motivation. Some youngsters start consuming drugs and alcohol in order to overcome negative emotions (such as stress, pressure, depression, frustration) and to excel in various fields. Several mediums like televisio...
Read More →Find the equation of the line passing through the point of intersection of the lines
Question: Find the equation of the line passing through the point of intersection of the lines 4x+ 7y 3 = 0 and 2x 3y+ 1 = 0 that has equal intercepts on the axes. Solution: Let the equation of the line having equal intercepts on the axes be $\frac{x}{a}+\frac{y}{a}=1$ Or $x+y=a$ $\ldots(1)$ On solving equations $4 x+7 y-3=0$ and $2 x-3 y+1=0$, we obtain $x=\frac{1}{13}$ and $y=\frac{5}{13}$. $\therefore\left(\frac{1}{13}, \frac{5}{13}\right)$ is the point of intersection of the two given lines....
Read More →Express each of the following integers as a product of its prime factors:
Question: Express each of the following integers as a product of its prime factors: (i) 420 (ii) 468 (iii) 945 (iv) 7325 Solution: TO EXPRESS: each of the following numbers as a product of their prime factors (i) 420 $420=2^{2} \times 3 \times 5 \times 7$ (ii) 468 $468=2^{2} \times 3^{2} \times 13$ (iii) 945 $945=3^{3} \times 5 \times 7$ (iv) 7325 $7325=5^{2} \times 293$...
Read More →Why is that once a person starts taking alcohol or drugs,
Question: Why is that once a person starts taking alcohol or drugs, it is difficult to get rid of this habit? Discuss it with your teacher. Solution: Drug and alcohol consumption has an inherent addictive nature associated with euphoria and a temporary feeling of well-being. Repeated intake of drugs increases the tolerance level of the bodys receptors, leading to more consumption of drugs....
Read More →Find :
Question: Find $\frac{d y}{d x}$ : $y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right), 0x1$ Solution: The given relationship is $y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ $y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ $\Rightarrow \sin y=\frac{1-x^{2}}{1+x^{2}}$ Differentiating this relationship with respect tox, we obtain $\frac{d}{d x}(\sin y)=\frac{d}{d x}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ ...(1) Using chain rule, we obtain $\frac{d}{d x}(\sin y)=\cos y \cdot \frac{d y}{d x}$ $\...
Read More →A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind.
Question: A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin? Solution: The merchant has 3 different oils of 120 liters, 180 liters and 240 liters respectively. So the greatest capacity of the tin for filling three different types of oil is given by the H.C.F. of 120,180 and 240. So first we will calculate H.C...
Read More →Find the equation of the lines through the point (3, 2)
Question: Find the equation of the lines through the point (3, 2)which make an angle of $45^{\circ}$ with the line $x-2 y=3$. Solution: Let the slope of the required line bem1. The given line can be written as $y=\frac{1}{2} x-\frac{3}{2}$, which is of the form $y=m x+c$ $\therefore$ Slope of the given line $=m_{2}=\frac{1}{2}$ It is given that the angle between the required line and line $x-2 y=3$ is $45^{\circ}$. We know that if $\theta$ isthe acute angle between lines $I_{1}$ and $I_{2}$ with...
Read More →Do you think that friends can influence one to take alcohol/drugs?
Question: Do you think that friends can influence one to take alcohol/drugs? If yes, how may one protect himself/herself from such an influence? Solution: Yes, friends can influence one to take drugs and alcohol. A person can take the following steps for protecting himself/herself against drug abuse: (a)Increase your will power to stay away from alcohol and drugs. One should not experiment with alcohol for curiosity and fun. (b)Avoid the company of friends who take drugs. (c)Seek help from paren...
Read More →During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each.
Question: During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy? Solution: We are given that during a sale, color pencils were being sold in packs of 24 each and crayons in packs of 32 each. If we want full packs of both and the same number of pencils and crayons, we need to find the number of each we need to buy. Given that Number of colo...
Read More →144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen.
Question: 144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have? Solution: Given that 144 cartons of coke cans and 90 cartons of Pepsi cans are to be stacked in a canteen. If each stack is of the same height and contains cartons of the same drink We need to find the greatest number of cartons, each stack would have Giv...
Read More →If three lines whose equations are
Question: If three lines whose equations are$y=m_{1} x+c_{1}, y=m_{2} x+c_{2}$ and $y=m_{3} x+c_{3}$ are concurrent, then show that $m_{1}\left(c_{2}-c_{3}\right)+m_{2}\left(c_{3}-c_{1}\right)+m_{3}\left(c_{1}-c_{2}\right)=0$. Solution: The equations of the given lines are y=m1x+c1 (1) y=m2x+c2 (2) y=m3x+c3 (3) On subtracting equation (1) from (2), we obtain $0=\left(m_{2}-m_{1}\right) x+\left(c_{2}-c_{1}\right)$\ $\Rightarrow\left(m_{1}-m_{2}\right) x=c_{2}-c_{1}$ $\Rightarrow x=\frac{c_{2}-c_{...
Read More →List the harmful effects caused by alcohol/drug abuse.
Question: List the harmful effects caused by alcohol/drug abuse. Solution: Alcohol and drugs have several adverse effects on the individual, his family, and the society. A.Effects of alcohol: Effects on the individual: Alcohol has an adverse effect on the body of an individual. When an individual consumes excess alcohol, it causes damage to the liver and the nervous system. As a result, other symptoms such as depression,fatigue, aggression, loss of weight and appetite may also be observed in the...
Read More →Two brands of chocolates are available in packs of 24 and 15 respectively.
Question: Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy? Solution: We are given that two brands of chocolates are available in packs of 24 and 15 respectively. If he needs to buy an equal number of chocolates of both kinds, then find least number of boxes of each kind he would need to buy. Given that Number of chocolates of1stbrand in one p...
Read More →Explain what is meant by metastasis.
Question: Explain what is meant by metastasis. Solution: The property of metastasis is exhibited by malignant tumors. It is the pathological process of spreading cancerous cells to the different parts of the body. These cells divide uncontrollably, forming a mass of cells called tumor. From the tumor, some cells get sloughed off and enter into the blood stream. From the blood stream, these cells reach distant parts of the body and therefore, initiate the formation of new tumors by dividing activ...
Read More →How is a cancerous cell different from a normal cell?
Question: How is a cancerous cell different from a normal cell? Solution:...
Read More →Find the value of p so that the three lines 3x + y – 2 = 0,
Question: Find the value of $p$ so that the three lines $3 x+y-2=0, p x+2 y-3=0$ and $2 x-y-3=0$ may intersect at one point. Solution: The equations of the given lines are 3x+y 2 = 0 (1) px+ 2y 3 = 0 (2) 2xy 3 = 0 (3) On solving equations (1) and (3), we obtain x= 1 andy= 1 Since these three lines may intersect at one point, the point of intersection of lines (1) and (3) will also satisfy line (2). p(1) + 2 (1) 3 = 0 p 2 3 = 0 p= 5 Thus, the required value ofpis 5....
Read More →A mason has to fit a bathroom with square marble tiles of the largest possible size.
Question: A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in cinches of the tile required that has to be cut and how many such tiles are required? Solution: A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10ft. by 8ft. We need to find the size in inches of the tile required that has to be cut and number of such tiles are required...
Read More →What is the mechanism by which the AIDS virus causes deficiency of immune system of the infected person?
Question: What is the mechanism by which the AIDS virus causes deficiency of immune system of the infected person? Solution: AIDS (Acquired Immuno Deficiency Syndrome) is caused by the Human immunodeficiency virus (HIV) via sexual or blood-blood contact. After entering the human body, the HIV virus attacks and enters the macrophages. Inside the macrophages, the RNA of the virus replicates with the help of enzyme reverse transcriptase and gives rise to viral DNA. Then, this viral DNA incorporates...
Read More →Find the area of the triangle formed by the lines
Question: Find the area of the triangle formed by the lines$y-x=0, x+y=0$ and $x-k=0$ Solution: The equations of the given lines are yx= 0 (1) x+y= 0 (2) xk= 0 (3) The point of intersection of lines (1) and (2) is given by x= 0 andy= 0 The point of intersection of lines (2) and (3) is given by x=kandy= k The point of intersection of lines (3) and (1) is given by x=kandy=k Thus, the vertices of the triangle formed by the three given lines are (0, 0), (k, k), and (k,k). We know that the area of a ...
Read More →Find :
Question: Find $\frac{d y}{d x}$ : $y=\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right), 0x1$ Solution: The given relationship is, $y=\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ $\Rightarrow \cos y=\frac{1-x^{2}}{1+x^{2}}$ $\Rightarrow \frac{1-\tan ^{2} \frac{y}{2}}{1+\tan ^{2} \frac{y}{2}}=\frac{1-x^{2}}{1+x^{2}}$ On comparing L.H.S. and R.H.S. of the above relationship, we obtain $\tan \frac{y}{2}=x$ Differentiating this relationship with respect tox, we obtain $\sec ^{2}\left(\frac{\mathrm{y...
Read More →What are the various routes by which transmission of human immunodeficiency virus takes place?
Question: What are the various routes by which transmission of human immunodeficiency virus takes place? Solution: AIDS (Acquired Immuno Deficiency Syndrome) is caused by the Human immunodeficiency virus (HIV). It has the following modes of transmission: (a)Unprotected sexual contact with an infected person. (b)Transfusion of blood from a healthy to an infected person. (c)Sharing infected needles and syringes. (d)From an infected mother to a child through the placenta....
Read More →15 pastries and 12 biscuit packets have been donated for a school fete.
Question: 15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain? Solution: We are given that 15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. We need ...
Read More →Draw a well-labelled diagram of an antibody molecule.
Question: Draw a well-labelled diagram of an antibody molecule. Solution:...
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...
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