Let a,b,c be complex numbers satisfying
a+b+c=abc=1
and
(ab+bc+ac)/3=1/a^2+1/b^2+1/c^2
The sum of absolute values of all possible ab+bc+accan be written as √n/m, (root (n)/m)where n and m are positive coprime integers. What is n+m?
The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10 n, on the n day of2007 (n = 1, 2, ..., 100), and then remains constant. On the other hand, the price ofOoty tea (in rupees per kilogram) is 89 + 0.15n, on the n day of 2007 (n = 1, 2, ...,365). On which date in 2007 will the prices of these two varieties of tea be equal?(1) May 21 (2) April 11(3) May 20(4) April 10(5) June 30 thth
🍻After the addition of 35 liters of water to a can of diluted milk, the concentration of milk in the can becomes 30%. Now, further 40 liters of water is added to the can and the concentration of milk in the can gets reduced by 10 percentage points. How many more liters of water must be added to the can now such that the concentration of milk in the can becomes 8%?
a man is sailing at a uniform speed, when a leaks developed in his boat he uses a bucket to empty water from it and reaches the shore in 30 mins just in time to avoid sinking . the leak fill boat as twice as man is able to empty it. Had the leak developed 10 mins earlier how much faster would the man has to work to avoid sinking????
Suppose a and b are positive integers satisfying 1≤a≤31, 1≤b≤31 such that the polynomial P(x)=x^3−ax^2+a^2b^3x+9a^2b^2 has roots r, s, and t.
Given that there exists a positive integer k such that
(r+s)(s+t)(r+t)=k^2,
compute the maximum possible value of ab.
Let N be the sum of all prime powers that can be written as 4^n+n^4 for some positive integer n.What are the last 3 digits of N?
Details and assumptions
A prime power is a number of the form p^k,, where p is a prime and k is a positive integer. Examples: 3,9,16.
For a parameter a, suppose u is the smallest positive real number such that
100cos^2 (u)−100sin^2 (u)=a+25
and v is the largest negative real number such that
25 root (3) sin(v)-75cos(v)=a
For how many integer values of a is u+v non-negative?
Let AB be the diameter of circle Γ1. In the interior of Γ1, there are circles Γ2 and Γ3 that are tangent to Γ1at A and B, respectively. Γ2 and Γ3 are also externally tangent at the point C. This tangent line (at C) cuts Γ1at P and Q, with PQ=20. The area that is within Γ1but not in Γ2 or Γ3 is equal to Mπ. Determine M.
Find the no of ways of selecting n articles out of 3n+1, out of which n are identical.
An ant is standing at the point (0,0) in the Cartesian plane.
The ant begins by walking a units in the negative y direction, for some positive integer a.
The ant then turns left and walks b units in the positive x direction for some positive integer b.
The ant again turns left and walks c units in the positive y direction for some positive integer c.
Finally the ant turns left once again and walks d units in the negative x direction, for some positive integer d.
An observer is watching the ant as it starts walking, but quickly gets bored of watching the ant.
The observer only watched the first 20 units of the ant's walk, and noticed that the ant did not visit the same point on the plane more than once in this time.
Q : How many different possible paths could the observer have seen?
NB :
If the observer only watched the first 4 units of the ant's walk, then a=b=c=d=1 is not a valid walk since the origin would be visited twice, namely at the start and at the end.
The ant walked for more than 20 units.
Let AB be the diameter of circle Γ1. In the interior of Γ1, there are circles Γ2 and Γ3 that are tangent to Γ1at A and B, respectively. Γ2 and Γ3 are also externally tangent at the point C. This tangent line (at C) cuts Γ1at P and Q, with PQ=20. The area that is within Γ1but not in Γ2 or Γ3 is equal to Mπ. Determine M.
Find it by yourself
in a triangle ABC, on the side AB 5 pts are marked, on the side BC 6 pts are marked, on the side AC 3 pts are marked(none of these pts on being the vertex of the triangle). How many triangle can be made by using these points?
Despite giving a discount of 11.11%,a shopkeeper gains a profit of 14.28%,so what is the percentage markup over cost price??
seven diff objects must be divided among 3 people. In how many ways it can be done if one or two of them can get no objects.
A team of miners planned to mine 1800 tons of ore during a certaain no. of days.Due to technical difficulties in one third of the planned no. of days the team was able to achieve an output of 20 tons of ore less than the planned output.To make up for this the team overachieved for the rest of the days by 20 tons.The end result was that the team completed the task one day ahead of time.How many tons of ore did the team initially plan to ore per day???
- 100 tons
- 150 tons
- 200 tons
- 50 tons
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When working alone A, B and C can complete a piece of work in 8, 12 and 30 days respectively. At the most only two people can work on each day and nobody works for more than two consecutive days. What is the minimum number of days that they will take to finish the work?
Q14 Ten workers started a job and worked on it for 10 days. Eleventh day onwards, a new worker joined them every day till the job was finished. It took exactly 20 days to finish the entire job. The same job has to be done starting with the minimum possible number of workers when it is known that a worker will quit the job every day after the first day. How many days will it take to finish the job in such a manner
Q14 Ten workers started a job and worked on it for 10 days. Eleventh day onwards, a new worker joined them every day till the job was finished. It took exactly 20 days to finish the entire job. The same job has to be done starting with the minimum possible number of workers when it is known that a worker will quit the job every day after the first day. How many days will it take to finish the job in such a manner
A and B are the two opposite ends of a swimming pool and the distance between them is 420 metres. Ankur and Manu start swimming towards each other at the same time from A and B, with speeds in the ratio 5 : 9 respectively. As soon as any of them reaches an end, he turns back and starts swimming towards the other end. At what distance (in metres) from A will they meet when Manu is in his 13th round? Note: A to B is considered one round and B to A another round.
A bus covers a journey at an average speed of 90 km/h. If the time for which the bus stops during the journey is excluded, then the average speed for the journey becomes 108 km/h. For how much time, on an average, does the bus stop every hour?
post approach as well