The given equation can be write as

=> 4k(x+1) ²+3(y+2) ²=12

on expanding wee get x² coefficient as 4k

and y² coefficient as 3

but in equation of circle x² coefficient is equal to y² coefficient

therefore 4k=3

=> k=3/4

Explanation : A = {(a, a², a³-1) (b, b², b³-1) (c, c², c³-1)}

=> {(a, a², a³) (b, b², b³) (c, c², c³)} - {(a, a², 1) (b, b², 1) (c, c², 1)} = 0

=> abc{(1, a, a²) (1, b, b²) (1, c, c²)} - {(a, a², 1) (b, b², 1) (c, c², 1)} = 0

=> abc{(a, a², 1) (b, b², 1) (c, c², 1)} - {(a, a², 1) (b, b² 1) (c, c², 1)} = 0

=> (abc-1){(a, a², 1) (b, b², 1) (c, c², 1)} = 0

abc - 1 = 0

=> abc = 1

There will equal n opposite signed terms that is 500 +ve one and 500-ve one therefore it's value comes to zero.

Parallelogram is cut by two sets of m parallel lines to its sides.

then we have  2 sets of (m+2) parallel lines ( 2 lines of the parallelogram)

so parallelogram is formed by taking 2 lines from each set

 = m+2C2 * m+2C2

 = [(m+2)(m+1)/2 ]2

 this also include 1 original parallelogram

 so total number of new parallelogram formed is  = (m + 2)²(m + 1)²/4

The probability of a year being a leap year is 1/4 and being non-leap is 3/4.A leap year has 366 days or 52 weeks and 2 odd days. The two odd days can be {Sunday,Monday},{Monday,Tuesday},{Tuesday,Wednesday}, Wednesday,Thursday},{Thursday,Friday},{Friday,Saturday},{Saturday,Sunday}.So there are 7 possibiliyies out of which 2 have a Sunday. So the probability of 53 Sundays in a leap year is 2/7.

So, the probability of 52 sundays is 1-2/7 = 5/7.

If x, y and z respectively represent AM, GM and HM between two numbers a and b, then

y² = xz

Here x = 5, y = 4

then 16 = 5 x z

z = 16/5