Karan T.

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Geometry

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Question:

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

Karan T.

Answer:

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

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Question:

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

Karan T.

Answer:

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

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Question:

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

Karan T.

Answer:

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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