Maths

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putting every number in the first 4 blanks

then there is a chance the addition might be in the form of 3n;2n+1,3n+2

then we can put the last number accordingly to make it divisible by 3

which turns out to be three in every case

there fore there are 3 chances to put nay number in the last blank

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here the venn diagram approach would be much more easier

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https://youtu.be/3ztEBbO1vjs?t=30342

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Question 30

Answer 30

key is to not equate the two equations but equate teh ratio

Question 31

what should be the approach for these questions?

Answer 31

assume a parametric point on the parabola and then find a tangent on that point

that tangent will be parallel to the line on which we want the point

as that will be perpendicular to the normal which is perpendicular and hence is shortest distance

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Question 38

https://imgur.com/rxT1TMC

Answer 38

https: //imgur.com/TaMxR0U
https: //imgur.com/CqUs0x3

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Question 54

25 feb shift 2

Answer 54

upload correct q

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Question 58

Answer 58

https://youtu.be/3ztEBbO1vjs?t=28026

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Question 61

Answer 61

https://youtu.be/3ztEBbO1vjs?t=3983

there is a difference of 3 in the binomial sum and we would need something to cancel out the remaining stuff which isn’t a multiple of three

substituting a cube root of unity will help as if we add them! we can cancel em out by adding and get the result
if there was a difference of 4 , we would have taken a 4th root of unity

Question 62

Answer 62

https: //imgur.com/ngK0Zjn
https: //imgur.com/WUTH4bS

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circles with equal radius will have the common tangent parallel to the line joining the centres

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Question 68

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if we have |Z1 + Z2| = |Z1| + |Z2|

it means Z1 and Z2 lies on the same line or they are in same direction

Question 69

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Question 71

Answer 71

subtracting the chances of them sitting together from the all possible possibilities

also when we count the sitting together of AA and BB there is a chance AA,BB be beside and they might not be which is counted in both of them! which i subtracted once and hence we get the answer

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Question 74

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