The Source of the Cosmos - The Official Website of the Sri ...srividya.org/wp-content/uploads/2013/01/Blossom-20-Petal-2.pdf · chara puja.” By exposing pancha puja fully I created

June 2016Blossom 20 Petal 2

Sri ChakraThe Source of the Cosmos

The Journal of the Sri Rajarajeswari Peetam, Rush, NY

The Sri Rajarajeswari Peetam ~ 6980 East River Road ~ Rush, NY 14543 ~ Phone: (585) 533 - 1970

Devi willing, the next issue of the Sri Chakra will be up on the temple’s website at the beginning of September 2016.

This magazine cannot keep publishing without contributions! Articles, poems, stories and photos about any spiritual topic are welcomed.

The next deadline for article submission is August 9, 2016. Please e-mail us with your contributions or feedback about this issue at [email protected] or talk to Kamya or Abhi at the temple.

Sri Gurubhyo Namaha!

Our special thanks and gratitude to this issue’s volunteers: Aiya, Vilas Ankolekar, Shambhavi

Dinakar, Colin Earl, and Virroshi Sriganesh.

Event Date/time LocationAnnual Alankara Utsavam July 1-4 Sri Rajarajeswari Peetam, full property

Guru Poornima July 19 Sri Rajarajeswari Peetam, full property

Vibhuti Saivaite Immersion (camp) July 30 - August 6 Sri Rajarajeswari Peetam, full property

Aadi Amavasyai August 2 Sri Rajarajeswari Peetam, outdoors

Aadi Puram August 5 Sri Rajarajeswari Peetam, yajnashala

Varalakshmi vratham August 12 Sri Rajarajeswari Peetam, yajnashala

Upcoming Events

In Three Months

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June Newsletter

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We all love hanging out with Aiya, drinking tea, and sometimes getting some guidance on how to travel down our own paths to the Devi. The following questions were asked to and answered by him at a workshop abroad.

Q & Aiya

Q: What is the significance of the pasham and ankusham?

Aiya: Raga-svarupa pashadya, krodha-kara_ankushojvala. Raga in this context means attachment, and the Devi is the one who cre-ates attachment between father and son, son and mother, mother and daughter, brother and sister, and so on. If there was no attachment, loka-karyam could not take place; everyone would be sitting in a cave somewhere. That is why she is called Lokamaya. So that pasham represents the attachments you have in life.

Anger is represented by the ankusham, which is the elephant goad.

Q: How long should your daily puja (nitya utsavam) be in your home?

Aiya: Let me tell you something from my 40+ years of doing this—don’t make your nitya utsavam long in your home. In 10 minutes, you should go in to the puja area, do your thing, and get out. If you sit there for 2 hours every day, after a few weeks or a few months, your body is going to find excuses not to do it at all. Maybe you have a backache, maybe you’re too tired—and that is the beginning of the end. Once you fall off the bandwagon, you will never climb back on again. So keep your daily rituals to a minimum.

Q: Where is Devi worship per-formed the most?

Aiya: Devi worship is most devel-oped in Andhra Pradesh—Lalita worship. Durga worship is devel-oped in Bengal; in some areas it takes the form of Kali worship. In Maharashtra, Ganesha worship is perfected. In Karnataka, it is Shiva worship. In Kerala, Krishna worship is perfected. And in Tamil Nadu, Murugan worship is per-fected. You can’t go anywhere in Tamil Nadu without seeing some-thing having to do with Murugan. The same way, everywhere you look in Andhra, shops and signs and temples all have to do with the Devi. And in Andhra, there is a place called East Godavari District. If you hear the pandits there chant-ing, you will cry because their dic-tion and pronunciation and attention to detail is so perfect.

Q: Why is archana done so often in temples?

Aiya: Ahhh. When you go into a temple, the first thing you see is a large board with all the prices and different levels of archanas they can perform for you. This is a normal situation—you walk into any temple in the world and this is what’s there. You

think it’s a business? That’s what we have brought it down to. Let me ask you a question. Say there is the annual fes-

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Gurus, students, and a whole lotta numbers

tival coming up at the local temple, and I decide to give $3000 because I’m under the influence of seven-and-a-half years of Saturn. And then there’s a young man who’s on a student Visa from India and he comes to the temple during the fes-tival. He only has $3 in his pocket that he can spare. Do you think the person who gave $3000 is going to get more grace than the person who gave $3? For Ganapathi, all that is pieces of paper. Whether it is

$3 million or $3, he doesn’t care. What he’s looking for is the attitude with which you give. If you give it thinking, “This community must enjoy this temple,” or, “Everybody here should get a share in Ga-napathi’s grace,” then yes, you have the proper attitude to receive his grace. But that’s not how our people think. We think, “I gave $3000 so I should be standing closer than that guy who gave $300!” I was talking to someone earlier about those people on the boards that make decisions about temples—they will only go to a temple if they know they’re going to be garlanded. Why? Does that make you better than anyone else, or will it give you moksha? We have brought our reli-gion and religious practices down

to a business. And it is actively—pardon me for saying this—it is actively being encouraged by the temple boards and the priests and everyone else who manages a tem-ple. I used to get very angry when I was 16 or 17 years old and had been reading Karl Marx and Lenin, and was fascinated with the concept that everyone should own everything. My mother used to tell me to sing the Thevaram, and I told her, “Bring Shiva down here and

I’ll ask him some choice questions before singing for him. You look everywhere around you—all the good people are suffering while the fellows who are crooks are doing well. What kind of a God is this?” That’s when Amma told me, “If you’re not a communist before you’re 21, you have no heart.

If you’re still a communist after 21, you have no head.”

Every temple (even the Rajarajeswari Peetam) performs archanas, because people want a quick and easy way to receive blessings, even if the blessings are miniscule. Here, archana was performed on Vijayadasami day, 2010, before the Navarathri chariot.

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The following article contains excerpts of several posts taken from the blog of Colin Earl, which have been compiled into one piece and republished here.

It really bothers me when people ask me to e-mail or record things that were taught to me orally. Oral traditions are taught that way for a reason. The pace of modern life is a reason to hold on to tradi-tion more, not sacrifice it. More often these same peo-ple want the information bundled that way so that they can repackage it more easily for reasons of name and fame, disguised as “wanting to share.” This happens again and again on many fronts and makes me less and less willing to share. If you are serious about developing your own practice I will share with you using the same methods by which I was taught. If you want to teach and make a name

for yourself, please go elsewhere. Teaching should be a natural evolution of the flowering of

The Ideal Studentby Colin Earl

your own experience, not repeating of your teacher’s words. One should travel many miles down the road before giving a tour of the first mile post. My parama-guru (guru’s guru), Guruji, always said his problem is those who want to practice don’t want to teach and those who want to teach don’t want to practice. What I learned was en-coded—you had to apply yourself to reap the fruits; it was not spoon-fed off of the silver platter that everyone asks for today. I never understood why Guruji never fully explained pancha-upachara puja, and more complicated pujas were taught first. Aiya has always said “I don’t give shortcuts because that is all they will ever do.” Now I get it—by doing longer pujas they come to un-derstand the essence of the simpler ones themselves through experi-ence. By explaining the simple well, I robbed them of this experience and

gave them the sense that they “have it,” instead of earning it and a false sense of empowerment.

Aiya says, “Pancha puja is like McDonald’s, you get fed but do not have the experience of a nutritious meal like Sodasa Upa-chara puja.” By exposing pancha puja fully I created fast food chains, which I now regret. I have always said that puja is like coding. You must learn how to frame your request, to get the result that you want. I sit here studying design patterns and digesting the jargon that I need to know to be able to communicate with other develop-ers and discuss these principles intelligently. I liken it the need to understand Sanskrit in regards to puja. Students often complain about the use of Sanskrit terminology, however puja without Sanskrit is like programming without jargon. So much can be communi-cated quickly with the right nama. It is like any other science, one must learn the lingo to be effective. Dur-

ing my time at Devipuram I was able to grasp and under-stand complex ideas quickly, because I was able to under-stand the terminology. Even when my teacher’s language was Telugu and mine English, we could understand each other clearly and precisely through Sanskrit terms. Puja is such a compressed science and you have to take the time to learn the code to be effective. It takes a com-

mitment, a life-long one with con-sistency, not with stops and starts. Often, people come to me

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asking me to teach them this, or teach them that, or e-mail questions about this or that. It is almost as if I am a mail-order course catalog or spiritual ATM machine. This is not the way I learned, so why should I teach in this matter? Would this be fair to the body of knowledge that is the Vidya? I learned from watching my teachers and absorbing what they were offering in the moment they offered it. It was not on-demand training how, what and when I wanted. It was organic, it was natu-ral. It produced a background and a context to support all those juicy tidbits that so many want to pluck from the tree, without watering and tending the roots. That is not to say that there were not well-timed and placed questions that inspired pieces to be shared; there were many. However these questions come from having the background, and the nature of the backlighting of these questions inspired the knowledge to come forth. These same questions asked in a differ-ent moment would not yield the same results. Most of what I learned was sitting around the Guru with my spiritual brothers and sisters having tea, not in work-shops or structured teaching. You had to look at the group and know how your question would be answered depending on the audience and ask the right ques-tion at the right time. Often people say, my

busy life doesn’t allow for such and such; my response is, then change your life. We all make sacrifices and choices and those sacrifices and choices are ours.

Advice

What I am trying to say is attend rituals, watch, and learn. Ask ques-tions at the right times, when the audience feels correct, not following a pre-planned agenda. Learn what is offered when it is offered. These are important skills for the spiritual student to have, and they pay great dividends. The second point is don’t learn with re-teaching in your head as you are learning. You can tell when people are doing this by the questions that they ask. Teaching should be the natural progression of the experience you have had in

working with the knowledge. It should be from your own experi-ence, not the rehashing of your teachers. Work with things for a few years on your own before teaching so you have a background of experi-ence to draw from.

Conclusion

In conclusion, watch and learn, then practice and internalize. Less ambi-tion, more sadhana. Guruji said, “Those who want to teach don’t want to practice and those who want to practice don’t want to teach.” Be in the second group, absorb and learn. Your teacher will tell you when it is time to teach.

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As human beings, our supreme gift over all other forms of life is consciousness. Conscience is the great observer. It is the very criteria for any kind of awareness to manifest. Awareness can, over time, build up and lead to choices, which in turn, lead to preferences. This is harmless if there was just one observer. Most people get their first dose of politics in the form of sibling rivalry or as camaraderie among friends. As time progresses, the stakes get higher and the degree of its impact on one’s life increases proportionately. It is only then that people truly recognize “politics.” It can progress from two individuals to two groups and so on. Politics has always been there in our lives. The more stubborn our opinions, the more politics we will have in our lives. Even within realms of spirituality, politics has always existed. The ever-blessed Adi Shankaracharya, thought to be Shiva incarnate, blessed on numerous occasions by the divine mother herself had to witness (the play of Maya) the politics played out among his

If You were Totakaby Shambhavi Dinakar

disciples. Adi Shankara-charya had many disciples. Most of the records per-tain to only four of them—Sureshwara or Mandana Mishrar (the eloquent), Has-tamalaka (the dexter-ous), Padmapada (the ever cheerful) and Totaka (the dullard). Every disciple was extraordinary. During Shankara’s discourses, Totaka would often close his eyes and listen to the Guru as though in a

trance. Whatever wavelength Totaka operated on was known only to the Guru i.e. Adi Shankara. The other disciples knew only what they saw, i.e. Totaka sleep-ing during the lessons imparted by the Guru. Adi Shankara was a very evocative speaker. He is known to have caused a rain shower of gold by his moving hymn to Goddess Lakshmi whom he identified in an impoverished woman who gave him all her eatables in charity. So in defense of his other disciples, all they knew was that Totaka was a fool to miss hearing those beautiful words. With what they knew, they formed opinions on Totaka’s merit and worth. Over time, their beliefs were strengthened. With each ser-mon from the Guru, they assumed that their knowl-edge had expanded while Totaka, who wasn’t paying attention remained ignorant. Let us think from a logical perspective now. Can a disciple ever judge another disciple’s worth? Most people would answer with a resounding “no.” But if you were in the same position as Sureshwara, Hastamalaka, or Padmapada and loved your Guru

Politics and Choices

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dearly and felt an anguish that amongst you there was someone un-worthy, would you honestly answer the question in the same way. Would your emotions sway your logic? Can you be dispassionate and stick with common logic? The true worth or merit of a disciple can only ever be determined by the Guru. Why? Because we don’t know what the Guru knows. The Guru’s awareness is not that of a single individual. Totaka continued in his single-pointed devotion to his Guru, being utterly unaware of other disciples. Adi Shankara eventually silenced internal debates of merit and worth by bestowing his grace and imparted all the Shastras to Totaka in a single glance. No one knows what came of that lesson for the other disciples. It is simply assumed that they were disciplined enough to be able to see their ego in

their actions and thoughts. If we were in Totaka’s shoes, what would we do? Could we be utterly unaffected by the other disciples and what they would say? A clear distinction should exist between compassion for others on their journey and for the ultimate goal of emancipation. Through compassion the water can be brought from the river and put in the horse’s mouth, but only the horse can swallow it. If there is a collision of purpose between these two ideologies viz. compassion and liberation, no one is benefitted—you or the benefac-tors of your compassion. Even in group worship we can chant the same things and perform the same rituals but we cannot decide what each of it means for anyone else but ourselves. Nor can we judge them for their defini-tion. We can however influence the

nature of group wor-ship for the better or for worse. Therein lies discretion of what we bring to the group. Do we bring harmony? Are we helping, hurt-ing, indifferent or invested/attached? For Totaka, if the focus was not cen-tered on the Guru, his reality could be very different. It is also worth noting that ultimately, it was only Totaka that could determine his fate. The others, al-though they seemed to have power, nev-er really did. We can only ever influence the fate of others but never decide it. That always depends on the individual alone.

By some miracle, While the other disciples were seated around Shankaracharya, Totaka was often by the river, listening to the discourse while washing the guru’s clothes

here in the northeast of a continent many miles from India, many have found the grace of the Guru and a magnanimous lineage. It can mean only what you choose for it to mean. And you can only choose for your-self. Never for another. We can look back at these stories and place ourselves in the shoes of each character and be aware of more than one paradigm/context only if we choose to pause the ego of stubborn opinions. Choices are not something that Maya will easily allow you to es-cape. With every choice, She opens the possibility of many others. We must consistently choose towards our ultimate goal. Every time.

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M A T H E M A T I C A L L E N S

The

Maha Meru Virroshi Sriganesh

Sri Vidya In Sanatana Dharma, more widely known to the West as Hinduism, there are numerous paths towards spiritual fulfillment and self-‐realization. One such system of prayer conceptualizes the ultimate universal energy in the form of the Divine Mother and is called Sri Vidya, which means “knowledge of the Goddess.” This worship of the Sacred Feminine has been prevalent in India for centuries, and continues to be practiced all around the world today. Sri Rajarajeswari Peetam, Sri Vidya Temple Society (SVTS) is a prominent teaching temple that propagates this form of worship in Canada and the United States. The images found in this document were taken during various festivals conducted at SVTS.

Maha Meru In the Sri Vidya tradition of Hindu worship, the energy of the universe is depicted as geometric designs formed by a number of interlocking polygons, often triangles, emanating from a central point and forming a series of enclosures. Each such arrangement is called a Yantra, and pertains to a particular Hindu deity. If this diagram is projected into three dimensions, each Yantra becomes the Meru for the corresponding deity. The most important and widely worshipped of these are the Sri Yantra and its corresponding Maha Meru. The Sri Yantra is formed by nine isosceles triangles of different sizes emanating from a central point. Four have upward apices, denoting the Male Principle, while five have downward apices, denoting the Female Principle. Thus, Hindus worship the Sri Yantra and the Maha Meru as the union of both the universal male and female energies. These sacred designs are considered powerful and thus widely worshipped by devotees for material and spiritual blessings alike.

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SPRING 2016 THE LOREM IPSUMS MATHEMATICAL LENS – MAHA MERU VIRROSHI SRIGANESH

The Mathematical Meru … Below are mathematical questions related to the Sri Chakra and the Maha Meru. Complete the following questions.

1. Figure 1 shows a Yantra created strictly using flowers, rice and betel leaves. As displayed, on each side of the outer frame (created by betel leaves), there is an opening to enter the Yantra. Suppose you are asked to create a box to store this masterpiece for later use. You are given that an entire side length is 2.4 𝑚𝑚 long. The opening on each side is a square with sides that are 0.3 𝑚𝑚 long. If the box must be 0.4 𝑚𝑚 high, calculate its surface area, which is an unusual rectangular prism.

2. Figure 2 displays a Meru created strictly uses 1008 conches. Your classmate was present during this festival and decided to measure the distance between the five concentric circles shows in Figure 2. She discovered that the distance between each one of those circles is 28 𝑐𝑐𝑐𝑐. She also measured the length of the conch, which happens to be 20 𝑐𝑐𝑐𝑐.

a. If the innermost circle has 58 conches, the second innermost circle has 72 conches, the third has 86, fourth has 100 and the fifth has 114 conches, determine the circumference of each circle.

b. Using your results from part a., determine whether there is a relationship between the circumferences of each circle and its radius.

3. Look at the pyramid-‐like shape in the centre of the Meru in Figure 2. The bottom layer in the pyramid has 43 conches. The layer above has 36. The layer above that has 29 conches; then 22 conches; then 15 conches; then 8 conches. Determine whether the layer to the number of conches on each layer is a linear relation, without graphing. Construct a table of values, if necessary.

Figure 1. A Yantra

Figure 2. A Meru

4. You volunteer yourself to help create the Meru displayed in Figure 2. You are assigned to add rice into each one of the conches in a specific manner. (Note: There are 1008 conches). You are asked to start with the one of the conches in the very top layer and work your way downwards, then outwards. Your task is to double the amount of grains of rice you add for each conch, starting with 1 grain for the first conch in the top layer. For example, Conch 1 = 1 grain, Conch 2 = 2 grains, Conch 3 = 4 grains, Conch 4 = 8 grains, etc. Is this linear? Why or why not? Graph this relationship.

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SPRING 2016 THE LOREM IPSUMS MATHEMATICAL LENS – MAHA MERU VIRROSHI SRIGANESH

5. You have assisted in creating the Yantra in Figure 1. Unfortunately, you come to realize that it was created in an inconvenient space within the temple. Hence, instead of dismantling it, you and your peers choose to carry the Yantra to its appropriate location (assuming it was created on a platform instead of tiles as shown). You realize that this must be done at a very slow pace since the flower petals and the betel leaves might fly off. The equation 𝑑𝑑 = 600 − 20𝑡𝑡 represents your distance away from the new position, which is initially 600 𝑐𝑐𝑐𝑐 away, and the distance you travel every minute, which is 20 𝑐𝑐𝑐𝑐/𝑚𝑚𝑚𝑚𝑚𝑚. Using an algebraic approach, determine how long it will take you to get half way through the distance.

10. Figure 3 shows a cake that has been frosted to resemble a Yantra. You volunteer to frost the entire cake since you love frosting. After 2 hours you have completed frosting 20% of the cake. After 6 hours you have completed frosting 70% of the cake. What is the average rate of change in your work?

6. Consider the circle created by the pink flowers in Figure 1.

a. Suppose this circle, on a Cartesian plane, is centered at the origin and passes through the point (6, 0). Determine the equation of the circle.

b. Suppose this circle, on a Cartesian plane, is centered at (6,28) and has a radius of 7. Determine the equation of the circle.

7. Look at the triangles filled with dark-‐colour petals in the centre of the Yantra in Figure 1. Consider one of the triangles and denote ∆𝐴𝐴𝐴𝐴𝐴𝐴.

a. Suppose ∡𝐴𝐴 = 95° , ∡𝐵𝐵 = 41° , 𝑏𝑏 = 20 𝑐𝑐𝑚𝑚 . Determine the length of 𝑎𝑎.

b. Suppose ∡𝐶𝐶 = 100° , 𝑏𝑏 = 5 𝑐𝑐𝑐𝑐 , 𝑐𝑐 = 15 𝑐𝑐𝑐𝑐 . Determine ∡𝐵𝐵.

8. Look at the triangles filled with dark-‐colour petals in the centre of the Yantra in Figure 1. Consider one of the triangles and denote ∆𝐴𝐴𝐴𝐴𝐴𝐴.

a. Suppose ∡𝐴𝐴 = 21° , 𝑏𝑏 = 150 𝑐𝑐𝑐𝑐 , 𝑐𝑐 = 60 𝑐𝑐𝑐𝑐. Determine the length of 𝑎𝑎.

b. Suppose 𝑎𝑎 = 11.8 𝑐𝑐𝑐𝑐 , 𝑏𝑏 = 8.6 𝑐𝑐𝑐𝑐 , 𝑐𝑐 = 6.2 𝑐𝑐𝑐𝑐. Determine ∡𝐴𝐴.

9. Convert your answers from Question 7b and Question 8b from degrees to radians.

Figure 3. A Yantra cake

11. Consider the triangles filled with dark-‐colour petals in the centre of the Yantra in Figure 1. There are 30 of them. You are given 4 different colour petals and asked to replace the dark colour petals in any pattern you would like. How many different ways can this be done?

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SPRING 2016 THE LOREM IPSUMS

1. To do this, we shall find the surface area of the bigger square first. First, we shall calculate the area of the top and base faces: 2.4 × 2.4 = 5.76 𝑚𝑚! 5.76 × 2 = 11.52 𝑚𝑚! Then, calculate the area of the 4 side faces: 2.4 × 0.4 = 0.96 𝑚𝑚! 0.96 × 4 = 3.84 𝑚𝑚! The sum of these areas will give us the surface area of the bigger square: SAbigger square = 11.52+ 3.84 = 𝟏𝟏𝟏𝟏.𝟑𝟑𝟑𝟑 𝒎𝒎𝟐𝟐 Now, we shall calculate the surface area of the smaller squares on each side of the bigger squares made from the openings to the Yantra. (Note: The area calculate of only 5 sides is required for this portion as the 6th side’s area has already been calculated with the bigger triangle (the 6th face is shared with the sides of the bigger triangle). First, we shall calculate the area of the top and base faces: 0.3 × 0.3 = 0.09 𝑚𝑚! 0.09 × 2 = 0.18 𝑚𝑚! Then, calculate the area of the 3 side faces: 0.3 × 0.4 = 0.12 𝑚𝑚! 0.12 × 3 = 0.36 𝑚𝑚! The sum of these areas will give us the surface area of the smaller square: SAsmaller square = 0.18+ 0.36 = 0.54 𝑚𝑚! Since there are 4 of these smaller triangles, we shall multiply the surface area of the smaller square by 4: SAsmaller squares = 0.54 × 4 = 𝟐𝟐.𝟏𝟏𝟏𝟏 𝒎𝒎𝟐𝟐 The total surface area of the unusual rectangular prism is: SAtotal = 15.36+ 2.16 = 17.52 𝑚𝑚! ∴ the surface area of the unusual rectangular prism is 17.52 𝑚𝑚!.

2. a. Circumferenceinner = 58 conches × 20 𝑐𝑐𝑐𝑐 = 1160 𝑐𝑐𝑐𝑐

Circumferencesecond = 58 conches × 20 𝑐𝑐𝑐𝑐 = 1440 𝑐𝑐𝑐𝑐

Circumferencethird = 58 conches × 20 𝑐𝑐𝑐𝑐 = 1720 𝑐𝑐𝑐𝑐

Circumferencefourth = 58 conches × 20 𝑐𝑐𝑐𝑐 = 2000 𝑐𝑐𝑐𝑐

Circumferenceouter = 58 conches × 20 𝑐𝑐𝑐𝑐 = 2280 𝑐𝑐𝑐𝑐

b. If we analyze the differences between the circumferences of the five circles, we see that the second innermost circle is 280 cm larger than the innermost circle, the third (middle) circle is 280 cm larger than the second circle, and so on. Hence, the circumference increases by a constant value of 280 cm. From part a., we know that the distance between each pair of circles is the same (constant) at 28 cm. The circumference increases by a constant 280 cm when the radius of a circle is repeatedly increased by the same amount each time (28 cm). This implies that it will go up by the same amount each time (linear relationship). The increase in circumference is always 10 times the increase in radius.

3. Let Layer 1 represent the bottom layer with 43

conches. Let Layer 2 represent the second last layer with 36 conches. Let Layer 3 represent the layer with 29 conches. Let Layer 4 represent the layer with 22 conches. Let Layer 5 represent the layer with 15 conches. Let Layer 6 represent the layer with 8 conches.

Answer Key

MATHEMATICAL LENS – MAHA MERU VIRROSHI SRIGANESH

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We shall create a table of values with the information given to visualize the relationship better. Layer # of

Conches First

Differences 1 43 2 36 -‐7 3 29 -‐7

4 22 -‐7 5 15 -‐7 6 8 -‐7

We see that the first differences for this relation are constant. In other words, the difference in the number of conches in each layer is constantly 7. ∴ this is a linear relationship.

4. Let Conch 1 represent the one with 1 grain of rice. Let Conch 2 represent the one with 2 grains of rice. Let Conch 3 represent the one with 4 grains of rice, and so on. We shall create a table of values with the information given to visualize the relationship better. Conch

# of Grains

First Differen

ces

Second Differen

ces

Third Differen

ces 1 1 2 2 1 3 4 2 1 4 8 4 2 1 5 16 8 4 2 6 32 16 8 4 7 64 32 16 8 8 128 64 32 16 9 256 128 64 32 10 512 256 128 64 ⋮ ⋮

⋮ ⋮ ⋮

100 6.34 × 10!"

⋮ ⋮ 1008

1.37 × 10!"!

We see that the first, second and third differences for this relation are not constant. ∵ the first difference are not constant, ∴ this relation is not linear.

Notice that the differences are all growing exponentially each time. Hence, it is evident that this relationship is exponential. If we denote 𝑛𝑛 to be the conch number, the number of grains in each conch can be represented as 2!!!. ∴ this is an exponential relation. Graph:

(The graph for our relation starts from the black point at !1, 1). The entire 2!!! graph, where n is the conch number, is made visible to view the behaviour when 𝑥𝑥 ∈ ℝ (for example, the asymptote).)

5. Half way through the distance: 𝑑𝑑 = 600 𝑐𝑐𝑐𝑐 ÷ 2 𝑐𝑐𝑐𝑐 = 300 𝑐𝑐𝑐𝑐

⇒ 𝑑𝑑 = 600− 20𝑡𝑡 300 = 600− 20𝑡𝑡 = 20𝑡𝑡 𝑡𝑡 = 15 ∴ it will take 15 minutes for you to get half way, 300 𝑐𝑐𝑐𝑐, through the distance.


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6. a. We know that the general equation of a circle is 𝑥𝑥! + 𝑦𝑦! = 𝑟𝑟!, where 𝑟𝑟 is the radius. ⇒ 𝑥𝑥! + 𝑦𝑦! = (6)! 𝑥𝑥! + 𝑦𝑦! = 36

∴ the equation of the circle that is centered at the origin and passes through the point (6, 0) is 𝑥𝑥! + 𝑦𝑦! = 36.

b. We know that the general equation of a circle that is not centered at the origin is (𝑥𝑥 − 𝑥𝑥!)! + (𝑦𝑦 − 𝑦𝑦!)! = 𝑟𝑟!, where 𝑟𝑟 is the radius.

⇒ (𝑥𝑥 − 6)! + (𝑦𝑦 − 28)! = 7! 𝑥𝑥! − 12𝑥𝑥 + 36 + 𝑦𝑦! − 56𝑦𝑦 + 784 = 49

𝑥𝑥! − 12𝑥𝑥 + 36 + 𝑦𝑦! − 56𝑦𝑦 + 784 − 49 = 0 𝑥𝑥! − 12𝑥𝑥 + 𝑦𝑦! − 56𝑦𝑦 + 771 = 0

∴ the equation of the circle that is centered at (6, 28) and has a radius of 7 is 𝑥𝑥! − 12𝑥𝑥 +𝑦𝑦! − 56𝑦𝑦 + 771 = 0.

7. a. This question requires the use of the Sine

Law.

⇒ 𝑎𝑎

𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛 = 𝑏𝑏

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

𝑎𝑎

𝑠𝑠𝑠𝑠𝑠𝑠95° = 20

𝑠𝑠𝑠𝑠𝑠𝑠41°

𝑎𝑎 = (𝑠𝑠𝑠𝑠𝑠𝑠95°)20

𝑠𝑠𝑠𝑠𝑠𝑠41°

≐ 30.37 𝑐𝑐𝑐𝑐 ∴ the length of side 𝑎𝑎 is approximately 30.37 cm.

b. This question requires the use of the Sine Law.

⇒ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑏𝑏 =

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠5 =

𝑠𝑠𝑠𝑠𝑠𝑠100°15

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = (5)𝑠𝑠𝑠𝑠𝑠𝑠100°15

𝐵𝐵 = 𝑠𝑠𝑠𝑠𝑠𝑠!! (5)𝑠𝑠𝑠𝑠𝑠𝑠100°15

≐ 19.16° ∴ ∡𝐵𝐵 is approximately 19.16°.

8. a. This question requires the use of the Cosine

Law. ⇒ 𝑎𝑎! = 𝑏𝑏! + 𝑐𝑐! − 2𝑏𝑏𝑏𝑏cos𝐴𝐴 = 150! + 60! − 2(150)(60)cos21° ≐ 9 295.55 𝑎𝑎 ≐ 96.41 𝑐𝑐𝑐𝑐

∴ the length of side 𝑎𝑎 is approximately 96.41 cm.

b. This question requires the use of the Cosine Law.

⇒ 𝑎𝑎! = 𝑏𝑏! + 𝑐𝑐! − 2𝑏𝑏𝑏𝑏cos𝐴𝐴 11.8! = 8.6! + 6.2! − 2(8.6)(6.2)cos𝐴𝐴 26.84 = −106.64cos𝐴𝐴

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 26.84−106.64

𝐴𝐴 = 𝑐𝑐𝑐𝑐𝑐𝑐!! −26.84106.64

≐ 104.58°

∴ ∡𝐴𝐴 is approximately 104.58°. 9. Question 7b:

𝐵𝐵 ≐ 19.16° = 19.16° ×

𝜋𝜋180°

≐ 0.10644𝜋𝜋 rad ≐ 0.3344 rad ∴ ∡𝐵𝐵 is approximately 0.3344 rad. Question 8b: 𝐵𝐵 ≐ 19.16° = 19.16° ×

𝜋𝜋180°

≐ 0.10644𝜋𝜋 rad ≐ 0.3344 rad ∴ ∡𝐵𝐵 is approximately 0.3344 rad.

10. 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑜𝑜𝑜𝑜 𝐶𝐶ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = !!!"#$ !" !

!!!"#$ !" !

= !"!"

= !"!!"!!!


15


7


= !"!

= 12.5% per hour ∴ the average rate of change in your work is 12.5% per hour. In other words, you complete frosting 12.5% of the cake every hour.

11. This is a permutation. Hence, we shall use the

formula nPr = !!!!! !

, where 𝑛𝑛 is the number of

triangles there are and 𝑟𝑟 is the number of different colour petals we have. nPr =

!!!!! !

30P4 = !"!

!"!! !

= !" × !" × !" × !" × !"!!"!

= 30 × 29 × 28 × 27 = 657 720 ∴ this can be done in 657 720 different ways.

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Sri Gurubhyo Namaha

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