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Karan T.
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Geometry
TutorMe
Question:

Convert the given polar equation into cartesian equation: $$r= 4cos\omega + 2sin\omega$$

Karan T.

Before going for solution, here is the concept. $$r^{2}$$=$$x^{2}$$ + $$y^{2}$$ in cartesian form. The $$x=rcos\omega$$ and $$y=rsin\omega$$. So coming back to the given equation $$r= 4cos\omega + 2sin\omega$$ ...(1). Multiply the equation (1) with $$r$$ on both sides therefore the resulting equation becomes $$r^{2}= 4rcos\omega + 2rsin\omega$$. ... (2) Now Substitute the value of $$r^{2} = x^{2} + y^{2}$$, $$x=rcos\omega$$ and $$y=rsin\omega$$ in equation (2). The resulting equation becomes $$x^{2} + y^{2}$$=$$4x +2 y$$ which can also be written as $$x^{2} -4x + y^{2}-2y =0$$. .....(3) Adding 4 and 1 on both left and right hand side of equation (3), then equation becomes $$x^{2} -4x +4 + y^{2}-2y +1 =5$$ which can also be written as $$(x-2)^{2} + (y-1)^{2} =5$$. Therefor the required cartesian equation is $$(x-2)^{2} + (y-1)^{2} =5$$.

Basic Math
TutorMe
Question:

If 3x - y =6, what is the value of $$\frac{8^{x}}{2^{y}}$$ ?

Karan T.

Look carefully on what is asked. In the given expression $$\frac{8^{x}}{2^{y}}$$ , numerator contains 8 which can be expressed as power of two as 8= $$2^{3}$$. Substitute this value of 8 in the numerator the given expression becomes, $$\frac{2^{3x}}{2^{y}}$$ which can also be written as $$2^{3x-y}$$ ...(1) . But it is given that 3x-y = 6, therefore substituting this value in equation (1) which results $$2^{6}$$ = 64. There ans is 64

Algebra
TutorMe
Question:

Let the given equation be 31x + 28y +30z=365. Find the value of x + y + z = ?

Karan T.

Just look at the co-efficients of the equaton, They look familiar to number of days defined for months in a year. And it is also that it sums up to 365 which is the number of days in a non leapyear. Thus x is number of months with 31 days, y is number of months with 28 days and z is number of months with 30 days in a leap year. So x =7 with no of months having 31 days, y = 1 which is month of Feburary and z = 4 with no if months having 30 days. So x+y+z= 7+1+4 =12. So 12 is the ans.

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